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Right-angled Triangle (rectangled triangle)
Note: To avoid any collision with built-in identifiers and reserved words in POV-Ray,
it's strongly recommanded to use only words beginning with capital letters
for all identifiers of variables declared by the user, i.e. use "Ri" instead of "r"
and use "H" instead of "h".
Dimensions and Names
The longest side is the side opposite to the right angle γ at Point C
is called hypotenuse c,
the other two sides are called legs or catheti (singular: cathetus) a and b. The angle α is at A and ϐ is the angle at B.
The median theorem:
A rule for all right triangles:
If MAB is the midpoint of the hypotenuse c, then CMAB = ½ c.
One can also say that point C is located on the circle with diameter [AB].
Conversely, if C is any point of the circle with diameter [AB],
then angle at C in the triangle ABC is a right angle.
A right-angled triangle
A right-angled triangle, the median theorem
Thales' theorem:
If AB is a diameter, then the angle at C is a right angle. | 677.169 | 1 |
A Course of Mathematics: In Two Volumes : for the Use of Academies ..., Volume 2
The other two angles may be found as before. The preference is, in this case, manifestly due to the former method.
Ex. 9. In an oblique-angled spherical triangle, are given two sides equal to 114° 40′ and 56° 30′ respectively, and the angle opposite the former equal to 125°20' to find the other parts. Ans. Angles 48°30′, and 62°55′; side, 83°12′, Ex. 10 Given, in a spherical triangle, two angles, equal to 48°30', and 125°20', and the side opposite the latter; to find the other parts
Ans. Side opposite first angle, 56° 40′; other side, 83°12 third angle, 62°54'.
Ex. 11. Given two sides, equal 114°30', and 56° 40′; and their included angle 62° 54′ to find the rest.
Ex 12. Given two angles, 125°20′ and 48°30', and the side comprehended between them 83°12′: to find the other parts. Ex. 13 In a spherical triangle, the angles are 48°31′, 62°56′, and 125°20' required the sides?
Ex. 14. Given two angles, 50° 12', and 58o 8′; and a side opposite the former, 62° 42'; to find the other parts.
Ans. The third angle is either 130°56′ or 156°14′. Side betw. giv. angles, either 119°4′ or 152°14′. either 79°12′ or 100°48'.
Side opp. 588',
Ex. 15. The excess of the three angles of a triangle, measured on the earth's surface, above two right angles, is 1 second; what is its area, taking the earth's diameter at 7957 miles?
Ans. 76-75299, or nearly 763 square miles.
Ex. 16. Determine the solid angles of a regular pyramid, with hexagonal base, the altitude of the pyramid being to each side of the base as 2 to 1.
Ans. Plane angle between each two lateral faces 126°52′11′′. between the base and each face 66°35'12"
Solid angle at the vertex 114-49768 The max angle
Each ditto at the base
222-34298
being 1000,
ON GEODESIC
ON GEODESIC OPERATIONS, AND THE FIGURE OF THE
EARTH.
SECTION I.
General Account of this kind of Surveying.
ART. 1. In the treatise on Land Surveying in the first volume of this Course of Mathematics, the directions were restricted to the necessary operations for surveying fields, farms, lordships, or at most counties; these being the only operations in which the generality of persons, who practise this kind of measurement, are likely to be engaged: but there are especial occasions when it is requisite to apply the principles of plane and spherical geometry, and the practices of surveying, to much more extensive portions of the earth's surface; and when of course much care and judgement are called into exercise, both with regard to the direction of the practical operations, and the management of the computations. The extensive processes which we are now about to consider, and which are characterised by the terms Geodesic Operations and Trigonometrical Surveying, arc usually undertaken for the accomplishment of one of these three objects. 1. The finding the difference of longitude, between two moderately distant and noted meridians; as the meridians of the observatories at Greenwich and Oxford, or of those at Greenwich and Paris. 2. The accurate determination of the geographical positions of the principal places, whether on the coast or inland, in an island or kingdom; with a view to give greater accuracy to maps, and to accommodate the navigator with the actual position, as to latitude and longitude, of the principal promontories, havens, and ports. These have, till lately, been desiderata, even in this country: the position of some important points, as the Lizard, not being known within seven minutes of a degree; and, until the publication of the board of Ordnance maps, the best country maps being so erroneous, as in some cases to exhibit blunders of three miles in distances of less than twenty.
3. The
3. The measurement of a degree in various situations; and thence the determination of the figure and magnitude of the earth.
When objects so important as these are to be attained, it is manifest that, in order to ensure the desirable degree of correctness in the results, the instruments employed, the operations performed, and the computations required, must each have the greatest possible degree of accuracy of these, the first depend on the artist; the second on the surveyor or engineer, who conducts them; and the latter on the theorist and calculator: they are these last which will chiefly engage our attention in the present chapter
2. In the determination of distances of many miles, whether for the survey of a kingdom, or for the measurement of a de gree, the whole line intervening between two extreme points is not obsolutely measured; for this, on account of the in equalities of the earth's surface. would be always very difficult, and often impossible. But, a line of a few miles in length is very carefully measured on some plane, heath, or marsh, which is so nearly level as to facilitate the measurment of an actual¬ ly horizontal line; and this line being assumed as the base of the operations, a variety of hills and elevated spots are selectcd at which signals can be placed, suitably distant and visible one from another: the straight lines joining these points constitute a double series of triangles, of which the assumed base forms the first side; the angles of these, that is, the angles made at each station or signal staff, by two other signal staffs, are carefully measured by a theodolite, which is carried successively from one station to another. In such a series of triangles, care being always taken that one side is common to two of them, all the angles are known from the observations at the several stations, and a side of one of them being given, namely, that of the base measured, the side of all the rest, as well as the distance from the first angle of the first triangle to any part of the last triangle, may be found by the rules of trigonometry. And so again, the bearing of any one of the sides, with respect to the meridian, being determined by observation, the bearings of any of the rest, with respect to the same meridian, will be known by computation. In these operations, it is always adviseable, when circumstances will admit of it, to measure another base (called a base of verification) at or near the ulterior extremity of the series: for the length of this base, computed as one of the sides of the chain of triangles, compared with its length determined by actual admeasurement, will be a test of the accuracy of all the operations made in the serics between the two bases.
R
3. Now, in every series of triangles, where each angle is to be ascertained with the same instrument, they should, as nearly as circumstances will permit, be equilateral. For, if it were possible to choose the stations in such manner, that each angle should be exactly 60 degrees; then, the half number of triangles in the series, multiplied into the length of one side of either triangle would, as in the annexed figure, give at once the total distance; and then also, not only the sides of the scale or ladder, constituted by this series of triangles, would be perfectly parallel, but the diagonal steps, marking the progress from one extremity to the other, would be alternately parallel throughout the whole length. Here too, the first, side might be found by a base crossing it perpendicularly of about half its length, as at H; and the last side verified by another such base, R, at the opposite extremity. If the respective sides of the series of triangles were 12 or 18 miles, these bases might advantageously be between 6 and 7, or between 9 and 10 miles respectively; according to circumstances. It may also be remarked. (and the reason of it will be seen in the next section) that whenever only two angles of a triangle can be actually observed, each of them should be as nearly as possible 45°, or the sum of them about 90°; for the less the third or computed angle differs from 90°, the less probability there will be of any considerable error. See prob. 1 sect. 2, of this chapter.
4. The student may obtain a general notion of the method employed in measuring an arc of the meridian, from the following brief sketch and introductory illustrations.
The earth, it is well known, is nearly spherical. It may be either an ellipsoid of revolution, that is, a body formed by the rotation of an ellipse, the ratio of whose axes is nearly that of equality, on one of those axis; or it may approach nearly to the form of such an ellipsoid or spheroid, while its deviations from that form, though small relatively, may still be sufficiently great in themselves, to prevent its being called a spheroid with much more propriety than it is called a sphere. One of the methods made use of to determine this point, is by means of extensive Geodesic operations.
The earth however, be its exact form what it may, is a planet, which not only revolves in an orbit, but turns upon an axis. Now, if we conceive a plane to pass through the axis of rotation of the earth, and through the zenith of any place on its surface, this plane, if prolonged to the limits of
the
the apparent celestial sphere, would there trace the circumference of a great circle, which would be the meridian of that place. All the points of the earth's surface, which have their zenith in that circumference, will be under the same celestial meridian. and will form the corresponding terrestrial meridian. If the earth be an irregular spheroid, this meridian will be a curve of double curvature, but if the earth be a solid of revolution, the terrestrial meridian will be a plane curve.
5. If the earth were a sphere, then every point upon a terrestrial meridian would be at an equal distance from the centre, and of consequence every degree upon that meridian would be of equal length. But if the earth be an ellipsoid of revolution slightly flattened at its poles, and protuberant at the equator; then, as will be shown soon, the degrees of the terrestrial meridian, in receding from the equator towards the poles, will be increased in the duplicate ratio of the right sine of the latitude; and the ratio of the earth's axes, as well as their actual magnitude, may be ascertained by comparing the lengths of a degree on the meridian in different latitudes. Hence appears the great importance of measuring a degree.
6. Now, instead of actually tracing a meridian on the surface of the earth,-a measure which is prevented by the interposition of mountains, woods, rivers, and seas,-a construction is employed which furnishes the same result. It consists in this.
Let ABCDEF, &c. be a series of triangles, carried on, as nearly as may be, in the direction of the meridian, according G
E
to the observations in art.3. These triangles are really spherical or spheroidal triangles; but as their curvature is extremely small, they are treated the same as rectilinear triangles, either by reducing them to the chords of the respective terrestrial arcs AC, AB, BC, &c. or by deducting a third of the excess, of the sum of the three angles of each triangle above two right angles, from each angle of that triangle, and working with the remainders, and the three sides, as the dimensions of a plane triangle; the proper reductions to the centre of the station, to the horizon, and to the level of the sea, having been previously made. These computations being made throughout | 677.169 | 1 |
...line fall on two parallel straight lines, it makes the alternate angles equal to one anotlier, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. Let the straight...
...3. The greater side of a triangle is opposite the greater angle. 4. If a straight line falling upon two other straight lines make the exterior angle equal to the interior, and opposite, upon the same side of the line ; or make the interior angles upon the same side together equal to two...
...straight line fall on two parallel straight lines it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. Show that the...
...XXVUL-XLI. (inclusive). 27.When are straight lines said to be parallel to one another ? Prove that if a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines must be parallel Same proposition. What...
...ridicule, he strolled quietly home, a triumphant Tory. Euclid. — i. What is meant by alternate angles ? If a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines shall be parallel. 2. How many sides...
...contradictories to an absurdity. One such proposition we will examine from a logical point of view : — " If a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." —...
...other, the base of that which has the greater angle shall be greater than the base of the other. 5. If a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another. 6....
...given lines must be greater than the length of the third line ? 6. If a straight line falling upon two other straight lines make the exterior angle equal to the interior and opposite upon the same side of the line, or make the interior angles on the same side together equal to two...
...oil two parallel straight lines BC, DE, it makes the alternate angles equal to one another, and tho exterior angle equal to the interior and opposite angle on the same side ; and also the two interior angles on the same side together equal to two right angles. 4. Parallelograms...
...may be used, but not symbols of operations, such as -, +, x.] 1. What is meant by alternate angles ? If a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines shall be parallel. 2. How many sides... | 677.169 | 1 |
Properties Of Circle
What Are The Properties Of CirclesTwo circles are congruent, if and only if they have equal radii.Two arcs of a circle are congruent if the angles subtended by them at the centre are equal.Two arcs subtend equal angles at the centre, if the arcs are congruent.If two arcs of a circle are congruent, their corresponding chords are equal.If two chords … [Read more...] about What Are The Properties Of Circles | 677.169 | 1 |
ﺽﮒﻣﻑﻛﻕ ... Schools , in giving to the young some knowledge of this important branch of science . STEPHEN BOYER , Principal of York Co. Academy . March 5 , 1845 . I cordially concur with the Rev. Mr. Boyer , in the opinion that MR . M'CURDY's plan ...
ﺥﻝﮞﺅﺉﻠﻣﻐ ﻕﻭﺅﮩﻭﻎﮩﮞﻕﮪﻕ
ﺽﮒﻣﻑﻛﻕ 90ﺽﮒﻣﻑﻛﻕ 117 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
ﺽﮒﻣﻑﻛﻕ 79 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
ﺽﮒﻣﻑﻛﻕ 133 - If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.
ﺽﮒﻣﻑﻛﻕ 13 - AB be the greater, and from it cut (3. 1.) off DB equal to AC the less, and join DC ; therefore, because A in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides DB, BC are equal to the two AC, CB. each to each ; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is< equal to the triangle (4. 1.) ACB, the less to 'the greater; which is absurd.
ﺽﮒﻣﻑﻛﻕ 83 | 677.169 | 1 |
Find the distance between the centres of the two circles, if their radii are 11 cm and 7 cm, and the length of the transverse common tangent is √301cm.
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Step by step video, text & image solution for Find the distance between the centres of the two circles, if their radii are 11 cm and 7 cm, and the length of the transverse common tangent is sqrt(301) cm. by Maths experts to help you in doubts & scoring excellent marks in Class 10 exams. | 677.169 | 1 |
Tag: hexTake 6 points on a circle such that every second edge (green chords) has length equal to the radius of the circle. Then the midpoints of the other three sides of the cyclic hexagon form an equilateral triangle | 677.169 | 1 |
Locus maths formulas book
Loci locus and its constructions solutions for icse board. Paper 1 noncalculator paper 80 marks weighed at 50% of total and paper 2 calculator paper 100 marks weighed at 50% of total. Trigonometry formulas righttriangle definitions, reduction formulas, identities, sum and difference formulas, double angle and half angle formulas, law of sines and cosines, area of triangle. This is usually, but not always, the same plane as the given geometric object. A circle is the locus of points on a plane that are a certain distance from a central point. Welcome to the cambridge o level mathematics syllabus d 4024 study guide. A locus is a set of points satisfying a certain condition.
Facts and formulas series by series understanding maths. This book was compiled by dr john shakeshaft and typeset originally by fergus gallagher, and currently by dr dave green, using the tex typesetting package. Great formulas explained physics, mathematics, economics. How to find equation of locus locus mathematics class 11. Length of perpendicular from a point to line and distance between two parallel lines. Cambridge o level mathematics syllabus d wikibooks, open. We can restate the definition of locus in biconditional form. Here is a stepbystep procedure for finding plane loci. For example, the locus of points that are 1cm from the origin is a circle of radius 1cm centred on the origin, since all points on this circle are 1cm from the origin. Revision summary of dse mathematics compulsory part. Download icse class 10 maths syllabus in pdf format. In this topic we shall study the loci involving the parabola, circles and straight lines. Free pdf download of icse class 10 mathematics revision notes and short keynotes to score more marks in your exams, prepared by our expert math teachers as per cisce guidelines.
Math formula shows how things work out with the help of some equations like the equation for force or acceleration. Ask your doubt of locus and get answer from subject experts and students on topperlearning. Jul 22, 2017 spm form 4 add maths locus in geometry coordinate duration. Mathematics sl formula booklet 4 topic 2functions and equations 2. But you may need to work with circle equations in your algebra classes. Cbse class 8 maths formulas available for chapter wise on. Eventually, formulas are used to provide mathematical solution for real world problems. Maths is a natural science concerned with the study of life and living organisms, including their structure, function, growth. Many geometric shapes are most naturally and easily described as loci. Many students go through the whole book to revise the formulas but instead, they end with. If the parameter varies, the intersection points of the associated curves describe the locus.
The locus of points 3 units from the given circle is two circles concentric with the original circle with radii of 7 and units. The most popular formulas this is a list of formulas which have most downloads. Ellipse the locus of all points where the sum of the distance to two fixed points is a constant. Students can access icse class 10 maths syllabus in our byjus website in accordance to the icse board. Loci locus and its constructions solutions for icse. Its very convenient for all students in high school or university and engineers to look for any easy or complicated formulas. Free download of step by step solutions for class 10 mathematics chapter 16 loci locus and its constructions of icse board concise selina publishers. Handbook of mathematical formulas and integrals, second. Let the two fixed points be a1, 1 and b2, 4, and px, y be the moving point. Maths formulas a musthave app for your smartphones and tablets. In analytic geometry, a curve on a graph is the locus of analytic points that satisfies the equation of the curve. Learn its meaning, definition, locus of points with examples for different shapes at byjus.
Write the given conditions in a mathematical form involving the coordinates x and y. The algebraic equation describing the geometrical path traced out by a variable point is called the locus of the point. The final figure shows the locus, and the caption gives its description. Icse class 10 mathematics revision notes free pdf download. The general form may be given in your book using different letters for the coefficients, but the. National council of educational research and training ncert class. Just download the formulas book from the given link below. Math formulas download maths formulas pdf basic math fomula. Female detective books engineering scholarships for high school. This lecture note covers the following topics in surface modeling. If you really want to crack the jee main then we strongly recommend you to buy our full jee main study material.
Math formulas download maths formulas pdf basic math. Each chapter is very specific, with a little theory in the beginning, and then the rest is just easy to use formulas. A locus can also be defined by two associated curves depending on one common parameter. Ye dkhiye apko is tarah ka content is book me milega. Laura received her masters degree in pure mathematics from. Yahan main apko kuch examples dikha raha hu ki aapko is notes me kis tarah ke maths tricks milengi. The locus of px,y such that its distance from a0,0 is less than 5 units is 1 x y 5 2 2 2 x y 10 3 x y 25 2 2 4 x y 20 2. These formula includes algebra identities, arithmetic, geometric and various other formulas. There are a total of 39 topics to be covered in this syllabus. Download the important maths formulas and equations pdf to solve the problems easily and score more marks in your class 8 cbse board exams. We will look at various examples of a locus of points, and we will look at an. The locus of points at a given distance from a given point is. Given a general triangle abc, point d lies somewhere on segment ac, point e lies somewhere on segment bc. Make a conjecture first as to what kind of shape the locus of points will define and then, see if it fits your conjecture.
To register for our free webinar class with best maths tutor in india. Problems involving describing a certain locus can often be solved by explicitly finding equations for the coordinates of the points in the locus. Locus 1a solutions maths 1b intermediate ts and ap youtube. Math formulas in algebra, analytic geometry, integrals. Formulas and concept for students studying in class 10th, class 10th maths formulas ebook is made to build a strong concept and to clarify the formulas at that last moment of the exam when it is very important for revision. Number and algebra page 1 polynomials like terms and unlike terms e.
Icse class 10 mathematics chapter 16 loci revision notes. Transformation of axes, intermediate first year 1b chapter 2 problems with solutions mathematics intermediate 1a and 1b solutions for some problems are available here. In this one, we were to find out the locus of a point such that it is equidistant from two fixed points, which was the perpendicular bisector of the line joining the points. You can use the following fourstep solution method to solve a 2d problem. This section covers loci within geometry and measures. Cbse class 8 maths formulas, important math formulas for. Locus of a point math formulas mathematics formulas basic math formulas javascript is disabled in your browser. The cbook invites students to experiment geometric loci generated by. Functions, mathematical inductions, addition of vectors, product of vectors and trigonometry. Mathematics hl and further mathematics hl formula booklet. Locus of a point math formulas mathematics formulas. Locus meaning, definition, points and examples byjus. Solutions for functions, mathematical induction, addition of vectors, trigonometric ratios upto transformations, trigonometric equations, hyperbolic functions, inverse.
A locus is a set of points which satisfy certain geometric conditions. Learn about and revise how to create loci and constructions using a compass with this bbc bitesize gcse maths edexcel study guide. Select any topic from the above list and get all the required help with math formula in detail. In the figure, the points k and l are fixed points on a given line m. In algebraic terms, a circle is the set or locus of points x, y at some fixed distance r from some fixed point h, k. To find the equation, the first step is to convert the given condition into mathematical form, using the formulas we have. Free pdf download of class 10 mathematics chapter 16 loci locus and its constructions. The locus of p x,y such that its distance from a0,0 is less than 5 units is 1 x y 52 2. Length of tangent, normal, subtangent and sub normal. Basically, in mathematics, a locus is a curve other shape made by all the points satisfying a particular equation of the relation between the coordinates. Kickstart your exam preparation by going through it and knowing the important topics that need to be focused more. The understanding maths series was written by experienced, qualified australian teachers to enhance understanding, confidence, enjoyment and results. For example, a circle is the set of points in a plane which are a fixed distance r r r from a given point p, p, p, the center of the circle problems involving describing a certain locus can often be solved by explicitly finding equations for the. It contains a list of basic math formulas commonly used when doing basic math computation.
Chapter 6 solving problems based on quadratic equations. If you think of a point moving along some path, we sometimes say that the path is the locus of the point. Each formula is explained gently and in great detail, including a discussion of all the quantities involved and examples that will make clear how and where to apply it. A real life saver when you just want to find something without having to wade through some lengthy derivations. Hyperbolic functions definitions, derivatives, hyperbolic. A point p is a point of the locus if and only if p satisfies the given conditions of the locus. This is my 7th year teaching high school math and i love it. Trigonometric equations, intermediate first year 1 a chapter 7 problems with solutions mathematics intermediate 1a and 1b solutions for some problems. I keep this book with me at all times and am never at a loss. Compiled and solved problems in geometry and trigonometry. Sometimes the idea of locus has a slightly different explanation.
In twodimensional locus problems, all the points in the locus solution lie in a plane. If possible, choose a coordinate system that will make computations and equations as simple as possible. Also find mathematics coaching class for various competitive exams and classes. In this book you will find some of the greatest and most useful formulas that the fields of physics, mathematics and economics have brought forth. As shown below, just a few points start to look like a circle, but when we collect all the points we will actually have a circle. How to solve a twodimensional locus problem dummies. | 677.169 | 1 |
Now, the equation of any straight line perpendicular to $~BC~$ can be written as $~x-y+k=0~~(k \neq 0)\rightarrow(5).$
Since the straight line passes through $~A(3,1)~$ , so
$~3-1+k=0 \Rightarrow k=-2.$
Hence, the equation of any straight line perpendicular to $~BC~$ is $~x-y-2=0.$
So, the equation of its altitude is $~x-y=2.$
3. Show that the product of the perpendiculars drawn from the two points $~(\pm \sqrt{a^2-b^2},0)~$ upon the sl $~\frac xa \cos\theta+\frac yb \sin\theta=1~$ is $~b^2.$
Solution.
We have the equation of the sl $~\frac xa \cos\theta+\frac yb \sin\theta=1~\rightarrow(1)$
The perpendicular distance of $~(1),~$ from the point $~(\sqrt{a^2-b^2},0)$
$=\frac{\left|$
Similarly, the perpendicular distance of $~(1),~$ from the point $~(-\sqrt{a^2-b^2},0)$
$=\frac{\left|-$
$\therefore~$ the product of the perpendiculars drawn from the two points $~(\pm \sqrt{a^2-b^2},0)~$ is
$ \\~~\times \\\=\frac{|b^2(a^2-b^2)\cos^2\theta-a^2b^2|}{b^2\cos^2\theta+a^2\sin^2\theta}\\=\frac{|-a^2b^2(1-\cos^2\theta)-b^4\cos^2\theta|}{b^2\cos^2\theta+a^2\sin^2\theta}\\=\frac{|-a^2b^2\sin^2\theta-b^4\cos^2\theta|}{b^2\cos^2\theta+a^2\sin^2\theta} \\=\frac{|-b^2(b^2\cos^2\theta+a^2\sin^2\theta)|}{b^2\cos^2\theta+a^2\sin^2\theta}\\=|-b^2|\\=b^2.$
4. One side of an equilateral triangle is the line $~5y=12x-3~$ and its centroid is at $~(2,-1);~$ find the length of a side of the triangle.
Solution.
The distance of the given sl $~12x-5y-3=0 \rightarrow(1)~$ from the point $~(2,-1)~$ is
Now, by $~(2)~$ we can clearly say that the point $~(h,k)~$ lies on the straight line $~11x-3y+11=0.$
Hence, we can finally say that any point on the straight line $~11x-3y+11=0~$ is equidistant from the straight lines $~12x+5y+12=0~$ and $~3x-4y+3=0.$
7. Find the equation of the straight line which is perpendicular to the sl $~3x-2y+5=0~$ and whose distance from the origin is equal to the perpendicular distance of the given line from the point $~(2,-1).$
Solution.
The equation of any straight line perpendicular to $~3x-2y+5=0~$ can be written as $~2x+3y+k=0~~(k \neq 0)~\rightarrow(1).$ | 677.169 | 1 |
8
Easy
Question
Given the two parallel lines cut by a transversal. Find the vertical angle to angle 4.
Angle 2
Angle 6
Angle 8
Angle 7
Hint:
Finding the angle by using the parallel lines property
The correct answer is: Angle 7
The vertical angle to ∠4 is ∠7 | 677.169 | 1 |
What Is the Sector of a Circle in Geometry?
The sector of a circle in geometry is a simple shape made up of two lines and an arc. It is the portion of a circle that is enclosed by two radii of the circle and the arc that connects them. It is also known as a wedge of the circle. In this article, we will learn more about the concept of sector of a circle and how it relates to other elements of geometry.
Understanding Arc, Radius, and Circumference
The arc of a circle is the curved line that makes up part of the sector of a circle. It is the line that connects the two radii and is measured in degrees. The radius of a circle is the line that connects the center of the circle to any point on the circumference. The circumference of a circle is the total length of the circle.
The arc length of a sector is equal to the central angle of the sector, multiplied by the circumference of the circle. The area of a sector is equal to one-half of the radius squared multiplied by the central angle of the sector.
Understanding Chords
The chord of a circle is a straight line connecting two points on the circumference of the circle. The length of the chord is equal to twice the radius of the circle. The chord of a sector is the line that connects the two points on the circumference of the sector. The length of the chord of a sector is equal to twice the radius of the circle, multiplied by the central angle of the sector.
The area of a sector can also be determined by finding the area of the triangle formed by the two radii and the chord of the sector. This triangle is called the triangle of a sector and can be found by using the formula for the area of a triangle.
Practice Problems
1. Find the area of a sector with a radius of 8 cm and a central angle of 60 degrees.
Answer: 96p cm2
2. Find the length of the arc of a sector with a radius of 10 cm and a central angle of 120 degrees.
Answer: 60p cm
3. Find the length of the chord of a sector with a radius of 6 cm and a central angle of 45 degrees.
Answer: 12 cm
4. Find the area of a sector with a radius of 10 cm and a chord of length 16 cm.
Answer: 80p cm2
5. Find the central angle of a sector with a radius of 4 cm and an arc length of 8p cm.
Answer: 90 degrees
6. Find the length of the chord of a sector with a radius of 7 cm and an area of 56p cm2.
Answer: 14 cm
Conclusion
The sector of a circle is the portion of the circle enclosed by two radii and the arc that connects them. It is important to understand the components of a sector, such as radius, circumference, arc, and chord, in order to correctly compute its area and arc length. With the right knowledge, you can easily calculate the area, arc length, or chord length of a sector.
We hope this article has helped you understand the concept of sector of a circle in geometry. If you have any doubts or questions, please feel free to ask in the comments section below.
FAQ
What is the formula to calculate the area of a sector of a circle?
The formula to calculate the area of a sector of a circle is A = (?/360) x pr2, where ? is the angle of the sector in degrees, and r is the radius of the circle.
What is the formula to calculate the arc length of a sector of a circle?
The formula to calculate the arc length of a sector of a circle is L = (?/360) x 2pr, where ? is the angle of the sector in degrees, and r is the radius of the circle. | 677.169 | 1 |
Context.arcNegative
Adds a circular arc of the given radius to the current path. The
arc is centered at (xc, yc), begins at angle1 and proceeds in
the direction of decreasing angles to end at angle2. If angle2 is
greater than angle1 it will be progressively decreased by
2*M_PI until it is less than angle1.
See cairo_arc() for more details. This function differs only in the
direction of the arc between the two angles.
Since 1.0 | 677.169 | 1 |
Pythagorean Theorem: Concept and Uses
To really understand the Pythagorean Theorem we have to be clear on some concepts. For example, it only applies to right triangles; in other words, triangles that have a right angle. We also have to know what the names given to the sides of a right triangle are: the sides that form the right angle are called legs, while the side opposite the right angle is called the hypotenuse.
Another important aspect of the Pythagorean Theorem has to do with its uses. This theorem is used for a large number of situations to find unknown measurements which otherwise couldn't be calculated exactly or which would take a long time to measure.
In the video included below, the Pythagorean Theorem is explained in great detail and in a very simple way, as well as all concepts related to this theorem that you have to know in order to understand it well. It also shows many uses and applications for the theorem and how it turns out to be very useful to solve a large number of problems. Now, on with the video! | 677.169 | 1 |
Working With Lines
The Distance Formula and Midpoints of Segments
The distance and the midpoint formulas give us the tools to find important information about two points.
Learning Objectives
Calculate the midpoint of a line segment and the distance between two points on a plane
Key Takeaways
Key Points
The Pythagorean Theorem relates the lengths of the three sides of a right triangle. If [latex]c[/latex] is the hypotenuse and [latex]a[/latex] and [latex]b[/latex] are the other two sides, then [latex]c^{2}=a^{2}+b^{2}[/latex].
Using the Pythagorean Theorem and two points [latex](x_{1},y_{1})[/latex] and [latex](x_{2},y_{2})[/latex], we can derive the distance formula: [latex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/latex].
The midpoint of a line segment given by two points [latex](x_{1},y_{1})[/latex] and [latex](x_{2},y_{2})[/latex] is [latex](\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})[/latex].
Key Terms
distance: The amount of space between two points, measured along a straight line
Pythagorean Theorem: States that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
midpoint: A point which divides a line segment into two lines of equal length
The Distance Formula
In analytic geometry, the distance between two points of the [latex]xy[/latex]-plane can be found using the distance formula. The distance can be from two points on a line or from two points on a line segment. The distance between points [latex](x_{1},y_{1})[/latex] and [latex](x_{2},y_{2})[/latex] is given by the formula:
[latex]\displaystyle
d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/latex]
This formula is easily derived by constructing a right triangle with the hypotenuse connecting the two points ([latex]c[/latex]) and two legs drawn from the each of the two points to intersect each other ([latex]a[/latex] and [latex]b[/latex]), (see image below) and applying the Pythagorean theorem. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Pythagorean Theorem: The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The image below names the two points, with the distance between them as the variable, [latex]d[/latex]. Notice that the length between each point and the triangle's right angle is found by calculating the difference between the [latex]y[/latex]-coordinates and [latex]x[/latex]-coordinates, respectively. The distance formula includes the lengths of the legs of the triangle (normally labeled [latex]a[/latex] and [latex]b[/latex]), with the expressions [latex](y_{2}-y_{1})[/latex] and [latex](x_{2}-x_{1})[/latex].
Distance Formula: The distance formula between two points, [latex](x_{1},y_{1})[/latex] and [latex](x_{2},y_{2})[/latex], shown as the hypotenuse of a right triangle
Example: Find the distance between the points [latex](2,4)[/latex] and [latex](5,8)[/latex]
Substitute the values into the distance formula that is derived from the Pythagorean Theorem:
[latex]\displaystyle
d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/latex]
[latex]\displaystyle
\begin{align}
d&=\sqrt{(5-2)^{2}+(8-4)^{2}}\\&=\sqrt{3^{2}+4^{2}}\\&=\sqrt{25}\\&=5
\end{align}[/latex]
Midpoint of a Line Segment
In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both end-points. If you have two points, [latex](x_{1},y_{1})[/latex] and [latex](x_{2},y_{2})[/latex], the midpoint of the segment connecting the two points can be found with the formula:
[latex]\displaystyle
(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})[/latex]
Another way to interpret this formula is an average: we average the [latex]x[/latex]-coordinates to find the [latex]x[/latex]-coordinate of the midpoint, and we average the [latex]y[/latex]-coordinates to find the [latex]y[/latex]-coordinate of the midpoint.
By looking at each coordinate, you can see that the [latex]x[/latex]-coordinate is halfway between [latex]x_{1}[/latex] and [latex]x_{2}[/latex], and the [latex]y[/latex]-coordinate is halfway between [latex]y_{1}[/latex] and [latex]y_{2}[/latex].
Midpoint of a Line Segment: The equation for a midpoint of a line segment with endpoints [latex](x_{1},y_{1})[/latex]and [latex](x_{2},y_{2})[/latex]
Example: Find the midpoint between [latex](2,4) [/latex] and [latex](5,8)[/latex]
Parallel and Perpendicular Lines
Parallel lines never intersect; perpendicular lines intersect at right angles.
Learning Objectives
Practice finding equations for lines that are parallel and lines that are perpendicular
Key Takeaways
Key Points
Any two lines are parallel if they have the same slope.
Two lines in the same plane are perpendicular if their slopes are negative reciprocals of each other. This means that one has slope of [latex]m[/latex] and the other has a slope of [latex]-\frac{1}{m}[/latex].
Two lines in the same plane are perpendicular if the product of their slopes equals [latex]-1[/latex].
Key Terms
parallel lines: Lines which never intersect even as they go to infinity. Their slopes are equal to each other.
reciprocal: Of a number, the number obtained by dividing [latex]1[/latex] by the given number; the result of exchanging the numerator and the denominator of a fraction.
perpendicular lines: Two lines whose intersection creates right angles. Their slopes are the negative reciprocal of each other.
Parallel Lines
Two lines in a plane that do not intersect or touch at a point are called parallel lines. The parallel symbol is [latex]\parallel[/latex].
For example, given two lines: [latex]f(x)=m_{1}x+b_{1}[/latex]and [latex]g(x)=m_{2}x+b_{2}[/latex], writing [latex]f(x)[/latex] [latex]\parallel[/latex] [latex]g(x)[/latex] states that the two lines are parallel to each other. In 2D, two lines are parallel if they have the same slope.
Given two parallel lines [latex]f(x)[/latex] and [latex]g(x)[/latex], the following is true:
Every point on [latex]f(x)[/latex] is located at exactly the same minimum distance from [latex]g(x)[/latex].
Line [latex]f(x)[/latex] is on the same plane as [latex]g(x)[/latex] but does not intersect [latex]g(x)[/latex], even assuming that the two lines extend to infinity in either direction.
Recall that the slope-intercept form of an equation is: [latex]y=mx+b[/latex] and the point-slope form of an equation is: [latex]y-y_{1}=m(x-x_{1})[/latex], both contain information about the slope, namely the constant [latex]m[/latex]. If two lines, say [latex]f(x)=mx+b[/latex] and [latex]g(x)=nx+c[/latex], are parallel, then [latex]n[/latex] must equal [latex]m[/latex].
For example, in the graph below, [latex]f(x)=2x+3[/latex] and [latex]g(x)=2x-1[/latex] are parallel since they have the same slope, [latex]m=2[/latex].
Parallel lines: [latex]f(x)=2x+3[/latex] in red is parallel to [latex]g(x)=2x-1[/latex] in blue; the slopes obtained from the graphs of the lines is the same as the slopes in their equations.
Perpendicular Lines
Two lines are perpendicular to each other if they form congruent adjacent angles. In other words, they are perpendicular if the angles at their intersection are right angles, [latex]90[/latex] degrees. The perpendicular symbol is [latex]\perp[/latex].
For example given two lines, [latex]f(x)=m_{1}x+b_{1}[/latex] and [latex]g(x)=m_{2}x+b_{2}[/latex], writing[latex]f(x)\perp g(x)[/latex] states that the two lines are perpendicular to each other.
For two lines in a 2D plane to be perpendicular, their slopes must be negative reciprocals of one another, or the product of their slopes must equal [latex]-1[/latex]. This means that if the slope of one line is [latex]m[/latex], then the slope of its perpendicular line is [latex]\frac{-1}{m}[/latex]. The two slopes multiplied together must equal [latex]-1[/latex]. However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).
Given two lines: [latex]f(x)=3x-2[/latex] and [latex]g(x)=\frac{-1}{3}x+1[/latex], note the values of the slopes. Since [latex]3[/latex] is the negative reciprocal of [latex]-\frac{1}{3}[/latex], the two lines are perpendicular. Also, the product of the slopes equals [latex]-1[/latex].
Perpendicular lines: The line [latex]f(x)=3x-2[/latex] in red is perpendicular to line [latex]g(x)=\frac{-1}{3}x+1[/latex] in blue. The values of their slopes are negative reciprocals of each other; therefore, the angle of intersection is [latex]90[/latex] degrees.
Writing Equations of Parallel and Perpendicular Lines
Example: Write an equation of the line (in slope-intercept form) that is parallel to the line [latex]y=-2x+4[/latex] and passes through the point [latex](-1,1)[/latex]
Start with the equation for slope-intercept form and then substitute the values for the slope and the point, and solve for [latex]b[/latex]: [latex]y=mx+b[/latex]. The value of the slope will be equal to the current line, since the new line is parallel to it. The point [latex](-1,1)[/latex] is substituted for [latex](x,y)[/latex].
[latex]\displaystyle
y=mx+b[/latex]
[latex]\displaystyle
\begin{align}
1&=-2(-1)+b\\
\\
1&=2+b\\
\\
b&=-1
\end{align}[/latex]
Therefore, the equation of the line has a slope ([latex]m[/latex]) of [latex]-2[/latex] and a [latex]y[/latex]-intercept ([latex]b[/latex]) of [latex]-1[/latex]. The equation is [latex]y=-2x-1[/latex].
Example: Write an equation of the line (in slope-intercept form) that is perpendicular to the line [latex]y=\frac{1}{4}x-3[/latex] and passes through the point [latex](2,4)[/latex]
Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal. The negative reciprocal of [latex]\frac{1}{4}[/latex] is [latex]-4[/latex]. Therefore, the new equation has a slope of [latex]-4[/latex], through the point [latex](2,4)[/latex]. Solve for [latex]b[/latex].
[latex-display]y=mx+b[/latex-display]
[latex]\displaystyle
\begin{align}
4&=-4(2)+b\\
\\
4&=-8+b\\
\\
b&=12
\end{align}[/latex]
Therefore, the equation of the line perpendicular to the given line has a slope of [latex]-4[/latex] and a [latex]y[/latex]-intercept of [latex]12[/latex]. The equation is [latex]y=-4x+12[/latex].
Linear Inequalities
A linear inequality is an expression that is designated as less than, greater than, less than or equal to, or greater than or equal to.
Learning Objectives
Solve problems involving linear inequalities
Key Takeaways
Key Points
When two expressions are connected by any of the following signs: [latex]<[/latex], [latex]>[/latex], [latex]\leq[/latex], [latex]\geq[/latex], or [latex]\ne[/latex] we have an inequality. For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable. This just means that you need to find the values of the variable that make the inequality true.
A linear inequality is solved very similarly to how we solve equations. The difference is that the answers are more than one true value, they can be any of the following: [latex]<[/latex], less than the found solution, [latex]>[/latex], greater than the found solution [latex]\leq[/latex], contains values equal and less than the found solution, [latex]\geq[/latex], contains values equal and greater than the found solution.
When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.
Key Terms
inequality: A statement that of two quantities of which one is specifically less than or greater than another. Symbols: [latex]<[/latex] or [latex]\leq[/latex] or [latex]>[/latex] or [latex]\geq[/latex], as appropriate.
linear equation: A polynomial equation of the first degree (such as [latex]x=2y-7[/latex]).
real numbers: The smallest set containing all limits of convergent sequences of rational numbers.
Linear Inequalities
When two linear expressions are not equal, but are designated as less than ([latex]<[/latex]), greater than ([latex]>[/latex]), less than or equal to ([latex]\leq[/latex]) or greater than or equal to ([latex]\geq[/latex]), it is called a linear inequality. A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
For inequalities that contain variable expressions, you may be asked to solve the inequality for that variable. This just means that you need to find the values of the variable that make the inequality true.
A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign. The same properties for solving an equation are used to solve an inequality; however, when solving an equation there is one solution (or one value that makes the equation true), but when solving an inequality there are many solutions (or values that make the statement true).
Solutions of Linear Inequalities
Example: Graph the solutions of the inequality: [latex]x>4[/latex]
The solutions to this inequality includes every number that is greater than [latex]4[/latex] as shown below.
Inequality: Solutions to [latex]x>4[/latex] are graphed in yellow on the number line. Notice the open circle means that the value of [latex]4[/latex] in not a solution to the inequality since [latex]4>4[/latex] is a false statement. If the inequality was [latex]x\geq 4[/latex], then [latex]4[/latex] would be a solution and there would be a closed circle over the [latex]4[/latex] on the number line.
Solving Linear Inequalities
Solving the inequality is the same as solving an equation. There is only one rule that is different: When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.
Example: Solve the inequality: [latex]-7x+3+x \leq 1-4x-10[/latex]
Step 1, combine like terms on each side of the inequality symbol:
[latex]\displaystyle
-6x+3\leq-4x-9[/latex]
Step 2, since there is a variable on both sides of the inequality, choose to move the [latex]-4x[/latex], to combine the variables on the left hand side of the inequality.
Adding [latex]4x[/latex] yields:
[latex]\displaystyle
-2x+3\leq-9[/latex]
Step 3, this is similar to solving a two step equation. Subtract [latex]3[/latex]:
[latex]\displaystyle
-2x\leq-12[/latex]
Finally, divide both sides by [latex]-2[/latex] (remember to reverse the inequality symbol):
[latex]\displaystyle
x\ge 6[/latex]
To read this answer, read from right to left, [latex]x\geq6[/latex]. This reads "[latex]x[/latex] is greater than or equal to 6". | 677.169 | 1 |
RHombus
Trapezium
1 pair of opposite parallel sides
Kite
2 pairs of equal adjacent sides
1 pair of opposite equal angles
Now that you know about 6 different types of quadrilaterals. Move the points (vertices) around and see how many different quadrilaterals you can find in the interactive below. (Watch this video if you would like to see this interactive in action -)
Worked Examples
Question 1
For each of the shapes below, choose the most precise classification.
A quadrilateral is illustrated whose two opposite sides are congruent indicated by the same hash marks on the sides and its sides are perpendicular to each other indicated by small squares on all four corners.
Trapezium
A
Rectangle
B
Square
C
A quadrilateral is illustrated whose all four sides are congruent indicated by the same hash marks on the sides and are perpendicular to each other indicated by small squares on all four corners.
Square
A
Trapezium
B
Rectangle
C
A quadrilateral is illustrated whose opposite horizontal sides are parallel indicated by the same arrowmarks on the sides. Its right side is perpendicular to the two opposite horizontal sides of the quadrilateral as indicated by the small squares in two right corners. Its top base is longer than the bottom base.
Trapezium
A
Rectangle
B
Square
C
A quadrilateral is illustrated whose opposite sides are congruent indicated by the same hash marks on the sides, and all four sides are perpendicular to each other indicated by small squares on all four corners.
Trapezium
A
Rectangle
B
Square
C
A quadrilateral is illustrated whose opposite horizontal sides are parralel indicated by the same arrowhead marks on the sides. The top base is longer than the bottom base. None of the sides are perpendicular to each other.
Trapezium
A
Square
B
Rectangle
C
Question 2
Patricia draws a quadrilateral, and covers it up. She tells Glen that the quadrilateral consists of right angles only. From this information, Glen knows that the quadrilateral is definitely a:
Rhombus
A
Square
B
Trapezium
C
Rectangle
D
Solving problems
When solving angle problems in geometry one of the most important components is the reasoning (or rules) you use to solve the problem. You will mostly be required in geometry problems to not only complete the mathematics associated with calculating angle or side lengths but also to state the reasons you have used. Read through each of these rules and see if you can describe why and draw a picture to represent it. | 677.169 | 1 |
X CBSE NCERT Maths Chap 11 - Constructions Solved Questions
1. Draw a line segment AB of length 4.4cm. Taking A as centre, draw a circle of radius 2cm and taking B as centre, draw another circle of radius 2.2cm. Construct tangents to each circle from the centre of the other circle.
2. Draw a pair of tangents to a circle of radius 2cm that are inclined to each other at an angle of 90. 3. Construct a tangent to a circle of radius 2cm from a point
on the concentric circle of radius 2.6cm and measure its length. Also,
verify the measurements by actual calculations. (length of tangent
=2.1cm) 4. Construct an isosceles
triangle whose base is 7cm and altitude 4cm and then construct another
similar triangle whose sides are 3/2 times the corresponding sides of
the isosceles triangle. 5. Draw a line segment AB of length 8cm. taking A as center, draw a circle of radius 4cm and taking B as centre, draw another circle of radius 3cm. Construct tangents to each circle from the center of the other circle. Section-B
PRACTICE EXERCISE
4. Construct a D ABC with BC = 6 cm,
<A = 60° and median AD through A is 5 cm long. Construct a DA'BC'
similar to DABC with BC = 8 cm. 5. Construct a DABC similar to a
given equilateral triangle PQR with side 5 cm such that each of its
sides is 6/7 th of the corresponding sides of DPQR. 6. Construct a DABC, BC = 6.5 cm,
<B = 45° and <A = 100°. Construct another triangle similar to the
triangle ABC whose sides are 6/5 times of the triangle ABC 7. Construct a tangent to a circle of radius 4 cm from a point
on the concentric circle of radius 6 cm and measure its length. Also,
verify the measurement by actual calculation. (Ans. 4.5 cm approx) 8. Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter each at a distance of 8 cm from its centre. Draw tangents to the circle from these two points A and B. 9. Draw a line segment AB of length 11 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle. 10. Let ABC be a right triangle in
which AB = 6 cm, BC = 8 cm and <B = 90°. BD is the perpendicular from
B on AC. The circle through B, C, D is drawn. Construct the tangents
from A to this circle. | 677.169 | 1 |
Trigonometry Table – Formula, Function, Identities, Graph & Examples
Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. Central to this study is the trigonometry table, a valuable tool that lists the values of trigonometric functions such as sine, cosine, and tangent for various angles. These functions are fundamental in understanding the properties of right-angled triangles and the unit circle.
Formulas in trigonometry provide the mathematical framework to solve problems involving triangles and periodic phenomena. The primary trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—are defined through ratios of a triangle's sides and extend to the unit circle for broader applications.
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. These identities, such as the Pythagorean identity and angle sum formulas, simplify complex expressions and solve equations more efficiently.
Graphs of trigonometric functions illustrate their periodic nature and key properties, such as amplitude, period, and phase shift. These visual representations are crucial in fields ranging from physics to engineering.
Examples and applications of trigonometry abound in real life, from calculating heights and distances to analyzing sound waves and electrical currents. Understanding the trigonometry table, along with its associated formulas, functions, identities, and graphs, is essential for mastering this versatile and practical area of mathematics.
A trigonometry table is an essential mathematical tool that provides the values of trigonometric functions for different angles. Trigonometry, a branch of mathematics, explores the relationships between the angles and sides of triangles. The primary trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot)—are crucial for solving problems related to angles and distances. These tables are particularly important for board exams and competitive exams.
Trigonometry Table
A trigonometry table provides the values of fundamental trigonometric functions such as sine (sin), cosine (cos), and tangent (tan) for commonly used angles, typically in degrees or radians. These tables are structured with angles like 0°, 30°, 45°, 60°, and 90°, allowing users to quickly reference the corresponding function values without needing to calculate them manually.
Angle (Degrees)
0°
30°
45°
60°
90°
Angle (Radians)
0
π/6
π/4
π/3
π/2
sin
0
1/2
√2/2
√3/2
1
cos
1
√3/2
√2/2
1/2
0
tan
0
1/√3
1
√3
Undefined
The sine (sin) function represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The cosine (cos) function represents the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent (tan) function represents the ratio of the length of the opposite side to the length of the adjacent side. These functions form a fundamental table in trigonometry, crucial for solving problems involving right triangles and applicable across physics, engineering, and mathematics.
Trigonometric Table
Trigonometric tables today may extend beyond standard functions to include hyperbolic trigonometric values, which are derived from hyperbolas rather than circles.
With the widespread availability of digital tools, trigonometric functions in tables can now be swiftly computed using calculators, computers, and various software applications. These resources ensure precise and reliable calculations for trigonometric functions across all angles, whether in degrees or radians.
Trigonometry Table Values
Degrees (°)
Sine (sin)
Cosine (cos)
Tangent (tan)
Cosecant (csc)
Secant (sec)
Cotangent (cot)
0°
0
1
0
undefined
1
undefined
30°
1/2
√3/2
√3/3
2√3/3
2
√3
45°
√2/2
√2/2
1
√2
√2
1
60°
√3/2
1/2
√3
2/√3
2
1/√3
90°
1
0
undefined
1
undefined
0
120°
√3/2
-1/2
-√3
-2/√3
-2
-1/√3
135°
√2/2
-√2/2
-1
-√2
-√2
-1
150°
1/2
-√3/2
-√3/3
-2√3/3
-2
-√3
180°
0
-1
0
undefined
-1
undefined
210°
-1/2
-√3/2
√3/3
-2√3/3
-2
√3
225°
-√2/2
-√2/2
1
-√2
-√2
1
240°
-√3/2
-1/2
√3
-2/√3
-2
1/√3
270°
-1
0
undefined
-1
undefined
0
300°
-√3/2
1/2
-√3
2/√3
2
-1/√3
315°
-√2/2
√2/2
-1
√2
√2
-1
330°
-1/2
√3/2
-√3/3
2√3/3
2
√3
360°
0
1
0
undefined
1
undefined
Please note: In this table, "undefined" denotes that the trigonometry formula is not applicable for that specific angle. Additionally, the values displayed are rounded to several decimal places for simplicity.
Sin Cos Table
Angles (In Degrees)
0°
30°
45°
60°
90°
180°
270°
360°
Angles (In Radians)
0°
π/6
π/4
π/3
π/2
π
3π/2
2πTrigonometry Table for Class 10
Trigonometry is introduced to students in class 10 across most Indian education boards, often posing a challenge to those entering their final year of high school. This branch of mathematics appears daunting at first due to its unfamiliarity. To aid students in grasping these concepts, a trigonometry table has been devised specifically for standard angles: 0°, 30°, 45°, 60°, and 90°. This table provides the values of Sin and Cos for these angles, crucial for understanding and mastering trigonometric principles. Memorizing this table not only facilitates success in school examinations but also establishes a solid groundwork for advanced studies. Below is the trigonometry table tailored for class 10:
Angle
Sin Value
Cos Value
0°
0
1
30°
1/2
√3/2
45°
√2/2
√2/2
60°
√3/2
1/2
90°
1
0
In exam situations, students often face nervousness which can cause them to forget memorized information. To counter this, a helpful technique for learning trigonometry tables comes into play. This trick allows you to derive trigonometric ratios without needing to memorize them outright.
Instead of rote memorization, follow these steps to find the values for standard angles:
List the standard angles from 0 to 90 degrees.
Write whole numbers starting from 0 beneath these angles: 0 for 0 degrees, 1 for 30 degrees, and so forth.
Divide these numbers by 4, the largest value.
Take the square root of the divided results.
These results give you the values of sine (Sin) for the standard angles.
Write down these sine values in reverse order to obtain the cosine (Cos) values.
With sine and cosine values at your fingertips, you can easily determine the values of other trigonometric ratios. This method ensures you can calculate trigonometric values confidently, even if you forget the exact trigonometry table during your exam.
Trigonometric Functions Table
The sine (sin), cosine (cos), and tangent (tan) are essential trigonometric functions used in mathematics and science. A standard table for these functions provides their values at significant angles, typically in degrees and sometimes in radians, within a right-angled triangle. Below is a commonly referenced table showing sin, cos, and tan values at angles of 0°, 30°, 45°, 60°, and 90°:
Angle (°)
Sin
Cos
Tan
0
0
1
0
30
1/2
√3/2
1/√3
45
√2/2
√2/2
1
60
√3/2
1/2
√3
90
1
0
Undefined
These values originate from the characteristics of right-angled triangles and the unit circle. They form the basis of trigonometry and find extensive application in mathematics, engineering, and physics. It's crucial to recognize that the tangent function becomes undefined at 90° due to division by zero (tan θ = sin θ / cos θ, and cos 90° = 0).
Trigo Table
Understanding the trigonometric ratios is crucial for students as it simplifies learning the Trigonometry Table. These ratios depend on specific trigonometric values. To facilitate understanding, we provide both the trigonometric table and its values below. Students are encouraged to familiarize themselves with the ratios and their relationships before delving into the steps of memorizing the trigonometric table. Familiarity with these ratios beforehand will significantly ease the process of learning trigonometry for students.
Trigonometry Table 0-360 Value
sin x = cos (90° – x)
cot x = tan (90° – x)
sec x = cosec (90° – x)
cos x = sin (90° – x)
tan x = cot (90° – x)
cosec x = sec (90° – x)
1/sin x = cosec x
1/tan x = cot x
1/cos x = sec x
Trigonometry Table Tricks To Learn Trigonometric Table
Learning the trigonometric table can be quite manageable, especially with the help of mnemonic devices and visualization techniques. Here's how you can approach it:
Mnemonic Devices:
Mnemonics such as "Some People Have Curly Brown Hair" are great for quickly recalling sine values like 0, 1/2, √2/2, √3/2, and 1 for angles 0°, 30°, 45°, 60°, and 90° respectively.
Visualize the Unit Circle:
Understanding the unit circle conceptually maps trigonometric values as coordinates on a circle. This visualization aids in remembering sine and cosine values for common angles.
Relate to Special Triangles:
Memorize the properties of 30°-60°-90° and 45°-45°-90° triangles. These triangles directly correlate with sine, cosine, and tangent values, simplifying memorization.
Regular Practice:
Consistent practice through exercises and quizzes reinforces memory retention. This practice is crucial for quick recall during exams or practical applications.
Choosing the best method depends on your learning style, but combining these approaches can significantly enhance your understanding and memorization of the trigonometric table.
How to Create a Trigonometry Table?
To create a trigonometric table, follow these steps:
Step 1: Construct the table with angles (0°, 30°, 45°, 60°, 90°) in the top row and trigonometric functions (sin, cos, tan, cot, sec, cosec) in the first column.
Can you give examples of using trigonometric functions?
Examples include finding heights using angles of elevation or depression, analyzing oscillatory motion using sine functions, and calculating forces in mechanical systems using cosine functions.
conclusion
Trigonometry Tables serve as indispensable tools in mathematics, offering a comprehensive reference for trigonometric functions across standard angles. They provide precise values for sine, cosine, tangent, and their reciprocal functions, facilitating calculations in fields ranging from engineering and physics to astronomy and beyond.
Understanding trigonometric formulas, such as Pythagorean identities and sum/difference identities, enhances problem-solving capabilities by simplifying equations and relationships involving angles. These identities, combined with the graphical representation of trigonometric functions on Cartesian planes, illustrate their periodic nature and essential properties, aiding in visualizing and analyzing various phenomena.
Real-world applications demonstrate the practical utility of trigonometry, from determining distances and angles in surveying and navigation to modeling oscillatory behavior in physics and engineering. By leveraging trigonometric identities and functions, one can derive solutions to complex problems involving angles, triangles, and periodic phenomena, thereby advancing scientific understanding and technological | 677.169 | 1 |
Therefore, the new endpoints of the line segment after dilation are ( (4, \frac{7}{2}) ) and ( (3, \frac{7}{2}) ), and the length of the dilated line segment is ( | 677.169 | 1 |
What is a decimal coordinate?
Decimal degrees are the latitude and longitude geographic coordinates as decimal fractions. Positive latitudes are north of the equator and less than zero latitudes are south of equator. Positive longitudes are east of the prime meridian and less than zero longitudes are to the west of the prime meridian.
Can 0.5 be a coordinate?
Can you have a decimal coordinateCan a decimal be in a coordinate plane?
A point on the x axis is represented by it's distance from the origin (0,0) in a negative or positive direction. E.g. The decimal 2.4 could be represented by the point (+2.4,0) on the xy− plane. The point x=2.4 on the x− axis. Thus all real numbers can be represented as points on the real line in in same way.
How do you convert decimal coordinates to minutes?
Here's How to Do the Conversion
The whole units of degrees will remain the same (e.g., if your figure is 121.135 degrees longitude, start with 121 degrees). Multiply the decimal portion of the figure by 60 (e.g., . 135 * 60 = 8.1). The whole number becomes the minutes (8).
How do you write coordinate?
Coordinates are written as (x, y) meaning the point on the x axis is written first, followed by the point on the y axis. Some children may be taught to remember this with the phrase 'along the corridor, up the stairs', meaning that they should follow the x axis first and then the y.
How do you write a point from a coordinate plane?
Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its y-coordinate, or vertical displacement from the origin. Together we write them as an ordered pair indicating the combined distance from the origin in the form (x,y) .
How do you write coordinates on a coordinate plane?
Coordinates are written as (x, y) meaning the point on the x-axis is written first, followed by the point on the y-axis. Some children may be taught to remember this with the phrase 'along the hall, up the stairs', meaning that they should follow the x-axis first and then the y.
Can coordinates be zero?
Can 0 be a coordinate?
Coordinates of Zero
Points with a y -coordinate equal to 0 are on the x-axis, and have coordinates (a,0) . Points with a x -coordinate equal to 0 are on the y-axis, and have coordinates (0,b) . The point (0,0) is called the origin.
What is the rule for coordinates?
The order in which you write x- and y-coordinates in an ordered pair is very important. The x-coordinate always comes first, followed by the y-coordinate. As you can see in the coordinate grid below, the ordered pairs (3,4) and (4,3) are two different points!
What is 1 hour as a decimal?
One hour is equal to 1.0, one decimal minute is equal to 1/60, and one second is equal to 1/3600. Decimal time is convenient for mathematical calculations, but often needs to be converted back to hours and minutes for everyday usage. | 677.169 | 1 |
Circle vs. Ellipse: What's the Difference?
A circle is a shape with all points equidistant from its center; an ellipse has two focal points and is elongated.
Key Differences
A circle and ellipse, both closed curves, vary distinctly in their geometric properties. A circle is a simple shape where every point on its perimeter is at an equal distance from the center. Imagine a rubber band being stretched uniformly in all directions; it forms a circle. In contrast, an ellipse appears as a "flattened" circle, having two axes of different lengths: a longer major axis and a shorter minor axis.
While every circle has a center, an ellipse boasts two focal points. For any point on an ellipse, the sum of its distances to these two foci remains constant. If these foci were to merge, the ellipse would transform into a circle. Furthermore, circles possess a consistent curvature across their entirety, providing them with perfect symmetry. Ellipses, however, exhibit varying curvature depending on the location, making them asymmetrical in comparison to circles.
While both figures are special cases of ellipses, the circle stands out as unique due to its constant radius. An ellipse, on the other hand, lacks a uniform radius, making it more complex and versatile in terms of its shape and properties.
Comparison Chart
Definition
All points equidistant from center
Oval shape with two focal points
Symmetry
Perfectly symmetrical
Asymmetrical
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Curvature
Consistent
Varies
Key characteristics
Single center, constant radius
Two foci, major and minor axes
Circle and Ellipse Definitions
Circle
A round plane figure.
The sun appears as a bright circle in the sky.
Ellipse
Oval shape with major and minor axes.
The racetrack was designed as an ellipse for variety.
Circle
Represents wholeness or completeness.
The circle of life continues endlessly.
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Ellipse
A flattened circle with two focal points.
The orbits of planets are often in the shape of an ellipse.
Circle
All points equidistant from a center.
She drew a perfect circle with her compass.
Ellipse
A closed curve derived from intersecting a cone with a plane.
The shadow of the earth on the moon during an eclipse forms an ellipse.
Circle
Can symbolize infinity.
Their love was like a never-ending circle.
Ellipse
Sum of distances from any point to two foci is constant.
When using string and pins to draw an ellipse, the string length remains unchanged.
Circle
A closed curve without corners.
The kids formed a circle to play a game.
Ellipse
Special case of conic sections.
An ellipse becomes a circle when its foci merge.
Circle
A plane curve everywhere equidistant from a given fixed point, the center.
Ellipse
A conic section whose plane is not parallel to the axis, base, or generatrix of the intersected cone.
Circle
A planar region bounded by a circle.
Ellipse
The locus of points for which the sum of the distances from each point to two fixed points is equal.
Circle
Something, such as a ring, shaped like such a plane curve.
Ellipse
Ellipsis.
FAQs
Are all round shapes circles?
No, round shapes can be circles, ellipses, or other forms.
What defines a circle's shape?
A circle is defined by all its points being equidistant from a central point.
Are all ellipses elongated?
No, when the major and minor axes are equal, an ellipse is a circle.
How is an ellipse formed in conic sections?
An ellipse is formed by cutting a cone with a plane at an angle to its base.
Why do planets have elliptical orbits?
Planetary elliptical orbits result from the gravitational forces acting between celestial bodies.
Which shape, circle or ellipse, is perfectly symmetrical?
A circle is perfectly symmetrical.
How do the major and minor axes relate to an ellipse?
An ellipse has a longer major axis and a shorter minor axis, determining its shape.
In what scenarios is an ellipse preferred over a circle?
Ellipses are preferred in situations requiring elongated shapes or when representing angled views of circles.
How does an ellipse differ from a circle in terms of foci?
A circle has one center, while an ellipse has two focal points.
Which shape has varying curvature: circle or ellipse?
An ellipse has varying curvature, while a circle's curvature remains consistent.
What's a real-world example of an ellipse?
Planetary orbits often resemble ellipses.
How do artists use ellipses in perspective drawing?
Artists use ellipses to depict circles viewed at an angle.
Can a circle be considered a type of ellipse?
Yes, a circle is a special case of an ellipse where the two foci merge into one.
In terms of foci, how can a circle be visualized?
A circle can be seen as an ellipse where both foci are at the same point, its center.
What happens when the two foci of an ellipse merge?
When the two foci merge, the ellipse becomes a circle.
How can one draw a perfect circle using an ellipse tool?
Using an ellipse tool, ensure the major and minor axes are equal to draw a perfect circle.
Which shape, circle or ellipse, can represent perfect wholeness?
A circle often symbolizes perfect wholeness or completeness.
Are all circles ellipses?
Yes, all circles are special cases of ellipses.
How do the diameters of a circle and ellipse compare?
A circle has a consistent diameter; an ellipse has varying diameters along its major and minor axes | 677.169 | 1 |
Lines and Angles - Sub Topics
The reading material provided on this page for Lines and Angles is specifically designed for students in grades 7 to 10. So, let's begin!
Lines and Angles
Ancient mathematicians introduced the concept of lines to represent one-dimensional objects without width or depth. Lines serve as fundamental building blocks in geometry, providing a straightforward understanding of straight objects. On the other hand, an angle is a geometric shape that arises from the intersection of two line segments, lines, or rays. When two rays intersect within the same plane, they form an angle. Angles provide a means to measure the amount of rotation or inclination between two intersecting lines or line segments.
Line
A line is a straight path that extends infinitely in both directions. Lines are often represented by a straight line with arrows on both ends to indicate that it goes on forever.
Ray
A line with one endpoint and another endpoint that extends infinitely in one direction is known as a ray.
Line Segment
A line segment is a part of a line that consists of two endpoints. It can be defined as a straight line with a starting and ending point.
Other Types of Lines
Curved Line
A line that bends and has a distinct shape, such as a circle or an ellipse.
Vertical Line
A line that runs straight up and down and is perpendicular to the horizon.
Horizontal Line
A line that runs straight across and is parallel to the horizon.
Perpendicular Lines
Perpendicular lines are two lines that intersect each other at a 90-degree angle. They form an "L" shape at their point of intersection.
Parallel Lines
Parallel lines are two or more lines that are always the same distance apart and never intersect, no matter how far they are extended in either direction.
Intersecting Lines
Intersecting lines are two or more lines that meet at a common point. The point of intersection is the point where the lines meet.
Properties of Lines
1. A line has an infinite length and no width or thickness. 2. A line is made up of an infinite number of points. 3. Two distinct lines cannot intersect at more than one point. 4. If two lines intersect, then the opposite angles formed are equal. 5. If two lines are parallel, then the corresponding angles formed are equal. 6. If two lines are parallel, then the alternate angles formed are equal. 7. If a line intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary (add up to 180 degrees). 8. The sum of the three angles in a triangle is always 180 degrees. 9. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Angles
An angle is the measure of the amount of rotation between two lines that intersect at a point. Angles are often represented by a small arc, with a point at the vertex (the point where the end point of rays meet) and two rays (lines) coming out of the vertex. The angle is measured in degrees (°) or radians (rad).
Classification of Angles
Zero Angle: An angle that measures exactly 0°.
Acute Angle: An angle that measures less than 90°, typically between 0° and 89°.
Right Angle: An angle that measures exactly 90°.
Obtuse Angle: An angle that measures greater than 90° and less than 180°, typically between 91° and 179°.
Straight Angle: An angle that measures exactly 180°.
Reflex Angle: An angle that measures greater than 180° and less than 360°, typically between 181° and 359°.
Complete Angle: An angle that measures exactly 360°.
Types of Angles based on Relation of Angles
1. Complementary Angles
Two angles whose measures combine to form 90°. If two angles a and b are complementary, then their sum is equal to 90°, i.e., ∠ a + ∠ b = 90°.
In this case, ∠ a is the complement of ∠ b, and ∠ b is the complement of ∠ a.
2. Supplementary Angles
Two angles whose measures combine to form 180°. If two angles a and b are supplementary, then their sum is equal to 180°, i.e., ∠ a + ∠ b = 180°.
In this case, ∠ a is the supplement of ∠ b, and ∠ b is the supplement of ∠ a.
3. Vertically Opposite Angles
Two angles are formed by intersecting lines that are opposite each other and have the same measure.
4. Linear Pair of Angles
Two angles are adjacent and supplementary.
Angles formed by Parallel Lines and Transversal Lines
1. Alternate Interior Angles: Two angles that are on opposite sides of the transversal and inside the alternate interior angles.
∠ 4 = ∠ 5 ∠ 3 = ∠ 6
2. Alternate Exterior Angles: Two angles that are on opposite sides of the transversal and outside the alternate interior angles.
∠ 2 = ∠ 7 ∠ 1 = ∠ 8
3. Corresponding Angles: Two angles that are on the same side of the transversal and correspond to each other.
∠ 1 = ∠ 5 ∠ 2 = ∠ 6 ∠ 3 = ∠ 7 ∠4 = ∠ 8
4. Co-interior Angles: Two interior angles that are on the same side of the transversal make 180° together.
∠ 3 + ∠ 5 = 180° ∠ 4 + ∠ 6 = 180°
5. Exterior Angle Property: The exterior angle property states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it | 677.169 | 1 |
Knowledge Check
C1 is a circle with centre at the origin and radius equal to r and C2 is a circle with centre at (3r, 0) and radius t 2r. The number of common tangents that can be drawn to the two circles are
A1
B2
C3
D4
Question 2 - Select One
Let R1andR2 be the radil of the circles with centres at C1andC2. Statement 1: If C1C2≤r1+r2, then the two circles have two common tangents. Because Statemetn 2: For two common tangents the two circles must intersect in two points.
AStateme-1 is True, Statement-2 is True, Statemetn-2 is correct explanation for Statement-1
BStatement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
CStatement-1 is True, Statement-2 is False
DStatement-1 is False, Statement-2 is ture
Question 3 - Select One
Let C be the circle with centre (1,1) and radius 1 . If T is the circle centred at (0,y) , passing through origin and touching the circle C externally , then the radius of T is equal to | 677.169 | 1 |
Unit 10 circles homework 5 answer key
report flag outlined. Answer: 96 degrees. Step-by-step explanation: Since you know that arc are twice the angle measure, that means arc RT is 42 * 2 = 84. Then, you have to find the measure of RS. Remember that arc RT and RS have to add up to 180 degrees because they form a semicircle (half of a circle). To find the measure of RS, just do 180 ...4.8/5 Unit 10 Circles Homework 10 Answer Key, Desktop Engineer Resume Format Pdf, Sample Small Business Plan Entertainment Industry, Help Writing Your Personal Statement, Difference Research Paper Essay, Discursive Essay Planner, Worn Path Essay TopicsUnit 10 Test study (Circles) Topic i; of Circles 1. Using the diagram below, give an opic 2: Area & Circumference Name: Date: example of each cirde part a. Center: b. Radius: PM …
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Math; Geometry; Geometry questions and answers; Name: Date: Bell: Unit 10: Circles Homework 8: Equations of Circles ** This is a 2-page document ** Directions: Use the information given to write the equation of the circle.Charts: Kohl's (KSS) Stock Gaps Higher as Activists Circle...KSS Retailer Kohl's (KSS) is trading sharply higher Monday as the company announced that it had received a number o...Instructional materials by All Things Algebra (R) are well-designed, comprehensive, and high-quality. This saves teachers time in lesson planning and ensures that students are exposed to material that is engaging, rigorous, and attainable for all. Thousands of schools across the United States (and the world) are already using All Things Algebra ...We detail the Circle K cash back policy (including potential limits, fees, and more), plus other gas stations that do cash back. You can typically get up to $40 cash back at Circle... | 677.169 | 1 |
#4 of 4: Medium
Translations
<p>What is the translation from Point `G` `(6, ``-1)` to Point `H` `(``-1,4)`? <br><highlight data-color="#666" data-style="italic">Use positive numbers or zero.</highlight></p><selectivedisplay><p><highlight data-color="#666" data-style="italic">Students will specify the translation in terms of Left/Right and Up/Down.</highlight></p></selectivedisplay>
Translations are a type of transformation in which a shape is moved from one position to another without changing its size, shape, or orientation. In grade 8 transformations, students learn how to perform translations by sliding a shape along a straight line in a specified direction and distance. They also learn how to use coordinate notation to describe translations and how to apply translations to solve problems. Understanding tr...
Show all
Grade 8
Transformations
8 | 677.169 | 1 |
Centered at each lattice point in the coordinate plane are a circle radius \frac{1}{10} and a square with sides of length \frac{1}{5} whose sides are parallel to the coordinate axes. The line segment from (0,0) to (1001,429) intersects m of the squares and n of the circles. Find m+n. | 677.169 | 1 |
Question 1.
Name the type of the following triangles.
(a) ∆PQR with m∠Q = 90°
(b) ∆ABC with m∠B = 90° and AB = BC
Solution:
(a) One of the angles is 90°
It is a right-angled triangle
(b) Since two sides are equal.
It is an isosceles triangle. Also m∠B = 90°
It is an Isosceles right-angled triangle
Question 6.
(a) Try to construct triangles using matchsticks.
(b) Can you make a triangle with?
(i) 3 matchsticks?
(ii) 4 matchsticks?
(iii) 5 matchsticks?
(iv) 6 matchsticks?
Name the type of triangle in each case. If you cannot make a triangle think of the reason for it.
Solution:
(b) (i) With the help of 3 matchsticks, we can make an equilateral triangle. Since all three matchsticks are of equal length.
(ii) With the help of 4 matchsticks, we cannot make any triangle because in this case, sum of two sides is equal to the third side and we know that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
(iii) With the help of 5 matchsticks, we can make an isosceles triangle. Since we get two sides equal in this case.
(iv) With the help of 6 matchsticks, we can make an equilateral triangle. Since we get three sides equal in length.
Question 8.
Draw any line segment \(\overline{\mathbf{P Q}}\). Take any point R not on it. Through R, draw a perpendicular to \(\overline{\mathbf{P Q}}\).
Solution:
Construction:
(i) Drawn a line segment \(\overline{\mathbf{P Q}}\) using scale and taken a point R outside of \(\overline{\mathbf{P Q}}\).
(ii) Placed a set-square on \(\overline{\mathbf{P Q}}\) such that one arm of its right angle aligns along \(\overline{\mathbf{P Q}}\).
(iii) Placed a scale along the other edge of the right angle of the set-square
(iv) Slide the set-square along the line till the point R touches the other arm of its right angle.
(v) Joined RS along the edge through R meeting \(\overline{\mathbf{P Q}}\) at S.
Hence \(\overline{\mathbf{R S}}\) ⊥ \(\overline{\mathbf{P Q}}\). | 677.169 | 1 |
Define the following kinds of angles: zero degree, acute, straight, right, and obtuse.
State the Pythagorean Theorem and explain its usefulness.
Define the following terms related to circles: radius, diameter, arc, and circumference.
Give the equations for finding a circle's circumference and area if you know its radius.
Explain the use of the following measuring tools: calipers, micrometers, and protractors.
Demonstrate how to convert measurements from inches to millimeters and from millimeters to inches | 677.169 | 1 |
Learn everything you need to know about adjacent angles in this comprehensive article. From real-world examples to measuring angles, this article will provide step-by-step instructions to help you understand adjacent angles properties and how they can be used in geometry and your day-to-day life.
This article provides a comprehensive guide to understanding adjacent angles and their properties, applications, and identification, with historical context and the contributions of famous mathematicians. Examples, strategies, and implications in real-life scenarios are also discussed. | 677.169 | 1 |
First principles of Euclid: an introduction to the study of the first book of Euclid's Elements 11 - 11 брь фб 11.
УелЯдб 127 ... Let us suppose that A D is not parallel to B C. Then draw A E meeting BD in E. Join E. Proof . A B D ( b ) Let us suppose that A E is parallel to BC . Then AB C , E CB are two triangles on the same base B C , and between the same ...
УелЯдб 18 66 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let...
УелЯдб 34 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it.
УелЯдб 94 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of' the base, equal to one another, and likewise those which are terminated in the other extremity.
УелЯдб 104 - If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
УелЯдб 51 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line. | 677.169 | 1 |
$\begingroup$The wording is a bit awkward. The best interpretation (as you seem to suggest) is that "perpendicular to the $y$-axis" describes the plane of the semicircular cross section, while "parallel to the $x$-axis" describes the diameter of the semicircle itself. (For a plane, "perpendicular to $y$" already guarantees "parallel to $x$ (and $z$)", so using both to describe the cross-sectional region would be redundant.) I might've phrased things like this: "Cross sections, in planes perpendicular to the $y$-axis, are semi-circles (above the $xy$-plane) with diameters spanning the figure's base."$\endgroup$
$\begingroup$Well, yeah, why not simpler is exactly what I am saying as a student. This professor seems to try to make things way more complicated than they actually are to the point that I don't even think I have really learned any of the actual concepts we should be covering (too focused on dumb stuff like this). Not to mention the fact that the slices are parallel to z and perpendicular to both x and y not parallel to x. The base of the semicircle runs parallel to x, not the slice.$\endgroup$ | 677.169 | 1 |
Ellipse Drawing Tool
Ellipse Drawing Tool - Drag point c, the center of the ellipse, to see how changing the center of the ellipse changes the equation. Web the ellipse tool lets you draw ellipses such as the ones below. The currently selected brush is used for drawing the ellipse outline. The ellipse can be rotated. Web an ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve.
Z = 1 6 6. Web drawing › drawing & lettering aids $19997 free delivery monday, november 6. Change d to adjust the magnitude of h (distance from blue point) and k (the distance from the red point) 3. Using the major and minor axis you can easily. Drag point c, the center of the ellipse, to see how changing the center of the ellipse changes the equation. In the shape tool options bar, set mode, fill, stroke, w, h, path operation, path alignment, path arrangement,. Click and drag on the artboard.
72. Drawing Tool:Ellipse How to use ibisPaint
Adjust z to change the angle from the center of the ellipse. Web in this tutorial, we'll review 2 coreldraw tools for drawing circles and ellipses: Position the pointer on the canvas, then click and.
Lotter Elliptic Trammel ellipsograph precision ellipse drawing
Position the pointer on the canvas, then click and drag to draw. Web explore math with our beautiful, free online graphing calculator. An ellipse can be drawn anywhere in the image by defining its points.
Ellipse Drawing Tool The ellipse can be rotated. Web select the ellipse tool () from the toolbar. Web to draw an ellipse, identify the minor and major axis of the ellipse, measure out equal distances from the intersection of these lines, and visualize an ellipse outline through the measured markers. If you can't find the ellipse tool, click and hold the rectangle tool to. Click on the boxes in order to see the steps to graph the ellipse. | 677.169 | 1 |
In isosceles trapezoid ABCD, point E is the midpoint of the larger base AD
In isosceles trapezoid ABCD, point E is the midpoint of the larger base AD, ED = EC, angle BAD = 60 degrees, prove that quadrilateral ABCE is a rhombus.
Since the trapezoid is isosceles, the angle CDA, at the base, is equal to the angle BAD and is equal to 60.
Consider a triangle CED, in which, according to the condition, ED = EC, that is, an isosceles triangle, then the angle ECD = CDA = 60. Since the two angles are equal to 60, then the angle CED = 60.
Then the triangle CED is equilateral.
Since the angles BAD and CED are equal to each other, AB is parallel to EC, and since the bases of the trapezoid AD and BC are also parallel, then AE is parallel to BC, hence ABCE is a parallelogram.
Since point E is the middle of AD, then AE = AD = EC = AB, therefore, ABCE is a rhombus.
Q.E.D | 677.169 | 1 |
How Does Gauth AI Simplify Complex Geometry Problems Like Finding Measure of Angle? calculus, algebra, geometry, and more.
This tool is very user-friendly and aims to enhance learning by breaking down complex concepts into parts that are easily understandable and hence make maths concepts accessible to everyone. If you are facing the query, what is the measure of angle r? 37° 53° 90° 97° then no need to worry this tool will explain it simply and let us see how.
Solving Trigonometric Principles
Trigonometry is a field of math that concentrates on the association between triangle sides and points. It has sophisticated tools for using angles to solve geometry problems. We can use trigonometric concepts like the sine, cosine, and tangent ratios to figure out the angle r. These proportions contrast the points of a triangle with the lengths of its sides.
We are able to use trigonometric ratios to determine the measurement of angle r because we are aware of the measurements of the other angles in the triangle that have been provided.
Steps to Solve Trigonometric Problem
The measure of angle r is still up in the air through an orderly cycle that includes inputting the question, getting a free preliminary, sitting tight for handling, and lastly getting the response. Individuals can efficiently and conveniently solve for the measure of angle r by following these steps.
Step 1: Input the Query
Entering the query into a math-solving platform like Gauth is the first step in solving for the measure of angle r. The given angles that are connected to angle r in this case 37°, 53°, 90°, and 97°are included in this query. The math-solving platform will be able to process the data and offer a solution for the measure of angle r if these values are entered precisely.
You can upload the image of the question by selecting the Upload Image option in case there is a diagram.
Step 2: Get the Free Trial
The next step is to get a free trial of the math-solving platform after entering the query. People can now access the features and tools they need to solve for the measure of angle r thanks to this. Users can take advantage of the platform's capabilities and guarantee accurate calculations and solutions by signing up for a free trial.
Step 3: Wait for Processing
When the query is submitted and the free preliminary is procured, people need to sit tight for the math-tackling stage to handle the data. During this time, the stage uses complex calculations and numerical computations to decide the proportion of point r in light of the given points.
Step 3: Get your Answer
Once the wait is over, you will see the solution to your input query, you will be amazed to know that this tool provides a detailed solution so that you can understand it easily. Moreover, there are further two options available below the solution that are Need Improvement, in case you are not satisfied with the answer and want it to be more detailed.
The next option is Helpful for me which means you are satisfied with the solution and give the tool a thumbs up.
Tips for Using Gauth More Efficiently
Here are some of the tips for using Gauth more efficiently:
● Enhancing Problem Identification
The ability to accurately identify the kind of math problem that needs to be solved is one of the most important aspects of using Gauth effectively. Users are able to utilize the appropriate features and quickly obtain the desired solution thanks to this recognition. Users can save time and make better use of the various features by correctly identifying the issue.
● Utilizing Abbreviations
Utilizing time-saving strategies and shortcuts is yet another helpful tip for increasing Gauth usage efficiency. Users of Gauth are able to take advantage of keyboard shortcuts for common mathematical symbols, operations, and functions, allowing them to input equations or expressions more quickly.
Final Verdict
Using platforms for solving math problems like Gauth ensures accurate results and makes the process of solving math problems easier. Using the given angles' measurements and trigonometric principles, the measure of angle r can be determined. Geometry gives amazing assets to tackling mathematical issues and assumes a significant part in grasping the properties of triangles and points | 677.169 | 1 |
triangle with three equal sides is inscribed inside a
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23 Feb 2015, 12:30
area of triangle in circumcsribed circle=abc/4r, Eq triangle, so a=b=c. Area of triangle will be = a^3/4r If we draw the eq. triangle inside a circle, then a=2rcos30=sqrt3 r Probability= Area of Triangle/ Area of Circle = (3(sqrt3)(r^3)/4r)/(pi*r^2) = 3(sqrt3)/4piA polygon is inscribed in a circle means that all of its vertices are on the circumference.
Re: A triangle with three equal sides is inscribed inside a
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09 Mar 2023, 02:07
Expert ReplyRe: A triangle with three equal sides is inscribed inside a
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09 Mar 2023, 03:08
Bunuel wrote:Sir, you made my day!, geometry has been a huge pain point for me ever since high school!, have devoured the quant mega thread, that has been my go to source, along with quarter wit quarter wisdom blog.
Re: A triangle with three equal sides is inscribed inside a
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25 Apr 2024, 07 triangle with three equal sides is inscribed inside a [#permalink] | 677.169 | 1 |
Solution of triangles Questions and Answers
Which of the following statements must be true based on the diagram below Select all that apply Diagram is not to scale K L H J M I LM is a segment bisector OLM is a perpendicular bisector DM is the vertex of a pair of congruent angles in the diagram L is the midpoint of a segment in the diagram M is the midpoint of a segment in the diagram None of the above 4
PROPERTI AND CONGRUENCE m l m 2 m 5 m 2 m 3 m 2 m 7 m 3 m 7 Remember In order to use the Transitive Property the equivalent parts must each be an ENTIRE SIDE of the equation W Transitive Property of Equality Symmetric Property of m l m 3 m 5 m 2 m 3 Note In a real proof it would not really be very helpful to end up back where we started but we are just practicing using the properties to justify each step 4
Ron wants to build a ramp with a length of 14 ft and an angle of elevation of 26 The height of the ramp is about feet C 26 Part B 14 ft y x B Note Round your answer to the nearest tenth The length of the base of the ramp is about Note Round your answer to the nearest tenth feet
Priya is teaching her younger cousin to ride a bike She wants to stay on roads that are not too steep and easy enough for a new bike rider She has decided the roads must have an angle less than or equal to 7 degrees A 7 degree angle in a right triangle has a 3 25 ratio for the legs Select the legs of all right triangles that would be safe for a new bike rider
Given the information below determine the value of X X 11x 11 D 30 E 25 21 A F 77 B AABC AFED To set up your proportion corresponding sides go in the same fraction and parts from the same triangle go in the same location of the fractions both in the numerator top or both in the denominator bottom Use the similarity statement to determine the corresponding sides Remember the order of
also select the reasoning for your answer 33 21 Answer 1 not similar Answer 2 27 Answer 3 18 not congruent Answer 4 The figures are not similar because the corresponding angles are not congruent and the corresponding sides are not enough information A STU Ano similarity statement S 30 22 not enough information B T
In the triangle ABC side b 9 side c 7 and side a 6 what is the value of the cos A Round your answer to the nearest hundredth b B picture of triangle is purely for labeling use angles and side lengths specified in the problem
Fill in the blanks to describe the transformations that occur in the the graph from the Parent Function to the Graph Note this is set up exactly the same as in your lesson so you should be using the same language wording as you learned in the lesson Graph Parent Direction Vertex Opening
Question 1 1 point Name the postulate if possible that makes the triangles congruent T O O O a b C d A H a b C Od SSS ASA AAS SSS Question 2 1 point Name the postulate if possible that makes the triangles congruent R T S SAS AAS ASA Not Possible A A E M B P Question 3 1 point Name the postulate if possible that makes the triangles congruent
1 Aiden borrows a book from a public library He read a few pages on day one On day two he reads twice the number of pages than he read on day one On the third day he reads six pages less than what he read on the first day If he read the entire book that contains 458 pages in three days how many pages did he read on day three
Georgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 4 Part 2 The Segment Theorems Graphic Organizer In the remaining items of this task we will work with the relationships between the lengths of the segments created when these lines intersect Picture Type 2 tangents vertex outside 2 secants vertex outside Secant and tangent vertex outside 2 secants VERTEX INSIDE Theorem A
In Exercises 1 and 2 classify the triangle by its sides and by measuring Its angles 2 In Exercises 3 and 4 classify AQRS by its sides 3 Q 2 2 R 1 2 S 4 4 In Exercises 5 8 find the value of x 5 7 63 85 2x 6 4x 18 to 36 4 Q 1 3 R 3 2 S 2 1 1 A5OTAS 8 11 The figure shows the measures of various angles of a roof and its supports Find the measure of 21 the angle between an eave and a horizontal support beam 2x 61 3x 5 9 The measure of one acute angle of a right triangle is 12 more than 3 times the measure of the other acute angle Find the measure of each acute angle of the right triangle 10 Your friend claims that the measure of an exterior angle of a triangle can never be acute because it is the sum of the two nonadjacent angles of the triangle Is your friend correct Explain your reasoning 110
26 Metal Brace The diagram shows the dimensions of a metal brace used for strengthening a vertical and horizontal wooden junction Classify the triangle formed by its sides Then copy the triangle measure the angles and classify the triangle by its angles 10 5 in 7 7 in 7 14 in
A cruise ship maintains a speed of 19 knots nautical miles per hour sailing from San Juan to Barbados a distance of 600 nautical miles To avoid a tropical storm the captain heads out of San Juan at a direction of 11 off a direct heading to Barbados The captain maintains the 19 knot speed for 11 hours after which time the path to Barbados becomes clear of storms a Through what angle should the captain turn to head directly to Barbados b Once the turn is made how long will it be before the ship reaches Barbados if the same 19 knot speed is maintained Barbados 600 11 San Juan | 677.169 | 1 |
Reflected Triangles
Task
The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected.
IM Commentary
This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.
Solution
The line in question is the perpendicular bisector of any pair of corresponding points. We choose the two corresponding points labelled $A$ and $B$ for the illustration below, but either of the other two pairs of points works just as well.
To summarize the construction: We construct the circle of radius $\overline{AB}$ cenetered at $A$ and the circle of radius $\overline{AB}$ cenetered at $B.$ These intersect at two points, which we label $P$ and $Q$. Then $P$ and $Q$ are both equidistant from $A$ and $B$, and so lie on the perpendicular bisector of $\overline{AB}$. We conclude that line $PQ$ is precisely the perpendicular bisector, which is the line over which the first triangle was reflected to arrive at the second.
We remark that the above construction is valid even from the strict Euclidean perspective on straight-edge and compass constructions. In a more modern setting, in which one typically allows the use of a compass with memory, one has a little more flexibility. In particular, we could replace the two circles in the above construction with any pair of circles of equal radius at least $\frac{1}{2}\overline{AB}$, centered at $A$ and $B$.
Reflected Triangles
The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected. | 677.169 | 1 |
Compass and straightedge constructions book
An article about compass and straightedge constructions hand selected for the. Open the compass and mark two points of intersection between arcs from the given line. Study carefully the following constructions, and pay attention how the compass is used. When doing compass and ruler constructions, we are using two tools. However, using curves created by other means the greeks resolved all three problems. When constructing an inscribed polygon with a compass and straightedge, how should you start the construction. If n pq with p 2 or p and q coprime, an ngon can be constructed from a pgon and a qgon. The first six books of the elements of euclid, in which coloured diagrams and symbols are used instead of letters for the greater ease of learners. The chapters are ordered in ascending levels of dif. The compass equivalence theorem is an important statement in compass and straightedge constructions. How to bisect an angle with compass and straightedge or ruler.
Three problems in particular attracted the most attention. Compass and straightedge constructions springerlink. Were the axioms designed to formalize the process of using a compass and straightedge. Square and square root construction by compass and. Geometry construction with compass and straightedge or ruler or ruler great descriptions about the history of constructions. This is a beginning lesson on compass and ruler constructions, meant for 6th or 7th grade. Using only a pencil, compass, and straightedge, students begin by drawing lines, bisecting angles, and reproducing segments. When they finish, students will have been introduced to 4 geometric terms and will be ready to tackle formal proofs.
Compass and straightedge project gutenberg selfpublishing. Their use reflects the basic axioms of this system. A total rite of passage for geometry constructions using the compass. Since the earliest times, mankind has employed the primary geometr.
With the straightedge, we are permitted to draw a straight line of inde. A large part of the first book deals with compass and straightedge constructions. You can construct a scalene triangle when the length of the three sides are given. However, the stipulation that these be the only tools used in a construction is artificial and only has meaning if one views the process of construction as an application of logic. Later they do sophisticated constructions involving over a dozen steps. A length is constructible if it can be obtained from a nite number of applications of a compass and straightedge. Line segment geometrical construction construction problem compass point original angle. In practical constructions, however, the parallel lines. How do you know that each of these constructions is valid when made with a compass and straightedge. Compass and straightedge provides original content on geometric drawing using classical construction, also known as compass and straightedge drawing.
In fact, the very first result, proposition 1 of book i, is a demonstration of the construction of an equilateral triangle using a compass and straight edge. Square and square root construction by compass and straightedge. The compass can be opened arbitrarily wide, but unlike some real compasses it has no markings on it. Compassandstraightedge a site created by david eppstein with dozens of links to specific constructions. Lets rst be very careful about determining exactly what actions we can accomplish, and what it means to \construct something. Practical geometric constructions since the earliest times mankind has employed the simple geometric forms of straight line and circle. Construction in geometry means to draw shapes, angles or lines accurately. A straightedge is simply a guide for the pencil when drawing straight lines. Basic compass and ruler constructions 1 homeschool math. Constructions with compass and straightedge a thing constructed can only be loved after it is constructed. Nowadays, we use rigid compasses, which can hold a certain radius, but is has been shown that construction with rigid compass and straightedge is equivalent to construction with collapsible compass and straightedge.
An investigation of historical geometric constructions introduction. These designs are made using a series of constructions with a compass and straightedge. Christopher m freeman shows students how to draw accurate constructions of squares, octagons, and other shapes. Jan 22, 2014 a video presentation related to chapter 32 of john b. Fraleighs a first course in abstract algebra, 7th edition.
Learn how to construct a pentagon using just a compass and a straightedge. Greeks used collapsible compasses, which would automatically collapse. There are many other ways to do constructions, but the compass and straightedge were chosen as one set of tools that make a construction challenging, by limiting what you are allowed to do, just as sports restrict what you can do e. Compass and straightedge or ruler and compass construction is the construction of lengths, angles, and other geometric figures using only an. Im interested in learning the origin of compass straightedge constructions. Triangle, given all 3 sides sss triangle, given one side and adjacent angles asa.
The complexity of algorithms of constructions by compass and straightedge. Constructions with straightedge and compass, grades 46 at. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Oct 10, 2018 a total rite of passage for geometry constructions using the compass. Construct the semicircle with center in and radius. Freeman, i thought it might be an interesting supplement to the math my son has been working on in homeschool. It can only be used to draw a line segment between two points or to extend an existing. Compass and straightedge or ruler and compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass the idealized ruler, known as a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. A video presentation related to chapter 32 of john b. Straightedge and compass constructions springerlink. Draw a line segment that is as long as these two line segments together.
Straightedge and compass construction challenges the math. Find all the books, read about the author, and more. Cinderella multiplatform java system for compass andstraightedge construction, dynamic geometry demonstrations and automatic theorem proving. Constructions with straightedge and compass, grades 46 paperback january 1, 2010. In the previous chapters we developed the algebraic machinery for proving that the three famous geometric constructions are impossible. Were asked to construct an angle bisector for the given angle. Circles can only be drawn using two existing points which give the centre and a point on the circle. Euclidea is all about building geometric constructions using straightedge and compass. That means you can find all the points that are at a specified distance from some point the circles center point. This is a 57 question open book test and i just need these answered. With euclidea you dont need to think about cleanness or accuracy of your drawing euclidea will do it for you.
Freeman and a great selection of similar new, used and collectible books available now at great prices. Euclidea geometric constructions game with straightedge and. But remember you can only construct angles divisible by 15 with a compass and a straightedge e. The collapsing compass euclid showed that every construction that can be done using a compass with. Essentially anything that can be constructed with the traditional euclidean tools of compass and straightedge. Compass and straightedge i include here both pages about the classical greek compass and straightedge style of construction, other topics involving greek mathematicians such as pythagoras and euclid, as well as the three famous problems they found impossible to construct with these tools. Practical geometric constructions wooden books sutton, andrew on. The difference is that manual dexterity with the instruments is not necessary, and arguably of course was never really as important as knowing which constructions. Given the unit length, and the segment of length construct solution. The first chapter of the book goes through the first 4 books of the elements and does quite a lot of compass and straightedge constructions. So, what exactly is a compassandstraightedge construction. Compass and straightedge constructions february 25, 2018 february 25, 2018 simply put, a real number is constructible if, starting form a line segment of unit length, a line segment of length can be constructed with a compass and straightedge in a fintie number of steps. The compass can be opened arbitrarily wide, but unlike some real.
On the more theoretical side, you have hartshornes euclid and beyond. Construction with straightedge and compass the drawings of the ancient greek geometers were made using two instruments. Construction with only a compass who says that both a straightedge and a compass are needed. The artistic project includes a series of three designs with increasing difficulty level. Other constructions that can be done using only a straightedge and compass. The compass and straightedge of compass and straightedge constructions are idealizations of rulers and compasses in the real world. Compassandstraightedge construction project gutenberg. To bisect an angle means that we divide the angle into two equal congruent parts without actually measuring the angle. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. These constructions use only compass, straightedge i. If possible, turn the ruler over so you cannot see them. Hundreds of years ago, lorenzo mascheroni and georg mohr showed that it is possible to limit oneself to only a compass.
This material is part of the introduction to modern algebra 2 math 42375237. Angle trisection, from the geometry forum archives. It contains a variety of exercises and explains the following constructions. Instead of concentrating on paper and pencil, compass and straightedge constructions, current books tend to emphasize the use of dynamic computer software, such as geometers sketchpad. Key vocabulary compass andstraightedge construction anchor of a compass concentric circles materials math journal 1, pp. If p 2, draw a qgon and bisect one of its central angles. So, faced with the problem of finding the midpoint of a line, it was very difficult to do the obvious measure it and divide by two. Geometry construction art geometry constructions, math.
In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. Resource for learning straightedge and compass constructions. The tool advocated by plato in these constructions is a divider or collapsing compass, that is, a compass that collapses whenever it is lifted from a page, so that it may not be directly used to transfer distances. Using a compass and straightedge, construct the altitude from vertex j to ml. The directions for each construction are brief but sufficient for somebody familiar with these types of constructions. Straightedge and compass construction, also known as rulerandcompass construction or. Constructions with compass alone university of washington. Lets learn how to create and copy segments and angles.
A straightedge is a ruler without measurement units such as cm or in on it. This is a beginning lesson on compass andruler constructions, meant for 6th or 7th grade. Compass and straightedge constructions rationalwiki. Compassandstraightedge constructions we learn exactly what compassandstraightedge constructions are, and what they can do. In this chapter we take a closer look at some elementary and not so elementary constructions, culminating is a discussion of vietes construction of a circle tangent to three given circles. Pdf the complexity of algorithms of constructions by. Equivalently, r is constructible if and only if there is a closedform expression for r using only the integers 0 and 1 and the operations for addition, subtraction. However, by the compass equivalence theorem in proposition 2 of book 1 of euclids elements, no power is lost by using a collapsing compass. How to bisect an angle with compass and straightedge or. Some very difficult exercises explore exactly which such constructions are possible and which not.
A compass was strictly used to make circles of a given radius. When constructing perpendicular lines with a compass and straightedge, how should you start the construction. Thus, to understand euclidean geometry, one needs some idea of the scope of straightedge and compass constructions. In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length r can be constructed with compass and straightedge in a finite number of steps. Its designed to put the compass and the straight edge right in the childs hands so they can explore mathematical geometry with precision. An excellent reference of over 170 compass and straightedge constructions.
In particular, i am interested in the historical interplay between euclids axioms for plane geometry, and compass straightedge constructions. The proof consists of noting that straightedge and compass construction is based on a intersecting two lines b intersecting a line and a circle. Constructions with straightedge and compass, grades 46 9781593634186. I know most of the constructions in book 1 of the elements, but would appreciate it if anyone could point me in the direction of a book or other. This euclidean construction works by creating two congruent triangles. The straightedge and compass of straightedge and compass constructions are idealizations of rulers and compasses in the real world. Philosophy of constructions constructions using compass and straightedge have a long history in euclidean geometry.
When i saw this hands on geometry book, by christopher m. The sloppiness and inaccuracy of manmade constructions could be avoided by the use of technology. How do you know that each of these constructions is valid. A few constructions remained that the greeks were never able to give, and these remained mysteries until modern times.
Jun 29, 20 straightedge and compass construction challenges posted on june 29, 20 by brent i havent written here in quite a whileive switched into work on research for my dissertation really hard so that i can actually graduate mode, and with a 21month old in the mix that leaves very little time for blogging. Oct 21, 2019 there are many other ways to do constructions, but the compass and straightedge were chosen as one set of tools that make a construction challenging, by limiting what you are allowed to do, just as sports restrict what you can do e. Over the course of the next few weeks, we will be learning and exploring the ancient art of constructions. What are some of the good books for construction in geometry with. In early geometry, the tools of the trade were a compass and straightedge. Teachers and students from india can accessdownload robocompass geometrical constructions and related materials from the following websites geometric constructions made easy using robocompass from kendra vidyalaya sangathan ncert. Teachers and students from india can accessdownload robocompass geometrical constructions and related materials from the following websites geometric constructions made easy using robocompass from kendra vidyalaya sangathan. Using a compass and straightedge, construct a diameter of the circle. Geometric constructions gives a brief history and provides references and links to dozens of geometric constructions, each with detailed instructions. These are best if students have seen at least one or two basic constructions before, such as bisecting a line segment. And they want us to make a line that goes right in between that angle, that divides that angle into two angles that have equal measure, that have half the measure of the first angle. Compass and straightedge constructions are known for all known constructible polygons. I think on this evidence the article should be titled straightedge and compass construction rather than compass and straightedge construction. This seems very surprising at first consideration, but we are already part way to proving this theorem.
You can construct a right triangle given the length of its hypotenuse and the length of a leg. Some very difficult exercises explore exactly which such constructions. Surprising constructions with straightedge and compass. For compass and straightedge construction personally, i still learn best by manually doing what im learning, see, e. An investigation of historical geometric constructions. This led to the constructions using compass and straightedge or ruler. We must divulge that the given problems do not have solutions if we limit ourselves to constructions using only compass and straightedge. Main index geometry plane geometry geometric constructions subject index comment on the page. In this chapter we introduce some geometry and start to show the connection between algebra and the geometry of constructions. | 677.169 | 1 |
area of triangle trigonometry worksheet pdf
Area Of Triangle Pdf Worksheet – Triangles are among the most fundamental shapes of geometry. Understanding triangles is crucial to getting more advanced concepts in geometry. In this blog post it will explain the different types of triangles Triangle angles, how to calculate the perimeter and area of a triangle, and provide specific examples on each. Types of Triangles There are three types of triangles: equal isosceles, and scalene. Equilateral triangles are made up of three equal sides as … Read more
Area Of Triangles Worksheet Pdf – Triangles are among the most fundamental shapes in geometry. Understanding triangles is critical to developing more advanced geometric ideas. In this blog post this post, we'll go over the various types of triangles with triangle angles. We will also discuss how to determine the length and width of a triangle, and show instances of each. Types of Triangles There are three kinds for triangles: Equal isosceles, and scalene. Equilateral triangles include … Read more | 677.169 | 1 |
Device
Software
TI-Nspire Version
Geometry: Creating a Midpoint Quad
Objectives
Students will explore the parallelogram formed by the midpoints of any quadrilateral.
Students will further explore special outer and inner quadrilaterals formed by the connected midpoints. Area relationships will also be investigated.
Vocabulary
midpoint quadrilateral
parallelogram
convex quadrilateral
concave quadrilateral
midsegment
About the Lesson
This "create your own" activity is designed to be introduced by the teacher and completed by the students as they discover and justify their conjectures. The time varies for this activity depending on whether the TI-Nspire document (.tns file) is provided or created by the students | 677.169 | 1 |
Humanities
... and beyond
Cardioid Curves
Key Questions
Answer:
A cardioid
Explanation:
This is a cardioid.
We can observe that the positive/negative just flip the orientation of the figure since cosine goes from -1 to 1 either way. Let's just discuss the negative case for now, but remember that it doesn't really matter (it just rotates it by an angle of 180 degrees)
We observe that there is an angle where the value goes all the way to 0 (#theta = 0#). This is a cusp at the same angle every time. For every other value, the angle is positive, all the way up to the opposite side (#theta = pi#) where it is #2a# away from the center. This means that the function kinda looks like a heart (hence the name cardioid). Here's a plot of one:
Answer:
Please see below
Explanation:
Cardioid curve is some thing like a heart shaped figure (that is how the word 'cardio' has come). It is the locus of a point on the circumference of a circle that moves on another circle without slipping.
Mathematically it is given by the polar equation #r=a(1-costheta)#, at times also written as #r=2a(1-costheta)#, | 677.169 | 1 |
Option 1) Statements 1 and 4 are correct
Option 2) Statements 2 and 4 are correct
Option 3) Statements 3 and 4 are correct
Option 4) Only statement 4 is correct
Question
Option 1) Statements 1 and 4 are correct
Option 2) Statements 2 and 4 are correct
Option 3) Statements 3 and 4 are correct
Option 4) Only statement 4 is correct
✨ Free AI Tools for You
Powered by Chatterbot AI
Answer
:
The correct option is $\mathbf{3}$.
Statement 3: The angle of incidence is equal to the angle of reflection. This is defined as $ i = r $.
Statement 4: The incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.
These two statements correctly represent the laws of reflection. Therefore, the correct option is $\mathbf{3}$, indicating that Statements 3 and 4 are correct. | 677.169 | 1 |
The Elements of Descriptive Geometry ...
No interior do livro
Resultados 1-5 de 10
Página 51 ... Conical Surfaces ; 3 , Cylindrical Surfaces . Surfaces of Revolution are those which may be supposed to be described ... surface described is that of the common , or right , cone . Cylindrical surfaces are described by the motion ...
Página 77 ... cone is a right one , and the section is an hyperbola ; the true form and magnitude are determined by turning the sectional plane down on the vertical plane ; the one conical surface is alone developed in the figure . The principal object ...
Página 78 ... conic section is the intersection with the plane pqp ' of a plane tangential to the conical surface , touching it in the generatrix containing the point . Draw the tangents mr , ns to the base abc ... ; these will be the horizontal ...
Página 83 ... surfaces . Hence the intersection of these cones is composed of two parts * , one situated on that portion of the larger cone * The conical surface , it must be remembered , consists of two parts , one on each side of the vertex ; only ...
Página 94 ... conical surface , the axis of which would be parallel to the generatrixes of the cylinder . Hence it follows that the tangents parallel to a given plane must be parallel to the right lines , which would be formed by the section of this cone | 677.169 | 1 |
Examples of a Line in Real Life
A line can be simply defined as the shortest distance between two points plotted randomly on a 2D surface. In geometry, a line can be defined as a one-dimensional figure that extends in both directions to infinity and does not have any width or depth. This implies that a line does not have any endpoint, hence the length of the line cannot be measured easily. Generally, a line is confused with a line segment. The difference between a line and a line segment is that a line does not have endpoints, while a line segment has two endpoints. On the other hand, the similarity between the line and the line segment is that both the types of geometric figures do not have width and depth parameters. This means that the length of a line is indeterminable, while the length of a line segment is determinable and confined. A line can be easily represented with the help of a straight line having arrowheads on both sides. The arrowheads on both sides of the figure of a line indicate the ability of a line to extend to infinity on both sides. The name of a line is usually either represented by a single lower case alphabet or two upper case alphabets. Here, the upper case alphabets tend to denote the points present on the surface of the line. From a different point of view, a line can be observed to be a connection between multiple collinear points that are plotted on a one-dimensional plane.
Examples of Line
Some of the most common examples of lines in real life are listed below:
1. Railway Tracks
Railway tracks tend to form a prominent example of lines in real life. This is because the railway tracks tend to stretch upto infinity on both sides and the length of the tracks is almost undeterminable.
2. Electricity Wires
The wires that are used by the energy service providers to transmit the electrical energy from the substation to the consumer destination tend to form yet another example of lines in real life. Such cables and wires tend to extend to an immeasurable value.
3. Markings on Roads
The markings made on the roads usually with the help of white ir yellow coloured painting colour tend to denote the different lanes of the road, the distinction between the foothpath and the driving path, and other road signals. Such markings form another example of line in real life as they can be observed as a set of infinite interconnected collinear points plotted on the surface of the road.
4. Zebra Crossing Stripes Slope of Mountains
The slope of mountains is yet another examples of line in real life. This is because the length of the slope of a mountain cannot be measured directly and can extend to infinity on both sides.
5. Ruler Horizon
Horizon is the imaginary line that tends to connect the surface of the earth and the sky. The horizon line stretches limitlessly in the surroundings and cannot be measured directly. The horizon is also known as the line that seperates the celestial surface from the ground or the earth's surface.
6. Window Panes Length of a Water Body
The length of the boundary of a river or the path taken by a river to flow is another example of lines in real life. This because generally a river has two sides stretching upto infinity and the length of a river cannot be measured directly and easily with the help of generic measuring equipment and devices.
7. Pencil or Pen Fences
If the designed fences are dismantled and each part of the Fences is layed ahead of the another, a representation of a line can be created easily. The length of such a line is difficult to calculate. Also, the line formed by arranging the individual fences parts can extend towards infinity in both sides.
8. Curtain Rods Axis
Axis can be defined as an imaginary fixed reference line that can be used to locate the position of an object in space. Typically, there are three axis lines called x, y, and z that tend to extend from infinity to negative of infinity. This implies that the axis lines can stretch upto infinity on both sides and the length of the axis lines is immeasurable.
9. Incense Stick Tunnel
A tunnel is a perfect example of line that can extend to infinity on both sides and is difficult to be measured.
10. Traffic Light Pole Roads
The ability of roads to stretch to infinity on both sides qualifies them to be listed under the category of the real life examples of line geometric figure.
11. Racing Track Lines Roller Coaster Tracks
The tracks of a roller coaster are a prominent example of lines in real life. This is because the length of any randomly picked part of a roller coaster track is capable of extending in both sides to infinity. Also, the distance of the roller coaster tracks is difficult to estimate directly.
12. Railings Irrigation Channel in Fields
To properly water the crops in an agricultural field, a narrow lane to carry water between the fields is typically created. This narrow lane of land eases the process to carry and circulate water throughout the field. The lanes tend to extend from negative infinity to positive infinity and form a classic example of application of the concept of lines in real life. Also, the rows along which the seeds are sowed or the crops are planted tend to form an example of lines in everyday life.
13. Straws Maps
A map is a graphical and pictorial representation of a physical locality. A map typically comprises of multiple pathways and location tags. The length of the pathways drawn on maps can extend to infinity on both sides and is typically immeasurable directly. This is the reason why the pathways drawn on maps is another example of lines in real life.
14. Bridge Strings Conveyer Belt
The conveyer belt used in most factories and companies is spread throughout the premises. The main objective of such conveyer belts is to transport objects from one place to the other. If you observe any random portion of a conveyer belt installed in an organisation, it extends to both sides. This implies that the structure of a conveyer belt is quite comparable to a line.
15. Ladder Rungs and Frame
16. Hands of an Analogue Clock
17. Thermometer
18. Popsicle Stick
19. Skiing Items
20. Sword
21. Cricket Stumps
22. Base Ball Bat Playing Ground Boundary
The boundary of a play ground such as a golf course, baseball field, soccer, etc., if assumed to be stretched open linearly, tends to form yet another example of lines in real life. This is because in such a case, the boundary of the ground stretches to infinity on both sides and is nearly immeasurable.
23. Cricket Bat Blood Veins
Blood veins are arranged in the body of a living being randomly. If the veins of a living being are observed closely, they tend to stretch to infinity and can be flexibly listed under the category of line geometric one dimension figure.
24. Thread
A thread stretched and arranged straight in an open space is a classic example of lines in real life. This is because the length of a infinitely stretched thread is a next to impossible task. | 677.169 | 1 |
Steps of Construction:
1. Construct a line segment BC = 4.5 cm.
2. Taking B as centre and 6 cm radius construct an arc.
3. Taking C as centre and 5.5 cm radius construct another arc which intersects the first arc at point A.
4. Now join AB and AC
Therefore, ∆ ABC is the required triangle.
5. Construct a perpendicular bisector of AB and AC which intersect each other at the point O.
6. Now join OB, OC and OA.
7. Taking O as centre and radius OA construct a circle which passes through the points A, B and C.
This is the required circumcircle of ∆ ABC. | 677.169 | 1 |
Lines and Angles Worksheet
The Lines and Angles Worksheet are a guide to help you organize and do math in the classroom. This worksheet is designed to help students learn and apply basic mathematical concepts, to help them think logically about things, to work on independent learning, and to develop a sense of responsibility. Using this worksheet will also help students develop and improve their problem solving skills. Below are some tips on how to use this worksheet.
First, introduce the use of the Lines and Angles Worksheet to the class. Have each student bring in a small-sized crayon or pencil so that they can color their own areas on the page. Have each student color a different area on the page to make it easy for them to write in the Numbers, and Percentages box as well as the box when you are writing the main information in the Math Worksheet.
Lines and Angles Worksheet Along with Measuring Angles Worksheet
Second, let students do some exercises as part of the introduction. Students will need to select an area of the page and then fill in the word numbers and make sure to fill in the number of times the word occurs on the page as well. The students will need to perform these exercises so that they can figure out the answers to the next sections.
Third, use the Line and Angle Worksheet to explain how different angles can be expressed. This worksheet will help students practice new words and phrases that they learn as they read the book. In this way, they can develop their reading skills as well as their math skills by reading the book and using the book as a reference when they do the assignments.
Fourth, use the Lines and Angles Worksheet to explain how the line is measured. A straight line can be measured using a compass as well as with a ruler. Having students measure a straight line from one place to another will help them develop the ability to measure angles and convert them to other measurements. They will also learn the properties of a circle and how to use a pen and paper to solve math problems.
Fifth, use the Lines and Angles Worksheet to show students how to calculate the slope of a line. Slopes can be calculated using formulas, or students can do the calculation themselves. Either way, having students do the calculations is a fun activity that develops the understanding of a concept in a fun way.
Lines and Angles Worksheet Also 128 Best Mathematics Images On Pinterest
Sixth, use the Lines and Angles Worksheet to show students how to use the x and y coordinates to figure out slopes and directions. Having students solve a problem and then ask questions while doing so is a great way to reinforce this understanding. Students can use the blue and red to figure out slopes while using the red and blue to figure out directions.
These are just a few tips on how to use the Lines and Angles Worksheet. A student can use it in a number of ways as long as they remember that it is an effective worksheet to help students learn the basics of math. Remember that this worksheet is used for learning, not for being really good at it!
Lines and Angles Worksheet Along with 38 Best Geometry Lines and Angles Images On Pinterest
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Two plane mirror are inclined to each other such that a ray of light incident on the first mirror (M1) and parallel to the second mirror (M2) is finally reflected form th second mirror (M2) parallel to the first mirror (M1) the angle between the two mirrors will be :
A
600
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B
750
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C
900
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D
450
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Solution
The correct option is A600 Assuming angles between two mirrors be θ . we know that angle made by light ray for incidence and reflection with mirror 95 Same. Now by geometry Sum of angles in a triangle =180∘30=180∘θ=60∘ | 677.169 | 1 |
Trigonometry is a branch of mathematics that deals with the relationship between the angles, sides of a right-angled triangle. It was invented by the Greek mathematician named Hipparchus. From this page, you can learn about the basics of trigonometry such as trigonometric ratios(sine, cosine, tangent, secant, cotangent, cosecant), functions, tables, formulas, and more example questions.
Trigonometry Definition
Trigonometry is one of the branches of mathematics. The word trigonometry is derived from the Latin words Trigon means triangle and Metron means to measure. It is the study of the relationship between the sides and angles of a right-angled triangle. It is useful to measure the missing side or angle of a right-angle triangle. The quick links to get important topics regarding trigonometry are here.
The three basic trigonometric functions are sine, cosine, tangent and the remaining three functions are cotangent, secant, and cosecant. The basics of trigonometry are covered in the following sections.
In a right-angled triangle, we have the following three sides.
Perpendicular: It is the side opposite to the angle θ.
Base: This is the adjacent side to the angle θ.
Hypotenuse: This is the side opposite to the right angle.
Trigonometric Ratios | Trigonometry Functions
The trigonometric ratios of a right-angled triangle are called trigonometric functions. The six trigonometric ratios are sin, cos, tan, cot, sec, cosec. These ratios are evaluated in the case of a right-angled triangle. The important functions are calculated using the below formulas.
Trigonometric Table
Check the table for common angles which are used to solve the trigonometric problems involving these ratios.
Angles
0°
30°
45°
60°
90°
sin θ
0
\(\frac { 1 }{ 2 } \)
\(\frac { 1 }{ √2 } \)
\(\frac { √3 }{ 2 } \)
1
cos θ
1
\(\frac { √3 }{ 2 } \)
\(\frac { 1 }{ √2 } \)
\(\frac { 1 }{ 2 } \)
0
tan θ
0
\(\frac { 1 }{ √3 } \)
1
√3
∞
cot θ
∞
√3
√2
\(\frac { 1 }{ √3 } \)
1
sec θ
1
\(\frac { 2 }{ √3 } \)
√2
2
∞
cosec θ
∞
2
1
\(\frac { 2 }{ √3 } \)
0
Trigonometry Angles
Trigonometric angles are the angles in right-angled triangles which are different trigonometric functions. The most used angles in trigonometry are 0°, 30°, 45°, 60°, 90°. The trigonometric values for these angles can be observed directly in the table. The important angles in trigonometry are 180°, 270°, 360°. The trigonometry angles can be defined as:
θ = sin-1 (\(\frac { Perpendicular }{ Hypotenuse } \))
θ = cos-1 (\(\frac { Base }{ Hypotenuse } \))
θ = tan-1 (\(\frac { Perpendicular }{ Base } \))
Unit Circle and Trigonometric Values
Unit circle can be used to calculate the values of basic trigonometric functions (sine, cosine, tangent). The below-given circle shows how trigonometric ratios like sin, cos can be represented in a unit circle.
For suppose the length of the perpendicular is y and base is x. The length of the hypotenuse is the radius of the unit circle. Therefore, the trigonometry ratios are as follows:
sin θ = y/1 = y
cos θ = x/1 = x
tan θ = y/x
Trigonometric Formulas
The trigonometric identities or formulas are the equations which are true for right-angled triangles. Few special trigonometric identities are here:
Pythagorean Identities
sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
cot²θ + 1 = cosec²θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ
Sum and Difference Identities
For angles x and y, we have the following relationships:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
cos(x + y) = cos(x)cos(y) – sin(x)sin(y)
tan(x + y) = \(\frac { tan(x) + tan(y) }{ 1 – tan(x)tan(y) } \)
sin(x – y) = sin(x)cos(y) – cos(x)sin(y)
cos(x – y) = cos(x)cos(y) + sin(x)sin(y)
tan(x – y) = \(\frac { tan(x) – tan(y) }{ 1 + tan(x)tan(y) } \)
If A, B, C are angles and a, b , c are the sides of the triangle, then
Basics of Trigonometry
There are 3 basic functions in trigonometry. They are sine, cosine and tangent. Based on these 3 functions other three functions derived are cotangent, secant and cosecant. The formulas of these functions are listed here.
Problem 2:
Two friends started climbing a pyramid-shaped hill. The first person climbs 315 m and finds that the angle of depression as 72.3 degrees from his starting point. How high is he from the ground? | 677.169 | 1 |
Lesson
Lesson 5
Lesson Purpose
The purpose of this lesson is to introduce angles and to motivate a need for vocabulary to describe what they are and their size.
Lesson Narrative
In this lesson, students are introduced to angles. They learn that an angle can be defined in terms of the geometric parts they have been working with in this unit.
In previous grades, students have used "square corners" to describe right angles within two-dimensional shapes. They may have considered an angle as the space within a square corner or the "pointy" corner itself. Here, students learn that an angle is a geometric figure made up of two rays that share the same endpoint, which we refer to as the vertex of the angle.
Throughout the lesson, students use the vocabulary they have developed to describe other geometric figures to identify and describe angles. Monitor for the ways students reason about how to describe the size of angles. Students will compare and measure angles in future lessons.
Engagement
Learning Goals
Teacher Facing
Identify angles in two-dimensional figures.
Recognize angles as geometric figures that are formed wherever two rays | 677.169 | 1 |
Hint: Students don't panic by seeing so much data! This is very simple to solve. But first we will plot the points in order to know the end points of the diagonals. Then we will go to find the distance or length of the diagonal and at the end the actual problem that is the product of the diagonals.
Complete step-by-step answer: Now we are given four points. These are the vertices of the rhombus given. We will plot these points first.
Now we can see the end points of the diagonals. So we will measure the length of the diagonal. We have done this before in the very starting chapters of the number line. Now we will just measure the distance as the units between the vertices on Y axis and the same on X axis. On observing the graph, we came to notice that the distance between vertices A and D is 16 units that is from 8 to -8. This is the length of the diagonal. On the other hand the distance between vertices B and C is 10 units that is from -5 to 5. This is the length of the other diagonal. Now we have found the distances as 10 and 16 units. So the product of the lengths will be, \[{d_1} \times {d_2} = 10 \times 16 = 160\] Thus the correct option is C. So, the correct answer is "Option C".
Note: Note that we can solve the same problem by using the distance formula \[d = \sqrt {{{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{x_2} - {x_1}} \right)}^2}} \]. But since one of the coordinates of the vertices is zero we can easily find the length by measuring the unit distances so we used it here. But if the vertices given have both the coordinates given then definitely go for the distance formula. | 677.169 | 1 |
Unit Circle: Everything you need to know
Published on June 5th, 2024
Unit Circle | Key Concepts and Applications
Simply hit the Get Answer button and explore the Unit Circle role in trigonometry
Parts of a Circle
Understanding the various parts of a circle is fundamental to grasping the concept of the unit circle. Each part plays a critical role in defining the properties and functions of a circle, which in turn, lays the groundwork for more advanced topics in geometry and trigonometry.
Center
The center of a circle is the fixed point from which every point on the circumference is equidistant. In the context of the unit circle, the center is always at the origin (0,0) of the Cartesian coordinate system. This central point is crucial as it defines the position and size of the circle.
Radius
The radius of a circle is the distance from the center to any point on the circumference. For a unit circle, this distance is always one unit. The radius is significant because it is the defining measure of the unit circle and is essential in calculating the circumference and area.
Diameter
The diameter of a circle is twice the length of the radius, spanning from one side of the circle to the other, passing through the center. Therefore, in a unit circle, the diameter is 2 units. The diameter is important for understanding the overall size of the circle and is often used in various geometric calculations.
Chord
A chord is a line segment that connects two points on the circle's circumference without necessarily passing through the center. The length of a chord can vary, and it helps in understanding the properties of the circle and its segments.
Arc
An arc is a portion of the circle's circumference between two points. Arcs are typically measured in degrees or radians and are used extensively in trigonometric functions to represent angles.
Sector
A sector is a region of the circle bounded by two radii and an arc. It resembles a 'slice' of the circle, and its area can be calculated using the formula: Area of a sector=12𝑟2𝜃Area of a sector=21r2θ, where 𝑟r is the radius and 𝜃θ is the central angle in radians.
Segment
A segment is a region of a circle bounded by a chord and the arc connecting the chord's endpoints. Segments help in dividing the circle into different regions for various applications in geometry and trigonometry.
Tangent
A tangent to a circle is a straight line that touches the circle at exactly one point. This point is known as the point of tangency. Tangents are perpendicular to the radius at the point of tangency and are used in many geometric constructions and proofs.
Secant
A secant is a line that intersects the circle at two points. It effectively 'cuts' through the circle and helps in understanding the relationship between the circle and external lines.
Circumference
The circumference of a circle is the total distance around the circle. For a unit circle, the circumference is 2𝜋2π units. The formula for the circumference of any circle is 2𝜋𝑟2πr, where 𝑟r is the radius. The circumference is crucial for understanding the perimeter and the overall size of the circle.
Area
The area of a circle is the space enclosed within its circumference. For a unit circle, the area is 𝜋π square units. The formula for the area of any circle is 𝜋𝑟2πr2, where 𝑟r is the radius. Understanding the area is important for various applications in geometry, physics, and engineering.
Quadrant
A quadrant is one of the four sections of the Cartesian plane, divided by the x-axis and y-axis. Each quadrant in a unit circle helps in identifying the signs of the trigonometric functions. For example, in the first quadrant, both sine and cosine are positive, while in the second quadrant, sine is positive and cosine is negative.
These fundamental parts of a circle are essential for understanding more complex concepts in trigonometry and geometry. By mastering these basics, you lay the groundwork for exploring the unit circle and its applications in various fields.
What is a Unit Circle?
Definition - Cartesian Plane
A unit circle is a circle with a radius of exactly one unit, centered at the origin (0,0) of the Cartesian coordinate plane. The Cartesian plane, consisting of an x-axis and a y-axis, provides a framework for graphing the unit circle and analyzing its properties. The equation of the unit circle in the Cartesian coordinate system is 𝑥2+𝑦2=1x2+y2=1, representing all the points (x, y) that lie exactly one unit away from the center.
Significance in Trigonometry
The unit circle is a fundamental tool in trigonometry, significantly simplifying the understanding and computation of trigonometric functions. By defining sine, cosine, and tangent functions based on the coordinates of points on the unit circle, one can easily visualize and understand their periodic nature and relationships. For any angle 𝜃θ measured from the positive x-axis, the coordinates (x, y) of the corresponding point on the unit circle are (cos𝜃,sin𝜃)(cosθ,sinθ). This relationship is crucial for solving trigonometric equations and understanding concepts such as amplitude, frequency, and phase shift in periodic functions.
Using the unit circle, one can derive key trigonometric identities, such as:
sin2𝜃+cos2𝜃=1sin2θ+cos2θ=1
tan𝜃=sin𝜃cos𝜃tanθ=cosθsinθ
These identities are foundational for advanced studies in mathematics, physics, and engineering.
Historical Background
The concept of the unit circle dates back to ancient Greek mathematics. The Greek mathematician Hipparchus, known as the "father of trigonometry," made significant contributions to the development of trigonometric functions by using a circle with a fixed radius to define these functions. Later, in the 2nd century, Claudius Ptolemy expanded on these ideas in his work "Almagest," which included a comprehensive table of trigonometric values based on a circle of radius 60 units.
During the Renaissance, mathematicians such as Euler and others refined and expanded the use of the unit circle in trigonometry. Euler's formula, 𝑒𝑖𝜃=cos𝜃+𝑖sin𝜃eiθ=cosθ+isinθ, elegantly links complex numbers and trigonometric functions, further demonstrating the profound significance of the unit circle in mathematics.
In modern times, the unit circle continues to be an essential tool in education and research, providing a clear and intuitive way to explore and understand the properties of trigonometric functions. It is widely used in various fields, including physics, engineering, computer science, and even economics, to model periodic phenomena and solve complex problems.
The unit circle, with its simple yet powerful properties, serves as a cornerstone of trigonometry. Its definition on the Cartesian plane, its significant role in defining trigonometric functions, and its rich historical background all contribute to its importance in mathematics and beyond. By mastering the unit circle, one gains a deeper understanding of trigonometric relationships and their applications in a wide range of disciplines.
Basic Concepts of Unit Circle
Explanation of the Unit Circle's Structure and Components
The unit circle, a circle with a radius of one unit, is centered at the origin (0,0) of the Cartesian plane. Its equation, 𝑥2+𝑦2=1x2+y2=1, describes all points (x, y) that are exactly one unit away from the center. The simplicity of the unit circle's structure makes it a powerful tool for exploring trigonometric functions.
The unit circle intersects the x-axis at (1,0) and (-1,0), and the y-axis at (0,1) and (0,-1). These points are crucial as they correspond to the fundamental angles of 0°, 90°, 180°, and 270° (or 0, 𝜋/2π/2, 𝜋π, and 3𝜋/23π/2 radians, respectively). Each point on the unit circle represents an angle 𝜃θ, measured from the positive x-axis, with coordinates given by (cos𝜃cosθ, sin𝜃sinθ).
Quadrants: The Cartesian plane is divided into four quadrants by the x and y axes, each affecting the sign of the trigonometric functions.
Angles: Represented in both degrees and radians, angles define the positions of points on the circle.
Coordinates: Every point (x, y) on the unit circle corresponds to (cos𝜃cosθ, sin𝜃sinθ), providing a direct relationship between angles and trigonometric values.
Understanding Radians and Degrees in the Context of the Unit Circle
Radians and degrees are two units for measuring angles. Understanding both is essential for fully grasping the unit circle and its applications.
Degrees: Degrees are a more intuitive unit for many people, dividing a circle into 360 equal parts. Key angles in degrees are:
0° at (1,0)
90° at (0,1)
180° at (-1,0)
270° at (0,-1)
360° at (1,0)
Radians: Radians offer a more natural measure in mathematics, defined by the length of the arc that an angle subtends on the unit circle. One complete revolution (360°) equals 2𝜋2π radians. Thus, key angles in radians are:
0 or 2𝜋2π at (1,0)
𝜋/2π/2 at (0,1)
𝜋π at (-1,0)
3𝜋/23π/2 at (0,-1)
The conversion between degrees and radians is given by: Radians=Degrees×𝜋180Radians=Degrees×180π Degrees=Radians×180𝜋Degrees=Radians×π180
Understanding these conversions is crucial for solving trigonometric problems and understanding periodic functions.
Significance in Trigonometry: The unit circle simplifies the visualization and calculation of trigonometric functions. For instance, the sine of an angle 𝜃θ is the y-coordinate of the corresponding point on the unit circle, while the cosine of 𝜃θ is the x-coordinate. Tangent, defined as the ratio sin𝜃cos𝜃cosθsinθ, can also be easily understood using the unit circle.
By using radians, many trigonometric identities and formulas become more intuitive and easier to work with. For example, the identity sin(𝜃+2𝜋)=sin(𝜃)sin(θ+2π)=sin(θ) immediately shows the periodic nature of the sine function.
The basic concepts of the unit circle, including its structure and the understanding of radians and degrees, provide a foundational knowledge essential for mastering trigonometry. The unit circle not only simplifies the study of trigonometric functions but also helps in visualizing their properties and relationships. By integrating these basic concepts, students and professionals can develop a deeper understanding of trigonometry and its applications in various fields.
Relationship Between Angles and Points on the Unit Circle
Understanding the relationship between angles and points on the unit circle is fundamental to mastering trigonometry. Each angle corresponds to a specific point on the unit circle, and these points help define the values of trigonometric functions such as sine, cosine, and tangent.
Angles and Coordinates
In the unit circle, every angle 𝜃θ is measured from the positive x-axis in a counterclockwise direction. The point on the unit circle corresponding to an angle 𝜃θ has coordinates (cos𝜃,sin𝜃)(cosθ,sinθ). This means the x-coordinate of any point on the unit circle is the cosine of the angle, while the y-coordinate is the sine of the angle.
For example:
At 0∘0∘ (or 0 radians), the coordinates are (1,0)(1,0).
At 90∘90∘ (or 𝜋/2π/2 radians), the coordinates are (0,1)(0,1).
At 180∘180∘ (or 𝜋π radians), the coordinates are (−1,0)(−1,0).
At 270∘270∘ (or 3𝜋/23π/2 radians), the coordinates are (0,−1)(0,−1).
These key angles and their corresponding points form the basis for understanding more complex angles and their trigonometric values.
Quadrants and Signs of Trigonometric Functions
The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions:
Periodicity and Symmetry
Trigonometric functions exhibit periodicity and symmetry, which are evident on the unit circle. The sine and cosine functions repeat their values every 360∘360∘ (or 2𝜋2π radians). This periodicity means thatAdditionally, the unit circle shows the symmetry of trigonometric functions:
Symmetry about the y-axis: cos(−𝜃)=cos(𝜃)cos(−θ)=cos(θ) and sin(−𝜃)=−sin(𝜃)sin(−θ)=−sin(θ).
Symmetry about the x-axis: cos(𝜋−𝜃)=−cos(𝜃)cos(π−θ)=−cos(θ) and sin(𝜋−𝜃)=sin(𝜃)sin(π−θ)=sin(θ).
Tangent Function
The tangent function, defined as the ratio of sine to cosine (tan𝜃=sin𝜃cos𝜃tanθ=cosθsinθ), also has a clear representation on the unit circle. The tangent value corresponds to the y-coordinate divided by the x-coordinate of the point on the unit circle. For instance, at 45∘45∘ (𝜋/4π/4 radians), tan𝜃=1tanθ=1 because both sine and cosine are equal.
Practical Applications
Understanding the relationship between angles and points on the unit circle is crucial for solving real-world problems involving periodic phenomena, such as sound waves, light waves, and alternating current in electrical engineering. The unit circle is also essential in navigation, computer graphics, and even in fields like economics where cyclical patterns occur.
The relationship between angles and points on the unit circle is foundational for understanding trigonometric functions. By mastering this relationship, one can easily compute the values of sine, cosine, and tangent for any angle, as well as appreciate the periodic and symmetrical nature of these functions. This knowledge is not only vital for academic purposes but also for practical applications in various scientific and engineering disciplines.
Conversion Between Radians and Degrees
Understanding how to convert between radians and degrees is crucial for working with angles in trigonometry, especially when using the unit circle. Both units of measurement are used to express the size of an angle, but they do so in different ways. Degrees are more intuitive for everyday use, while radians provide a more natural measure for mathematical calculations.
Definition and Relationship
Degrees: A degree is a measure of angle equal to 13603601 of a full circle. One complete revolution around a circle is 360 degrees.
Radians: A radian measures the angle subtended by an arc of a circle that is equal in length to the radius of the circle. One full revolution around the circle is 2𝜋2π radians.
Common Angle Conversions
Practical Applications and Significance
Conversion between radians and degrees is not just an academic exercise; it has practical significance in various fields:
Physics: Radians are often used in physics to measure angular displacement, angular velocity, and angular acceleration. For instance, when analyzing rotational motion, it's more convenient to use radians because they simplify many mathematical expressions.
Computer Graphics: In computer graphics, angles are often measured in radians to perform rotations and transformations in a more mathematically straightforward manner.
Navigation and Astronomy: Degrees are commonly used in navigation and astronomy to express positions and directions.
Visual Understanding on the Unit Circle
The unit circle offers an excellent way to visualize the relationship between radians and degrees. By plotting key angles and their corresponding points on the circle, one can see how these units interrelate. For example, at 90 degrees (or 𝜋/2π/2 radians), the point on the unit circle is (0, 1), illustrating that sin(90∘)=sin(𝜋/2)=1sin(90∘)=sin(π/2)=1 and cos(90∘)=cos(𝜋/2)=0cos(90∘)=cos(π/2)=0.
Mastering the conversion between radians and degrees is essential for anyone studying trigonometry, physics, engineering, or computer graphics. By understanding both units and their interconversion, you can seamlessly work with angles in various contexts, enhancing both your academic and practical skills.
Angles on the Unit Circle
Understanding angles on the unit circle is fundamental to mastering trigonometry. Each angle on the unit circle corresponds to a unique point, which helps in defining the trigonometric functions. This section will explore the relationship between angles and points on the unit circle, as well as the conversion between radians and degrees, enhancing your comprehension of these critical concepts.
Relationship Between Angles and Points on the Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin (0,0) of the Cartesian coordinate plane. Each angle 𝜃θ measured from the positive x-axis in a counterclockwise direction corresponds to a specific point (𝑥,𝑦)(x,y) on the unit circle. The coordinates of this point are given by (cos𝜃,sin𝜃)(cosθ,sinθ).
For example:
0° (0 radians): The point on the unit circle is (1,0)(1,0).
90° (𝜋/2π/2 radians): The point is (0,1)(0,1).
180° (𝜋π radians): The point is (−1,0)(−1,0).
270° (3𝜋/23π/2 radians): The point is (0,−1)(0,−1).
These points represent the cosine and sine of the angle 𝜃θ, respectively. The unit circle also demonstrates the periodic nature of trigonometric functions, with cos𝜃cosθ and sin𝜃sinθ repeating every 360∘360∘ (or 2𝜋2π radians).
Quadrants and Signs of Trigonometric Functions
The unit circle is divided into four quadrants, each affecting the signs of the sine and cosine functions:
First Quadrant (0° to 90° or 0 to 𝜋/2π/2 radians):
Both sine and cosine are positive.
Second Quadrant (90° to 180° or 𝜋/2π/2 to 𝜋π radians):
Sine is positive, cosine is negative.
Third Quadrant (180° to 270° or 𝜋π to 3𝜋/23π/2 radians):
Both sine and cosine are negative.
Fourth Quadrant (270° to 360° or 3𝜋/23π/2 to 2𝜋2π radians):
Sine is negative, cosine is positive.
This division helps in quickly determining the signs of trigonometric functions based on the angle's quadrant.
Conversion Between Radians and Degrees
Radians and degrees are two units for measuring angles, each useful in different contexts. Understanding how to convert between these units is essential for working with trigonometric functions on the unit circle.
Degrees: A degree is a measure of angle equal to 13603601 of a full circle. One full revolution around the circle is 360 degrees.
Radians: A radian measures the angle subtended by an arc of a circle that is equal in length to the radius of the circle. One complete revolution around the circle is 2𝜋2π radians.
To convert from degrees to radians: Radians=Degrees×𝜋180Radians=Degrees×180π
To convert from radians to degrees: Degrees=Radians×180𝜋Degrees=Radians×π180
Common Angle Conversions:
30∘=𝜋630∘=6π radians
45∘=𝜋445∘=4π radians
60∘=𝜋360∘=3π radians
90∘=𝜋290∘=2π radians
180∘=𝜋180∘=π radians
270∘=3𝜋2270∘=23π radians
360∘=2𝜋360∘=2π radians
Visual Representation on the Unit Circle
The unit circle provides a visual and intuitive way to understand the relationship between angles and their corresponding points. For instance, at 45∘45∘ (𝜋/4π/4 radians), the coordinates are (2/2,2/2)(2/2,2/2), showing that both sine and cosine have the same positive value.
This visual tool also helps in understanding the periodicity and symmetry of trigonometric functions) illustrate the repeating nature of these functions.
The relationship between angles and points on the unit circle, combined with the ability to convert between radians and degrees, forms the cornerstone of trigonometry. This knowledge allows for a deeper understanding of trigonometric functions and their applications, making it indispensable for students and professionals in various fields. Mastering these concepts provides a robust foundation for further exploration of mathematical and real-world phenomena.
Trigonometric Functions on the Unit Circle
The unit circle is a powerful tool in trigonometry, providing a clear and visual way to understand the definitions and relationships of trigonometric functions such as sine, cosine, and tangent. These functions are fundamental in various fields, from mathematics and physics to engineering and computer graphics.
Definition of Sine, Cosine, and Tangent
Trigonometric functions describe the relationships between the angles and sides of a right triangle. On the unit circle, these functions are defined using the coordinates of points corresponding to angles measured from the positive x-axis.
Sine (sin𝜃sinθ): For any angle 𝜃θ, the sine is the y-coordinate of the point on the unit circle. Thus, sin𝜃=𝑦sinθ=y.
Cosine (cos𝜃cosθ): The cosine of an angle 𝜃θ is the x-coordinate of the point on the unit circle. Therefore, cos𝜃=𝑥cosθ=x.
Tangent (tan𝜃tanθ): The tangent is the ratio of the sine to the cosine of an angle 𝜃θ. It represents the slope of the line connecting the origin to the point on the unit circle. Hence, tan𝜃=sin𝜃cos𝜃=𝑦𝑥tanθ=cosθsinθ=xy.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in trigonometry, especially in the context of the unit circle. It states that for a right triangle with sides 𝑎a and 𝑏b, and hypotenuse 𝑐c:
𝑎2+𝑏2=𝑐2a2+b2=c2
In the unit circle, the hypotenuse is always 1 (the radius of the circle), and the sides 𝑎a and 𝑏b are the x and y coordinates of the point on the circle. Therefore, the equation becomes:
𝑥2+𝑦2=1x2+y2=1
This relationship is critical as it underpins the definition of sine and cosine on the unit circle, ensuring that for any angle 𝜃θ:
cos2𝜃+sin2𝜃=1cos2θ+sin2θ=1
This identity, known as the Pythagorean identity, is essential for simplifying and solving trigonometric equations.
How These Functions Relate to Points on the Unit Circle
The unit circle simplifies the visualization and calculation of trigonometric functions. By understanding how these functions relate to points on the unit circle, one can easily compute values and solve trigonometric problems.
Key Relationships:
At 0∘0∘ (0 radians), the point is (1, 0), so cos0∘=1cos0∘=1 and sin0∘=0sin0∘=0.
At 90∘90∘ (𝜋/2π/2 radians), the point is (0, 1), so cos90∘=0cos90∘=0 and sin90∘=1sin90∘=1.
At 180∘180∘ (𝜋π radians), the point is (-1, 0), so cos180∘=−1cos180∘=−1 and sin180∘=0sin180∘=0.
At 270∘270∘ (3𝜋/23π/2 radians), the point is (0, -1), so cos270∘=0cos270∘=0 and sin270∘=−1sin270∘=−1.
These key points illustrate how sine and cosine values are derived from the unit circle.
Using the Unit Circle for Trigonometric Functions:
For any angle 𝜃θ, draw a line from the origin to the circumference of the unit circle, creating an angle 𝜃θ with the positive x-axis.
The x-coordinate of the intersection point gives cos𝜃cosθ, and the y-coordinate gives sin𝜃sinθ.
To find tan𝜃tanθ, divide the y-coordinate by the x-coordinate of the point.
Visualizing Periodicity: The unit circle also helps in understanding the periodicity of trigonometric functions. Sine and cosine functions repeat their values every 360∘360∘ (or 2𝜋2π radians), which is evident as one completes a full rotation around the circleThe unit circle provides a clear and intuitive framework for understanding trigonometric functions such as sine, cosine, and tangent. By relating these functions to points on the unit circle, and leveraging the Pythagorean theorem, one gains a deeper comprehension of their properties and behaviors. This foundational knowledge is crucial for solving trigonometric equations and applying these functions in various scientific and engineering contexts.
What are the Applications of Unit Circle
The unit circle is not just a theoretical construct; it has numerous practical applications in various fields, particularly in solving trigonometric equations and understanding periodic functions. By leveraging the unit circle, one can gain deeper insights into the properties and behaviors of trigonometric functions, facilitating problem-solving and analysis in both academic and real-world contexts.
Solving Trigonometric Equations Using the Unit Circle
One of the primary applications of the unit circle is in solving trigonometric equations. The unit circle provides a visual and intuitive method for understanding the relationships between angles and trigonometric functions such as sine, cosine, and tangent. Here's how the unit circle can be used to solve trigonometric equations:
Identifying Solutions for Sine and Cosine Functions: To solve equations like sin𝜃=𝑘sinθ=k or cos𝜃=𝑘cosθ=k, one can use the unit circle to find the angles that correspond to these values. For example, to solve sin𝜃=12sinθ=21, locate the points on the unit circle where the y-coordinate is 1221. This occurs at).
Using the Inverse Trigonometric Functions: The unit circle helps in understanding the inverse trigonometric functions, which are used to determine the angles that correspond to specific trigonometric values. For instance, sin−1(12)=30∘sin−1(21)=30∘ and cos−1(−12)=120∘cos−1(−21)=120∘ (or 2𝜋/32π/3 radians).
Solving Equations Involving Multiple Angles: For equations like sin(2𝜃)=32sin(2θ)=23, the unit circle can be used to find all possible solutions within a given interval. By considering the periodic nature of sine and cosine, one can determine that) are solutions.
Visualizing Symmetry and Periodicity: The unit circle's symmetry properties help identify equivalent angles that produce the same trigonometric values. For example, sin(30∘)=sin(150∘)sin(30∘)=sin(150∘) and cos(45∘)=cos(315∘)cos(45∘)=cos(315∘), demonstrating the periodicity and symmetry of these functions.
Using the Unit Circle to Understand Periodic Functions
The unit circle is essential for understanding the periodic nature of trigonometric functions. Periodicity refers to the repeating patterns of these functions over regular intervals. The unit circle provides a clear visual representation of this concept:
Visualizing Sine and Cosine Waves: By tracing the coordinates of points on the unit circle as the angle 𝜃θ increases, one can plot the corresponding sine and cosine waves. These waves repeat every 360∘360∘ (or 2𝜋2π radians), illustrating the periodicity of the functions.
Understanding Amplitude and Phase Shift: The unit circle helps in understanding the amplitude (the maximum value of the function) and phase shift (the horizontal shift of the function). For instance, the function 𝑦=sin(𝜃+𝜙)y=sin(θ+ϕ) can be visualized on the unit circle by shifting the angle 𝜃θ by 𝜙ϕ radians.
Modeling Real-World Phenomena: Periodic functions modeled using the unit circle are applicable in various real-world scenarios, such as sound waves, light waves, and alternating current in electrical engineering. For example, the alternating voltage in an AC circuit can be represented as a sine wave, 𝑉(𝑡)=𝑉0sin(𝜔𝑡+𝜙)V(t)=V0sin(ωt+ϕ), where 𝑉0V0 is the amplitude, 𝜔ω is the angular frequency, and 𝜙ϕ is the phase angle.
Fourier Series and Transform: The unit circle is foundational in understanding the Fourier series and Fourier transform, which decompose periodic functions into sums of sine and cosine terms. This is crucial in signal processing, where complex signals are analyzed in terms of their frequency components.
The applications of the unit circle extend far beyond simple trigonometric calculations. By using the unit circle, one can effectively solve trigonometric equations and understand the periodic nature of trigonometric functions. These applications are vital in fields such as physics, engineering, and computer science, where periodic phenomena are commonly encountered. Mastery of the unit circle thus equips students and professionals with essential tools for both theoretical and practical problem-solving.
Inverse trigonometric functions, such as arcsine (sin−1sin−1), arccosine (cos−1cos−1), and arctangent (tan−1tan−1), are essential for determining angles corresponding to specific trigonometric values. The unit circle provides a clear geometric interpretation of these functions:
Arcsine (sin−1sin−1): This function returns the angle whose sine is a given value. For instance, sin−1(0.5)=30∘sin−1(0.5)=30∘ (or 𝜋/6π/6 radians). On the unit circle, this corresponds to the angle where the y-coordinate is 0.5.
Arccosine (cos−1cos−1): This function returns the angle whose cosine is a given value. For example, cos−1(0.5)=60∘cos−1(0.5)=60∘ (or 𝜋/3π/3 radians). On the unit circle, this is the angle where the x-coordinate is 0.5.
Arctangent (tan−1tan−1): This function returns the angle whose tangent is a given value. For instance, tan−1(1)=45∘tan−1(1)=45∘ (or 𝜋/4π/4 radians). On the unit circle, this involves finding the angle where the slope of the line from the origin to the point equals 1.
By using the unit circle, one can visualize how these inverse functions map specific trigonometric values back to their corresponding angles, aiding in the understanding of their properties and applications.
Exploring the Relationship Between Trigonometric Functions Using the Unit Circle
The unit circle is a powerful tool for exploring the intricate relationships between trigonometric functions. Key relationships include:
Pythagorean Identity: As derived from the Pythagorean theorem, cos2𝜃+sin2𝜃=1cos2θ+sin2θ=1. This identity is fundamental in trigonometry and can be easily visualized on the unit circle.
Tangent and Secant Functions: The tangent function is the ratio of sine to cosine, tan𝜃=sin𝜃cos𝜃tanθ=cosθsinθ. The secant function is the reciprocal of cosine, sec𝜃=1cos𝜃secθ=cosθ1. On the unit circle, tan𝜃tanθ represents the slope of the line through the origin and the point (cos𝜃,sin𝜃)(cosθ,sinθ).
Cosecant and Cotangent Functions: Similarly, the cosecant function is the reciprocal of sine, csc𝜃=1sin𝜃cscθ=sinθ1, and the cotangent function is the reciprocal of tangent, cot𝜃=1tan𝜃cotθ=tanθ1. These relationships can also be visualized on the unit circle.
Exploring these relationships using the unit circle enhances understanding of how trigonometric functions interact and complement each other, providing a deeper insight into their applications in various mathematical contexts.
Real-World Examples
Practical Applications of the Unit Circle in Various Fields
The unit circle has numerous practical applications across various fields, from navigation to engineering and physics:
Navigation: The unit circle is essential in navigation, particularly in understanding bearings and calculating distances. It helps in determining the shortest path between two points on the Earth's surface, crucial for air and sea navigation.
Physics: In physics, the unit circle is used to model periodic phenomena such as oscillations and wave motion. For example, the motion of a pendulum can be described using trigonometric functions derived from the unit circle.
Engineering: Engineers use the unit circle to analyze alternating current (AC) circuits. The sine and cosine functions represent the voltage and current waveforms, allowing engineers to design and troubleshoot AC systems effectively.
Computer Graphics: The unit circle is fundamental in computer graphics for rotations and transformations. It helps in creating smooth animations and accurately positioning objects within a graphical interface.
How the Unit Circle is Used in Navigation, Physics, and Engineering
Navigation: By using the unit circle, navigators can calculate the angle between the North direction and the direction to a destination point, aiding in precise navigation.
Physics: In physics, trigonometric functions based on the unit circle describe harmonic motion, such as the oscillations of springs and the propagation of sound waves.
Engineering: The unit circle assists engineers in analyzing signal phases and frequencies in telecommunications and control systems, ensuring efficient and reliable system performance.
Summary of Key Points Covered in the Content
Throughout this comprehensive guide, we have explored the fundamental concepts and applications of the unit circle, including:
Importance of the Unit Circle in Understanding Trigonometry
The unit circle is a crucial tool for understanding trigonometry. It simplifies the visualization and computation of trigonometric functions, aiding in solving complex mathematical problems and modeling real-world phenomena. Mastery of the unit circle equips students and professionals with essential skills for various scientific and engineering disciplines.
Further Resources for Continued Learning About the Unit Circle
To further enhance your understanding of the unit circle and its applications, consider exploring the following resources:
Unit circle negative?
Unit circle cotangentUnit circle secant?
The secant function on the unit circle is the reciprocal of the cosine function, defined as sec𝜃=1cos𝜃secθ=cosθ1.
Embedded math unit circle?
The embedded math unit circle refers to the use of the unit circle in various mathematical contexts to define and understand trigonometric functions.
Unit circle questions?
Unit circle questions are exercises or problems designed to test your understanding of the unit circle and its trigonometric values.
Unit circle art project?
A unit circle art project involves creating a visual representation of the unit circle, often as part of a mathematics assignment.
Unit circle projects?
Unit circle projects involve creating visual or interactive representations of the unit circle to help understand and memorize key angles and trigonometric values.
Unit circle table of values?
A unit circle table of values lists key angles in degrees and radians, along with their corresponding sine, cosine, and tangent values.
Reference angle unit circle?
A reference angle on the unit circle is the acute angle formed by the terminal side of the given angle and the x-axis, used to find trigonometric values.
Unit circle tattoo?
A unit circle tattoo is a design that includes the unit circle with key angles and trigonometric values, often chosen by math enthusiasts.
Tan on the unit circle?
To find the tangent of an angle on the unit circle, divide the y-coordinate (sine) by the x-coordinate (cosine) of the corresponding point.
Khan academy unit circle?
Khan Academy provides comprehensive lessons and interactive exercises to help learn and understand the unit circle and its applications in trigonometry.
How does the unit circle work?
The unit circle works by representing angles and their corresponding sine, cosine, and tangent values as points on a circle with a radius of one unit.
Unit circle memorization game?
A unit circle memorization game is an interactive activity designed to help users learn and memorize the unit circle, including key angles and trigonometric values.
Unit circle all values?
A unit circle with all values includes the coordinates and corresponding sine, cosine, and tangent values for all key angles.
Radius of a unit circle?
The radius of a unit circle is one unit.
Unit circle cosine?
The cosine values on the unit circle correspond to the x-coordinates of the points for various angles.
Standard position unit circle?
In the unit circle, an angle in standard position has its vertex at the origin and its initial side along the positive x-axis.
Unit circle with tangent labeled?
A unit circle with tangent labeled includes the tangent values for each key angle, helping to visualize and understand the relationship between sine, cosine, and tangent.
Unit circle blank pdf?
A blank unit circle PDF is a downloadable template used for practice, where you can fill in the key angles and their corresponding trigonometric values.
Tangent on the unit circle?
The tangent on the unit circle is the ratio of the y-coordinate (sine) to the x-coordinate (cosine) of the point corresponding to the angle.
Is cos x or y on the unit circle?
On the unit circle, the cosine of an angle corresponds to the x-coordinate of the point, while the sine corresponds to the y-coordinate.
Unit circle calc?
A unit circle calculator is a tool that helps you find the sine, cosine, and tangent values for any given angle.
Unit circle chart pdf?
A unit circle chart PDF is a downloadable and printable version of the unit circle, including key angles and their corresponding trigonometric values.
Circle unit chart?
A circle unit chart is another term for a unit circle chart, showing key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
All students take calculus unit circle?
The mnemonic "All Students Take Calculus" helps remember the signs of the trigonometric functions in each quadrant of the unit circle: All (positive in the first quadrant), Students (sine positive in the second quadrant), Take (tangent positive in the third quadrant), Calculus (cosine positive in the fourth quadrant).
Unit circle khan academy?
Khan Academy provides comprehensive lessons and interactive exercises to help learn and understand the unit circle and its applications in trigonometry.
Unit circle fill in?
A unit circle fill-in worksheet is a practice sheet where you complete the blank unit circle with key angles and their corresponding trigonometric values.
Unit circle positive and negative?
The unit circle includes both positive and negative angles, with positive angles measured counterclockwise and negative angles measured clockwise.
Unit circle pre calc?
In pre-calculus, the unit circle is used to introduce and explore the properties and applications of trigonometric functions.
Unit circle chart blank?
A blank unit circle chart is an empty template used for practice, where you can fill in the angles and their corresponding trigonometric values.
Memorize unit circle?
To memorize the unit circle, use mnemonic devices, flashcards, and regular practice to help memorize the key angles and their corresponding trigonometric values.
Unit circle project examples?
Unit circle project examples include creating visual representations of the unit circle, such as posters, models, or digital projects.
Tan values on unit circle?
Tan values on the unit circle are the tangent values for each angle, calculated as the ratio of the y-coordinate (sine) to the x-coordinate (cosine).
45 45 90 triangle unit circle?
The 45-45-90 triangle on the unit circle helps to visualize and understand the sine, cosine, and tangent values for the angle 45° (or 𝜋/4π/4 radians).
Unit circle trick?
A unit circle trick is a mnemonic or method used to help memorize the key angles and their corresponding trigonometric values on the unit circle.
Understanding the unit circle?
Understanding the unit circle involves learning its structure, how it is used to define trigonometric functions, and its applications in solving trigonometric problems.
Cot on unit circle3pi/4 on unit circle?
On the unit circle, the angle 3𝜋/43π/4 (or 135 degrees) corresponds to the coordinates (−2/2,2/2)(−2/2,2/2).
Cos on unit circle?
The cosine values on the unit circle correspond to the x-coordinates of the points for various angles.
Unit circle with angles?
A unit circle with angles shows key angles marked in both degrees and radians, along with their corresponding sine, cosine, and tangent values.
Unit circle print out?
A unit circle print out is a physical copy of the unit circle chart, used for study and practice, showing key angles and their trigonometric values.
Full unit circle with tan?
A full unit circle with tan includes all key angles and their corresponding sine, cosine, and tangent values.
Fill in unit circle?
Filling in the unit circle involves completing a blank unit circle template with key angles and their corresponding sine, cosine, and tangent values.
Unit circle filled?
A filled unit circle includes all key
please continue from where you left
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Unit circle filled?
A filled unit circle includes all key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
Hand trick unit circle?
The hand trick for the unit circle is a mnemonic device that helps in memorizing the key angles and their trigonometric values by using your hand as a visual guide.
Unit circle chart radians?
A unit circle chart in radians includes angles measured in radians, providing a natural mathematical representation.
Unit circle sheet?
A unit circle sheet is a practice worksheet or reference sheet that includes the complete unit circle with all key angles and trigonometric values.
Unit circle negative angle?
A negative angle on the unit circle is measured in the clockwise direction, showing the corresponding negative trigonometric values.
Cos and sin on unit circle?
The cosine and sine values on the unit circle correspond to the x and y coordinates of the points for various angles, respectively.
Cheat sheet unit circle trig functions?
A cheat sheet for unit circle trig functions provides a quick reference to the key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
Csc on unit circle?
The cosecant function on the unit circle is the reciprocal of the sine function, defined as csc𝜃=1sin𝜃cscθ=sinθ1.
Unit circle triangle?
Unit circle triangles are right triangles formed within the unit circle, used to visualize and understand the relationships between the trigonometric functions.
Sin on unit circle?
The sine values on the unit circle correspond to the y-coordinates of the points for various angles.
Unit circle trig identities?
The unit circle helps to understand and derive trigonometric identities such as sin2𝜃+cos2𝜃=1sin2θ+cos2θ=1.
How to read a unit circle?
To read a unit circle, identify the angle in degrees or radians, and use the coordinates of the corresponding point to find sine, cosine, and tangent values.
Sec on unit circle?
The secant function on the unit circle is the reciprocal of the cosine function, defined as sec𝜃=1cos𝜃secθ=cosθ1.
Terminal point on unit circle?
The terminal point on the unit circle is the point where the terminal side of an angle intersects the circle, given by the coordinates (cos𝜃,sin𝜃)(cosθ,sinθ).
Tangent unit circle chart?
A tangent unit circle chart shows the tangent values for each key angle, calculated as the ratio of sine to cosine.
Unit circle labeled?
A labeled unit circle includes the key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
How to read unit circle?
Read the unit circle by locating the angle of interest and using the x and y coordinates of the corresponding point to determine the trigonometric function values.
Unit circle with all values?
A unit circle with all values includes the coordinates for key angles and the corresponding sine, cosine, and tangent values.
Unit circle chart printable?
A unit circle chart printable is a downloadable and printable version of the unit circle, used for study and practice.
Radians on unit circle?
On the unit circle, angles can be measured in radians, where 360∘=2𝜋360∘=2π radians.
-pi/2 on unit circle?
On the unit circle, the angle −𝜋/2−π/2 (or -90 degrees) corresponds to the coordinates (0, -1).
Unit circle with triangles?
A unit circle with triangles includes right triangles formed within the unit circle, used to visualize and understand the relationships between the trigonometric functions.
Unit circle w tan?
A unit circle with tan includes the tangent values for each key angle, helping to visualize and understand the relationship between sine, cosine, and tangent.
Unit circle with pi?
A unit circle with pi includes angles measured in radians, where 360∘=2𝜋360∘=2π radians, providing a natural mathematical representation.
Tan in unit circle?
The tangent values in the unit circle are the ratios of the y-coordinates (sine) to the x-coordinates (cosine) for various angles.
Unit circle sin and cos?
The sine and cosine values on the unit circle correspond to the y and x coordinates of the points for various angles, respectively.
Unit of a circle?
The term "unit of a circle" typically refers to the unit circle, a circle with a radius of one used in trigonometry.
Unit circle for tangent?
The unit circle helps visualize and understand the tangent function, showing the ratio of the y-coordinate to the x-coordinate for various angles.
What is unit circle?
The unit circle is a circle with a radius of one unit, centered at the origin, used to define trigonometric functions and understand their properties.
How to use a unit circle?
Use a unit circle to find the sine, cosine, and tangent of angles by identifying the coordinates of the corresponding points.
ASTC unit circle?
The mnemonic "ASTC" (All Students Take Calculus) helps remember the signs of the trigonometric functions in each quadrant of the unit circle.
Inverse trig unit circle?
The inverse trig unit circle refers to using the unit circle to understand inverse trigonometric functions, such as arcsine, arccosine, and arctangent.
Filled unit circle?
A filled unit circle includes all key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
Which of the following explains why cosine 60 degrees = sine 30 degrees using the unit circle?
Using the unit circle, cosine 60 degrees equals sine 30 degrees because both points correspond to the coordinates (12,32)(21,23).
What is the value of tangent theta in the unit circle below?
To find the value of tangent theta in the unit circle, divide the sine value by the cosine value of the corresponding point.
Hand trick for unit circle?
The hand trick for the unit circle is a mnemonic device that helps in memorizing the key angles and their trigonometric values by using your hand as a visual guide.
Unit chart circle?
A unit chart circle is another term for a unit circle chart, showing key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
Unit circle finger trick?
The unit circle finger trick is a mnemonic device used to help memorize the key angles and their corresponding trigonometric values by using your fingers as a visual guide.
Unit circle flashcards?
Unit circle flashcards are a study tool used to help memorize the key angles, their coordinates, and the corresponding sine, cosine, and tangent values.
Unit circle calculus?
In calculus, the unit circle is used to define and understand the properties of trigonometric functions and their derivatives.
Unit circle coordinates calculator?
A unit circle coordinates calculator is a tool that helps you find the coordinates (sine and cosine values) for any given angle.
Chart:3pqvgs8fx98= unit circle?
This keyword likely represents a specific context or code related to a unit circle chart, showing key angles and their corresponding trigonometric values.
Unit circle chart with tangent?
A unit circle chart with tangent includes the tangent values for each key angle, helping to visualize and understand the relationship between sine, cosine, and tangent.
Unit circle including tangent?
A unit circle including tangent shows the tangent values for each key angle, calculated as the ratio of sine to cosine.
Unit circle radius?
The radius of the unit circle is always one unit.
Authors
Thomas M. A.
A literature-lover by design and qualification, Thomas loves exploring different aspects of software and writing about the same. | 677.169 | 1 |
Let A B C D be a parallelogram with \angle B A D<90^{\circ}. A circle tangent to sides \overline{D A}, \overline{A B}, and \overline{B C} intersects diagonal \overline{A C} at points P and Q with A P<A Q, as shown. Suppose that A P=3, P Q=9, and Q C=16. Then the area of A B C D can be expressed in the form m \sqrt{n}, where m and n are positive integers, and n is not divisible by the square of any prime. Find m+n. | 677.169 | 1 |
MATH 331 Homework Problem
1. How do the coordinates of point D relate to vectors u and v?
2. Find the distance from A to D.
3. Find the measure of angle EAD.
4. Move points A and B and see if you can generalize your formulas.
Try negative coordinates for #2.
Move counterclockwise when answering #3.
5. What do you notice about this problem? | 677.169 | 1 |
Finding the Rotated Vector: Solving for w in Various Cases
Rotating a Vector
Here's the problem: Let v denote a vector that has initial position v1 and final position v2.
We know these two numbers. Now suppose we rotate v counterclockwise by an angle
theta. The result is a new vector, denoted by w, that also has initial position v1 and final
position v2.
Let's say that we rotate vector v by an angle theta. If we do this, our new vector becomes
w. The problem we are going to discuss is how to find w.
Time for a warmup with a bit of cases.
There is a vector v that points in the x direction. V1 comma 0.
So, there we have it. V1 and a comma 0. And then when we rotate v, we get another
vector, w. OK, so what's w? This is actually very similar to problems we've solved before
because we know this angle, theta, and we know the length of w. The length of w is equal
to its starting point, so it's v1.
The length of w equals to the length of v, which is v1. Now
we want to find the components of w. The first component is v1 times the sine of theta, and
the second component is v1 times the cosine of theta.
This is it.
What if in the second case, let's assume that v is in the y-direction rather than in the
x-direction. So v is 0 comma v2.
Thus, there is a vector v. We can rotate this vector by an angle 0 and obtain a new vector
w. Please find w.
[Graph]
[Graph]
[Graph]
Cite this page
Finding the Rotated Vector: Solving for w in Various Cases. (2023, Aug 02). Retrieved from
Students looking for free, top-notch essay and term paper samples on various topics. Additional materials, such as the best quotations, synonyms and word definitions to make your writing easier are also offered here. | 677.169 | 1 |
A kite is a quadrilateral with two distinct sets of adjacent congruent sides. It looks like a kite that flies in the air.
Figure \(\PageIndex{1}\)
From the definition, a kite could be concave. If a kite is concave, it is called a dart. The word distinct in the definition means that the two pairs of congruent sides have to be different. This means that a square or a rhombus is not a kite.
The angles between the congruent sides are called vertex angles. The other angles are called non-vertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles.
Figure \(\PageIndex{2}\)
Facts about Kites
1. The non-vertex angles of a kite are congruent.
Figure \(\PageIndex{3}\)
If \(KITE\) is a kite, then \(\angle K\cong \angle T\).
2. The diagonal through the vertex angles is the angle bisector for both angles.
Figure \(\PageIndex{4}\)
If \(KITE\) is a kite, then \(\angle KEI\cong \angle IET\) and \(\angle KIE\cong \angle EIT\).
3. Kite Diagonals Theorem: The diagonals of a kite are perpendicular.
Figure \(\PageIndex{5}\)
\( \Delta KET\) and \(\Delta KIT\) are isosceles triangles, so \(\overline{EI}\) is the perpendicular bisector of \(\overline{KT}\) (Isosceles Triangle Theorem).
What if you were told that \(WIND\) is a kite and you are given information about some of its angles or its diagonals? How would you find the measure of its other angles or its sides?
For Examples 1 and 2, use the following information:
\(KITE\) is a kite.
Figure \(\PageIndex{6}\)
Example \(\PageIndex{1}\)
Find \(m\angle KIS\).
Solution
\(m\angle KIS=25^{\circ}\) by the Triangle Sum Theorem (remember that \angle KSI is a right angle because the diagonals are perpendicular | 677.169 | 1 |
93.
УелЯдб 2 ... AB , CB , is named the angle ABC , or CBA ; that which is con- tained by AB ... equal to one another , each of the angles is called a right angle ; and the ... equal to one another . XVI . And this point is called the centre of ...
УелЯдб 6 ... AB be the given straight line . It is required to describe on AB an equilateral triangle . CONSTRUCTION From the ... equal to AB ; ( def . 15 ) and because the point B is the centre of the circle ACE , therefore BC is equal to BA ...
УелЯдб 8 ... equal to AD ; ( def . 15 ) but the straight line C is likewise equal to AD ; ( constr . ) therefore AE and C are each of them equal to AD ; wherefore the straight line AE is equal to C. ( ax . 1. ) And therefore from AB , the greater of ...
УелЯдб 9 ... AB upon DE ; because AB is equal to DE , therefore the point B shall coincide with the point E ; and AB coinciding with DE , and the angle BAC being equal to the angle EDF , ( hyp . ) therefore AC shall coincide with DF ; and because AC is | 677.169 | 1 |
You are given a parallelogram ABCD. BE-bisector of angle ADC, CD = 8, BC = 12 Find the perimeter of the parallelogram.
According to the properties of a parallelogram, its opposite sides are equal:
AB = CD = 8 cm;
BC = AD.
Since the bisector of the angle of the parallelogram cuts off the isosceles triangle from it, the segment EC is equal to the side CD:
EC = CD = 8 cm.
The perimeter of a parallelogram is the sum of the lengths of all its sides:
P = AB + BC + CD + AD.
Let's find the length of BC and AD:
BC = AD = BE + EC;
BC = AD = 12 cm + 8 cm = 20 cm;
P = 20 cm + 8 cm + 20 cm + 8 cm = 56 cm.
Answer: The perimeter of this parallelogram is 56 | 677.169 | 1 |
I think the sixth option from the main menu (MidPoint/EndPoint) will do the trick for: "collinearity of three points", I just labeled it with a different name.
Regarding to conic sections: (circumference, parabola and ellipse) It's a great Idea, in fact I've already thought about implementing something like that, it's just that time happens to be my main limitation. perhaps it could be possible in a near future. | 677.169 | 1 |
Plane and Solid Geometry
From inside the book
Results 1-5 of 11
Page 373 ... tetraedron ; one of six faces , a hexaedron ; one of eight faces , an octaedron ; one of twelve faces , a dodecaedron ; and one of twenty faces , an icosaedron . It is instructive to construct the regular polyedrons as shown in the ...
Page 375 ... tetraedron 3 in . on an edge . Of a regular octaedron 4 in . on an edge . Of a regular icosaedron 6 in . on an edge ... tetraedron is e , show that its volume is given by the formula Ve3√2 . = 12 SOLUTION . VON × area of △ ABC ...
Page 378 ... tetraedron is its altitude above the base . Find the center of gravity of a regular tetraedron 8 in . on an edge . 6. The edge of a regular tetraedron is e . Find the edge of a regular tetraedron that has a volume n times as great as ...
Popular passages
Page 169 - In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.
Page 155 - ... any two parallelograms are to each other as the products of their bases by their altitudes. PROPOSITION V. THEOREM. 403. The area of a triangle is equal to half the product of its base by its altitude. | 677.169 | 1 |
Special Right Triangles - Video Tutorials & Practice Problems
45-45-90 Triangles
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Welcome back everyone. So up to this point, we've spent a lot of time talking about trigonometric functions, the Pythagorean theorem and how they all relate to the right triangle. Now, what we're going to be learning about in this video is some of the special and common right triangles that you're going to see. And in this video specifically, we're going to be talking about the 45 45 90 special tri. Now, the reason that these triangles are special is because it turns out that because they show up relatively frequently, there are actually some shortcuts that you can use to solve these triangles very fast. So if you don't like all the brute force work we've been doing with trigonometric functions of the Pythagorean theorem, you're going to learn some shortcuts for solving these triangles in this video. So without further ado let's get right into things. Now, when you have a triangle with 45 degree angles like this triangle down here, for example, this is going to be a situation where you have the special 45 45 90 triangle. And in these triangles, the two legs of the triangle are always going to be the same length. So if you ever see a situation where you have a right triangle and two of the legs are the same, that means you're dealing with this special triangle. Now, what we can do with this is we can actually solve for the hypotenuse of the triangle by simply taking a multiple of the lake length. And the multiple that you're going to look for is the square root of two. Because if you take a leg like five and you multiply it by the square root of two, this will give you the hypotenuse. And that's the answer we just solved for the long side of this triangle. So as you can see this shortcut right here, solving for the sides of the triangle really straightforward and really fast. Now, if you didn't remember this relationship, there is another strategy you can use, which is simply the long version of using the Pythagorean theorem. So let's say that we set this side to A, that side to B and then the hypotenuse equal to C. And we want to solve for the hypotenuse. Well, you could say that A squared plus B squared equals C squared. That's the paging theorem. And in this case, we said A and B are both. So we have five squared plus five squared is equal to C squared and five squared is 25. So we have 25 plus 25 equals C squared, 25 plus 25 is 50. And what we can do is take the square root on both sides of this equation to get that C is equal to the square root of 50. And the square root of 50 actually simplifies down to five times the square root of two. So notice when using the long version of this problem solving, we get to the same answer. But this is what's nice about the shortcut is it lets you get this answer without having to go through this long process. Now, to ensure we know how to solve these types of triangles, let's see if we can solve some examples where we have this special case. So for each of these examples, we're asked to solve for the unknown sides of each triangle. And we'll start with example a now notice we have 245 degree angles and two legs that are the same length. So that means we're dealing with a 45 45 90 triangle. Now recall that if we want to find the missing side or the hypotenuse, we just need to take one of the legs and multiply it by the square root of two. Well, one of the legs are 11 and then we multiply this by the square root of two. And that right there is the answer. Notice how quick it is using this method. See it's very straightforward and that's what's really nice about these special cases. But now let's take a look at example, B in this example, we have a 45 degree angle and we are given the hypotenuse. So how could we go about solving this? Well, first off, we need to figure out if we actually are dealing with a special case triangle. And it turns out that we are because since we have a 45 degree angle here and a 90 degree cusp there, we know by default, this has to be a 45 degree angle. You see all the angles in a right triangle have to add to 100 8090 plus 45 plus 45 equals 180. So this is a special case triangle. Now to solve for the missing sides, what we can do is use this relationship. Notice in this situation, we're given the hypotenuse or the long side. So what I'm going to do is take the hypotenuse instead of equal to the number we have, which is 13. And I'll say that that's equal to the leg multiplied by the square root of two. Now, to solve for the leg, what I can do is divide the square root of two on both sides of this equation. That'll get the square root of twos to cancel, giving us that the leg of this triangle is equal to 13 over the square root of two. Now, what I can do is rationalize the denominator here by multiplying the top and bottom by the square root of two, that'll get these square roots to cancel, giving us that the leg of this triangle is equal to 13 times the square root of 2/2. So what we're gonna end up with is 13 radical 2/2 for this side of the triangle and then 13 radical 2/2 for that side of the triangle. Because again, for a 45 45 90 triangle, these two sides have to have the same length. So that is how you can solve 45 45 90 triangles. And this is the shortcut that you can use. So hope you found this video helpful. Thanks for watching and let me know if you have any questions.
2
Problem
Problem
Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).
A
x=81x=81x=81
B
x=92x=9\sqrt2x=92
C
x=29x=2\sqrt9x=29
D
x=162x=\sqrt{162}x=162
3
Problem
Problem
Without using a calculator, determine all values of P in the interval [0°,90°)\left[0\degree,90\degree\right)[0°,90°) with the following trigonometric function value.
cscP=2\csc P=\sqrt2cscP=2
A
P=30°P=30\degreeP=30° only
B
P=45°P=45°P=45° only
C
P=60°P=60°P=60° only
D
P=30°,60° P=30°,60°P=30°,60°
4
concept
Common Trig Functions For 45-45-90 Triangles
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Hey, everyone. So in a recent video, we talked about the special 45 45 90 right triangles and we learned how you can use some shortcuts to quickly find the missing sides. Well, in this video, we're going to be learning about the common trig functions and how they are associated with the 45 45 90 triangle. And we're going to see if we can recognize any kinds of patterns for these trig functions. Because what we're going to learn for these special cases is there may be shortcuts when finding ratios for the trigonometric functions as well, meaning these triangles could be overall a lot easier to solve. So I'm always interested in finding shortcuts and hopefully, you are too. So let's get right into this. Now, we're going to start with the sine function and we know that sign is the option divided by the hypotenuse. Now, if we go to a 45 45 90 right triangle, we can go to either one of the two angles because they're both 45 degrees. So if we take a look at this angle, for example, and we go to the opposite side, we can see that the opposite is five. And if we divide this by the hypotenuse or along side, that's going to be five times the square root of two, we can cancel the fives here giving us one over the square root of two. And by rationalizing this denominator, I'll multiply the top and bottom by the square root of two. This will get the square roots to cancel giving us square root of 2/2. So that means for the sign, we'll end up with square root 2/2. But now let's try solving for the cosine for the cosine, we get adjacent, divided by hypotenuse. Now, if we go to one of our 45 degree angles, the adjacent side is five. And if we divide this by the hypotenuse, we'll have five times the square root of two, the five will cancel, giving us one over the square root of two. And we already figured out this is the same thing as square root 2/2. So this will just simplify to radical 2/2, which is the cosine for this right triangle. Now, let's take a look at the tangent for the tangent we have opposite over adjacent. So what I'm going to do is go over here to our right triangle, look at one of our angles. And if I go to the opposite side, we end up with five, if I go to the adjacent side, we also have five So we'll have 5/5, which is simply one, meaning the tangent for our 45 45 90 triangle is one. So let's see if we notice any patterns with these trigonometric functions. Well, something that I noticed is that the sine and cosine are the same and the tangent just comes out to one. So this is something that's pretty straightforward about these triangles is that you get the same sine and cosine values. But let's take a look at these other reciprocal identities like the co sequent sequent and cotangent. So for the code secret, we know that this is just one over the s of data, which means all we're going to do is flip this fraction that we got. So we'll have two over the square root of two. And then we can rationalize this denominator when doing this, the square roots will cancel on bottom of the fraction giving us two times the square root of 2/2. These are going to cancel right here, meaning all we're going to end up with is the square root of two for the cosecant. And because the sine and cosine were the same, that means one over the cosine, which is our second is also going to be the square root of two. So all we now have to solve for is the cotangent, which is just one over the tangent. But we learned that the tangent is one. So the cotangent will be 1/1, which is simply equal to one. So notice how really straight for all these trigonometric functions are, when you're looking at a special case 45 45 90 triangle, we can recognize that the sine and cosine will always be square root 2/2, that the cosecant and see it will always just be the square root of two and that the tangent and cotangent will always be one. So these are all of the common trig functions for the 45 45 90 triangle. Hope you found this video helpful and thanks for watching.
5
concept
30-60-90 Triangles
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Welcome back everyone. So in recent videos, we've been talking about special case right triangles and we recently learned about the 45 45 90 triangle and some shortcuts you can use for solving this special case. Well, in this video, we're going to be taking a look at the 30 60 90 triangle, which is another special case. And for these triangles, there are shortcuts you can use to solve them as well. And this is a type of triangle that is going to show up a lot, not only in this course but also in future math courses and possibly other courses too like physics or other science related courses because this situation just tends to happen a lot. So without further ado let's get right into some shortcuts, you can use to solve this triangle. Now, the thing that's special about this triangle is that you can relate the side lengths to the shortest leg on the triangle. Now keep in mind for the 30 60 90 triangle, it's a bit different than the 45 45 90 triangle because for the 45 45 90 case, we learned that two of the side lengths are the same for the triangle. And so you can use one relationship to relate all the sides. Well, for the 30 60 90 case, the two side lengths are not the same, we have two different lengths. So what we need to do here is keep track of which side relates to which other side. But we can do that using the shortcuts. So for example, if you want to find the hypotenuse of a 30 60 90 triangle, you can take the short leg of your triangle, the shortest side and you can multiply it by two. And if you want to find the long leg of triangle, you can take the short leg of the triangle again, and you can multiply it by the square root of three. Now again, I want to emphasize these shortcuts only work if you're dealing with a 30 60 90 triangle. And this is the special case we have in this example. So let's see if we can solve for the missing sides. Well, I can see that the hypotenuse is going to be the shortest leg multiplied by two. I can see the shortest leg is five. And if we multiply that by two, we're going to get 10, meaning the hypotenuse is 10. Now, I can see that the long leg of the triangle is equal to the short leg which we have down here. And we said that that's five multiplied by the square root of three So the long leg of the triangle is going to be five times the square root of three. So this is how you can find the missing sides of a 30 60 90 right triangle. So as you can see, it's still very straightforward when you have these shortcuts available, but to make sure we know how to use these shortcuts. Well, let's try some other examples that are a little bit more complicated. So for these examples, we're asked to solve for the unknown side of each triangle. And we're going to start with example a. Now, in this example, I see that we have the hypotenuse and a 30 degree angle. So what we first have to ask ourselves is, is this a 30 60 90 triangle? Well, it actually is because if this is 30 degrees and this is 90 degrees, then the other missing angle has to be 60 degrees because 60 plus 30 gives you 90 then you add another 90 you get to 180 all angles in the triangle have to add to 180 degrees. So this is a special case. So that means we can use the shortcuts we learned about, well, first I'm going to try solving for the short leg and I'm going to do this using this relationship up here because we have that the hypotenuse is equal to the short leg multiplied by two. Now, the hypotenuse we can see is the longest side which is eight. And so that's going to equal the short leg of the triangle multiplied by two. So if I want to find the short leg, all I need to do is divide both sides of this equation by two, that's going to get the twos to cancel on the right side, giving us that the short leg of our triangle is equal to eight divided by two, which is four. So that means this missing side is four. Now, if I want to find the long side of the triangle, I could use the Pythagorean theorem from here or I could use this relationship that says that the long leg is the short leg multiplied by the square root of three. So we have that the long leg is equal to the short, I'm just gonna write it as long, by the way, we have that's equal to the short leg, which we just figured out is four and that's multiplied by the square root of three. Meaning this missing, the triangle is four times the square root of three. So that is how you can use shortcuts to find the missing sides of a 30 60 90 triangle. But now let's try another example in this case example. B and for this example, we're asked to find the missing sides as well. And what I can see is that we have a 60 degree angle and the long side of the triangle given to us now because we have a 60 angle here. By default, this other angle has to be 30 degrees because all angles of the triangle have to add to 180. So this is a special case. So let's try using some of these relationships. Well, since I see that we have the long leg, I'm going to use this relationship right here, which is that the long leg is going to be equal to the short leg multiplied by the square root of three. Now, I can see that we have that the long leg of the triangle is one, so we'll have one, the long side is equal to the short side multiplied by the square root of three. So what I can do from here is divide the square root of three on both sides of this equation, which will give the square root of threes to cancel on the right side, giving us that the short leg of this triangle is equal to one divided by the square root of three. And I can go ahead and rationalize this denominator by multiplying the top and bottom by the square root of three. They'll get the square roots to cancel, giving us that the short leg of this triangle is equal to the square root of 3/3. So that's going to be the short leg of the right triangle. Now, our last step for solving this right triangle is to find the hypotenuse or the longest side. But I can see based on the relationship up here that the hypotenuse is going to be the short leg, which we already determined is square root 3/3, multiplied by two. So that means that this missing side of the triangle is going to be two times the square root of 3/3. So this is how you can find all the missing sides of a 30 60 90 triangle using some simple shortcuts. So that's how you solve these types of problems. Hope you found this video helpful. Thanks for watching.
6
Problem
Problem
Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).
A
x=10,y=55x=10,y=5\sqrt5x=10,y=55
B
x=3,y=4x=3,y=4x=3,y=4
C
x=53,y=10x=5\sqrt3,y=10x=53,y=10
D
x=5,y=52x=5,y=5\sqrt2x=5,y=52
7
Problem
Problem
Without using a calculator, determine all values of A in the interval [0,π2)\left[0,\frac{\pi}{2}\right)[0,2π) with the following trigonometric function value.
cosA=32\cos A=\frac{\sqrt3}{2}cosA=23
A
000 only
B
π4\frac{\pi}{4}4π only
C
π6\frac{\pi}{6}6π only
D
π3\frac{\pi}{3}3π only
8
concept
Common Trig Functions For 30-60-90 Triangles
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Welcome back everyone. So up to this point, we've been talking about special case right triangles and we recently took a look at the 30 60 90 case. Well, in this video, we're going to be taking a look at the trigonometric functions associated with the 30 60 90 triangle. Now, so far, the trigonometric functions we've learned about have been a bit tedious just because we have to find ratios for each of the situations and see how they relate to each other. But what we're hopefully going to figure out in this video is that for the special case triangles, there are patterns that show up when you solve for the trigonometric ratios and functions. So without further ado let's get right into this and see if we can find some shortcuts and patterns that show up. So we're going to start with finding the sign of a 30 60 90 triangle. And we're going to focus on the 30 degree angle for these trigonometric functions on the left side. Now recall that sine is opposite over hypotenuse. So if I go to a 30 degree angle, the opposite side of this triangle is five and the hypotenuse or the long side of the triangle is 10 and 5/10 reduces to one half, meaning that the sign of 30 degrees is one half. But now let's take a look at the cosine of our 30 degree angle for the cosine, it's going to be adjacent over hypotenuse and the adjacent side to the 30 degree angle is five times the square to three. And then this is divided by the hypotenuse which is 10 and again, 5/10 reduces to one half. So we'll end up with one times the square root of 3/2 and one times square to three is just square to three. So all we're gonna end up with is the square root of 3/2. And that's our cosine. But now let's take a look at the tangent for the tangent. It's going to be opposite over adjacent. Well, going to this triangle, we can see that the opposite side is five and then the adjacent side is five times the square root of three. Again, the fives will cancel, giving us one over the square root of three. And if I go ahead and rationalize this denominator by multiplying the top and bottom by the square root of three, we'll get the square roots to cancel, giving us the square root of 3/3 as our tangent. So this is what we get for the sine cosine and tangent of our 30 60 90 triangle. When specifically looking at the 30 degree angle. But now let's solve for the cosecant. And we're going to use reciprocal identities to solve these next three trigonometric functions. Because I noticed for the cosecant, it's the same thing as one over the sin of theta. And we said that the sin of theta is one half. So all we need to do is flip one half which is just going to be 2/1 or two. So the cosecant of our 30 degree angle is two. Now let's take a look at the second of our angle. Well, for the second, we can see that this is one over cosine. And we always said that the cosine is square root 3/2. So for the second, we're going to get two over the square root of three, which I will need to rationalize this denominator. So multiply the top and bottom by the square of three, cancel the square roots giving us two times the square root of 3/3, which is the result for our second of 30 degrees. But now let's take a look at our cotangent. The cotangent is going to be one over the tangent. And we figured out that the tangent is square root 3/3. So the code tangent is going to be three over the square root of three just flipping this fraction. And again, I'll need to rationalize this denominator. When doing this, the square roots will cancel giving us three times the square root of 3/3. And these three are going to cancel leaving us with just the square root of three. So that is how you can find the trigonometric functions for the 30 degree angle. Now, because this is a 36 90 triangle, we also need to find all the trigonometric functions for the 60 degree angle. So let's go ahead and do that on this right side. So for the sine function, if we're looking for 60 degrees, recall that this is opposite over hypotenuse. Now, the opposite side is five times the square root of three. And this is going to be divided by the hypotenuse which is 10, the five and 10 will reduce to give us just a two in the denominator. Meaning that all we're going to have is the square root of 3/2 for our sign. Now let's take a look at our cosine for the cosine of 60 degrees, we're called that this is adjacent over hypotenuse. So we're going to go to the adjacent side of the 60 degree angle which is five and then divide this by the hypotenuse which is 10, 5/10 reduces to one half, meaning that the cosine over a 60 degree angle is one half. Now let's take a look at the tangent which is opposite over adjacent the side opposite to 60 degrees is five times the square root of three and then adjacent side is five, the files will cancel, giving us just the square root of three, meaning that the square root of three is the tangent of 60 degrees. Now, for these last three trigonometric functions, we're going to use the reciprocal identities. Now, Kent is the same thing as one over S sign. So if I want to find the coin of 60 degrees, I just need to flip the sign of 60 degrees which would be two over the square root of three. Now, I can go ahead and rationalize this denominator, that'll get the square roots to cancel on the bottom, giving us two square root 3/3. So that is going to be the cosecant of our 60 degree angle. Now, what I can also do is find the second of our 60 degree angle and to do this, what I need to do is it's one over cosine. So I need to go ahead and flip the cosine of 60 degrees since the cosine of 60 degrees is one half, that means that the sequence of 60 degrees is going to be 2/1, which is just equal to two. So two is the sequin of 60 degrees. Now, lastly, we're going to take a look at the cotangent of 60 degrees and all I need to do is go ahead and flip the tangent that we got here. So we said the tangent is squared to three. So the cotangent is going to be one over the square root of three. I can go ahead and rationalize the denominator getting the square roots to cancel on bottom, giving the square root of 3/3 for the cotangent of 60 degrees. So these are all of the trigonometric functions evaluated for the 30 60 90 triangle. And you may notice that we have some patterns that emerge when we figure out what all these trigonometric values are because notice how the sine and cosine will switch places when looking at the different angles because the sine of 30 is one half, whereas the cosine of 30 is square root 3/2. But when we go to the 60 degree angle, the sine is square root 3/2 and the cosine is one half, we see the same kind of behavior in the coin and sequin since these are reciprocal identities, you just switch them when looking at the different angles. Now we also see this behavior in the tangent and code tangent because notice when looking at the 30 degree angle, the tangent is square root 3/3 and the code tangent is square root three. But when looking at the 60 degree angle, the tangent is just square root three and the code tangent is square root 3/3. So we see the same kind of switch when looking at this trigonometric function. So hopefully, this helps you to recognize some of the patterns and give you some general understanding as to how you can find the trigonometric functions for a 30 60 90 triangle. I hope you found this video helpful. Thanks for watching. | 677.169 | 1 |
Questions tagged [geometric-construction]
Questions on constructing geometrical figures using a limited set of tools. The compass and straightedge are almost always allowed, while other tools like angle trisectors and marked rulers (neusis) may be allowed depending on context.
The most common use of "geometric construction" refers to the "compass and straightedge" constructions in classical Euclidean geometry. The notion has been extended also to (a) compass/straightedge constructions in non-Euclidean geometries and (b) allowing different sets of tools such as a marked straightedge (neusis) or origami.
Given any triangle $\triangle ABC$, let us draw its orthocenter $D$. By means of this point, we can draw three circles with centers in $A,B,C$ and passing through $D$.
These circles intersect in the points $E,F,G$, which can be seen as the vertices…
I want to preface this by saying I have been trying for a while to prove myself wrong because my results appear to contradict the work of some previous work by people who have studied much more than me. Anyway. I have what I believe to be trisection…
Given the following segments how would you construct a segment with length of $\frac{2a}{a+b^2}$?
Given the three line segments below, of lengths a, b and 1, respectively:
For example if I wanted to construct a segment $z$ satisfying an equation…
Given 3 heights : $h_1=5\mathrm{cm}$ ; $h_2=7\mathrm{cm}$ ; $h_3=8\mathrm{cm}$ ... It is required to draw that triangle using only compass and ruler !
N.B.: It is not allowed to calculate the area then the sides: the measure of the sides won't be…
Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are based on the length of a side, but have not seen…
Hi, recently I've been researching path synthesis in aircraft navigation and stumbled across a few images of constructions used to join different navigation paths together. In the image linked above, I will know the values of c,g,p1,p2&p3 and will… | 677.169 | 1 |
A rounded rectangle is a rectangle with rounded corners. The figure is defined by the rectangle itself, along with the width and height of the ovals forming the corners (called the diameters of curvature), as shown in Figure 13-5. The corner width and corner height are limited to the width and height of the rectangle itself; if they are larger, the rounded rectangle becomes an oval.
Figure 13-5
A rounded rectangle
A polygon is defined by a sequence of points representing the polygon's vertices, connected by straight lines from one point to the next. You define a polygon by specifying an array of x and y locations in which to draw lines and passing it as a parameter to | 677.169 | 1 |
ICSE Grade X Mathematics Demo Videos
Hi students in this module from the chapter Heights and Distances Let's show a sum based on Application Of trigonometry It's going to be very interesting Let's see the sum first which is Two vertical poles are on either side of a road A 30 metre long ladder Is placed between the two poles When the ladder Rest against 1 pole It's makes angle 32 degree and 24 minutes with the pole and When it is turned to rest Against another pole It makes an angle 32 degree and 24 minutes with the road Calculate the width off the road Now this is what the sum says Right Let's understand what are we supposed to draw It says the two vertical poles are on either side of the road You have two poles on either side of the road A 30 metre long ladder Is placed between the two poles You have a 30 metre long ladder Length of the ladder is given to us as 30 metres Now this is placed in such a way that when ladder rests to 1 pole It makes angle of 32 degree and 24 minutes With the pole Right when it is placed On one pole The angle made by the ladder with the pole The angle there Its nothing but It is 32 degree and 24 minutes And when it is turned To rest against the another pole Now its rest on the other pole Here here It makes 32 degree and 24 minutes with the road Right very important Thing to be noted there When it is rested like this The angle made by the ladder with the road Its nothing but 32 degree and 24 minutes Very important thing right first with the road then with the pole Right The sum says find Calculate the width of the road So we need to find the width of the road And we say the width of the road We are saying find the length of BD That is what we need to find So this is what we have in the sum Let's think how I going to get this It's going to be very easy Very simple Let's see the sum Right here we going to describe First will say that AB and DE represents the two poles That is what we have there Right Then we have AC and CE It represents the two positions of the ladder Right AC Is the first position and CE is the next position they represents the two positions of the ladder Right and we have AC=CE=30m This is known to us And we have those two angles there angle BAC is 32 degrees and 24 minutes and angle ECD has 32 degrees and 24 minutes That is what we have we need to find BD observe BD look at BD BD is the width of the road right BD is nothing but made up of two things BC and CD so we need to first find the value BC then the value of CD lets find out BC you know the BC belongs to the right angle triangle ABC consider that right angle triangle ABC infact we have an acute angle yes let's look at that for that acute angle BC something which we need to find that appears in the opposite side and what given to us is AC is the hypotenuse we are taking about opposite side of hypotenuse right opposite side and hypotenuse which ratio comes to your mind which trigonometry ratio yes its nothing but sin so were going to use sin for that angle so you're going to use sin of angle BAC = BC/AC so you know the sin of BAC means it is sin of 32 degree and 24 mins that's equal to BC if you don't know keep it as it is AC we know that is 30 so BC upon 30 right beautiful now we are going to get that value of sin 32 degree and 24 mins from the trigonometric table from natural sines from that table if you see in that row which contains 32 and in the column which headed by 24 mins see there the intersect of the two will give you the value of sin 32 degree and 24 mins which nothing but 0.5358 so we get that value substitute the value there so we got the value of sin 32 degree and 24 mins that is 05358 substitute that now the value of BC you don't know keep it as it is and now we know that it BC/30 now BC = 0.5358 * 30 so now we just multiply it and the product which we get is the value of BC that is nothing but 16.074m you got the value of BC now we need to find the value of CD observe CD CD belongs to the right angle triangle CDE right we going consider it right triangle CDE in that right angle triangle CDE we have an acute angle that is 32 degree and 24 mins for that acute angle we need to find the adjacent side and what is known to us is the hypotenuse so we are taking about adjacent side and hypotenuse when talking about adjacent side and hypotenuse which trigonometric ratio comes to your mind yes it is nothing but the trigonometric ratio is cos so your going to use cos for that particular angle so we right cos of angle ECD = adjacent side that is CD upon the hypotenuse that is CE now we just need to substitute the value so we write cos of 32 degree and 24 mins = CD/CE which is nothing but 30 so now we need the value of cos 32 degree and 24 mins this is something which we can get from the table of natural cosines so now look there in that row consisting of 32 and that column headed by 24 mins observe the intersection of that two gives you the value of 32 degree and 24 mins which 0.8443 so you substitute the value there 0.8443 = CD/30 now it's so simple now we get CD = 0.8443 * 30 now we get CD = 25.329m so you got the value of CD too wasn't that easy now isn't it easy to get the value of BD yes it is so simple now we can say that BD = BC + CD the value of BC if you see its simple it is 16.074 + the value of CD we know it is 25.329 add those two and we get the value of BD as 41.403 and if you round it off now we can say that BD = 41.4m that means width of the road is 41.4m was that easy a very important sum but a very simple one.
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...TRIGONOMETRY. SECTION I. 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure hag sides. 2. Equal triangles, upon equal bases in the same straight line, and towards the same parts,...
...accuracy of the previous work. Moreover, since the sum of all the interior angles of any polygon is equal to twice as many right angles as the figure has sides, lessened by four ; as the given figure has five sides, the sum of all its interior angles must be 2x5...
...Hence it follows that the sum of all the inward angles of the polygon alone, A + B -f- C + D + E, is equal to twice as many right angles as the figure has sides, wanting the said four right angles. QED Corol. 1. In any quadrangle, the sum of all the four inward...
...two right angles. COR. 1. All the interior angles of any rectilineal figure together with four right angles, are equal to twice as many right angles as the figure has sides. COB. 2. All the exterior angles of any rectilineal figure, made by producing the sides successively...
...as there are sides of the polygon BCDEF. Also, the angles of the polygon, together with four right angles, are equal to twice as many right angles as the figure has sides (Prop. XXVIII., BI); hence all the angles of the triangles are equal to all the angles of the polygon,...
...triangle, &c. QED COR. 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. For any rectilineal figure ABODE can be divided into as many triangles as the figure has sides, by...
...to two right angles, taken as many times, less two, as the polygon has sides (Prop. XXVI.); that is, equal to twice as many right angles as the figure has sides, wanting four right angles. Hence, the interior angles plus four right Let the sides of the polygonSection.) Section I. 1. All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides. 2. If the square described upon one side of a triangle be equal to the sum of the squares described... | 677.169 | 1 |
Category Mans71576
Maths chart on shapes
GM2-4: Identify and describe the plane shapes found in objects. As a class, brainstorm and record on the class chart, all shape and attribute language The ability to accurately identify shapes is a foundational mathematical skill, and it Feel free to give hints using the "Need to Know" section in the chart above. math vocabulary in regards to the shapes and their attributes. need to keep this chart for future conversations with students about shapes. (SMP 1,6). Part II:.
Use these colorful posters of shapes for your classroom walls, or for use in math centers! They can also be printed for individual student use or RTI! The paper size is 11 x 17 inches (tabloid size to be printed at Kinko's or another printer) but they can also be printed at a smaller size (8.5 x 11 3D Shapes (solid shapes) • First solids • Basic solids • Sphere • Cube • Cone • Cylinder • Prisms • Pyramids • Platonic solids • Faces, edges, vertices, nets • Views, cross-sections • Surface area and volume Lines Angles • Angles • Angle Pairs • Measuring angles Position • First position words • Coordinates • Compass points 2D means 2 Dimensional, and includes shapes like triangles, squares, rectangles, circles and more! Here we show the moost common 2D shapes. Show Ads. Hide Ads About Ads. 2D Shapes. Regular Polygons. A polygon is a plane (2D) shape with straight sides. HelpingWithMath.com provides free, printable resources for parents who want to tutor their children with math. There are worksheets, tables, charts, number lines, games, and more. Formula chart for Geometry. Use these Geometry formulas to calculate perimeter, area, base area, lateral area, and surface area for various Geometric shapes along with the distance formula, and equation of a circle. Over 70 formulas included. The geometry formula sheet is also available for download.
Geoboards. Children make shapes on the geoboard with rubberbands. geoboard math. Geoboard Cards. Children duplicate the geometric design on one of
math vocabulary in regards to the shapes and their attributes. need to keep this chart for future conversations with students about shapes. (SMP 1,6). Part II:. Tons of fun math activities included and a FREE pattern block symmetry activity! Create a detailed anchor chart for each 2D shape you are teaching. Draw the A printable chart showing types of shapes including squares, rectangles, triangles, parallelograms, trapezoids, rhombi, pentagons, hexagons, and octagons. Level A.2: Sort, describe and name 3-D shapes including cube, cuboid, sphere and chart' (a large piece of card or paper on which the teacher acts as a scribe to commonly used in infant maths activities (small plastic teddy bears, transport 15 Mar 2012 We generally do one of these charts per day (it only takes about five minutes and can usually be done at the beginning of math or during calendar)
If you like, you could play this with a friend, taking turns to draw shapes. If you can 't draw a shape, pass and see whether your partner can, the winner is the one ^ Kendall, D.G. (1984). "Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces". Bulletin of the London Mathematical Society. 16 (2): 81–121. doi:
4 Jan 2018 The math student measured each side of the nonagon until he had measurements for all nine edges. Octagon. The sectional shape is a quarter of In this lesson you will learn how to describe the distribution of data by analyzing the shape of a graph. | 677.169 | 1 |
finding angles of a triangle with only sides
Finding Missing Angles In Triangles Worksheet Pdf Grade 8 – Triangles are one of the most fundamental patterns in geometry. Understanding triangles is crucial to understanding more advanced geometric principles. In this blog post this post, we'll go over the various types of triangles Triangle angles, how to calculate the dimension and perimeter of the triangle, and give details of the various. Types of Triangles There are three types of triangulars: Equilateral isosceles, as well as … Read more | 677.169 | 1 |
Determine if the given conjecture is true or not. Give a counterexample if it is false. Given: ∠𝑃 𝑎𝑛𝑑 ∠𝑄 are complementary. ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary. Conjecture: ∠𝑃 ≅ ∠R
The correct answer is: As ∠𝑃 and ∠R have equal measures. Hence, the given conjecture i.e. " ∠𝑃 ≅ ∠R " is true.
Hint: Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture. Counterexample: It is an example which shows that the conjecture is false. Solution It is given that ∠𝑃 𝑎𝑛𝑑 ∠𝑄 are complementary. ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary. Draw the figure which shows that ∠𝑃 𝑎𝑛𝑑 ∠𝑄 and also in which ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary. In this figure ∠𝑃 + ∠𝑄 = 90o …..(1) and ∠Q + ∠R = 90o …..(2) Equating (1) and (2) ∠𝑃 + ∠𝑄 = ∠Q + ∠R ∠𝑃 = ∠R We know that two angles are congruent when they have equal measures. So ∠𝑃 ≅ ∠R. Final Answer: As ∠𝑃 and ∠R have equal measures. Hence, the given conjecture i.e. " ∠𝑃 ≅ ∠R " is true | 677.169 | 1 |
Engage NY Eureka Math 4th Grade Module 4 Lesson 1 Answer Key
Eureka Math Grade 4 Module 4 Lesson 1 Problem Set Answer Key
Question 1A C}\). e. Draw a point not on \(\overline{A B}\) or \(\overline{A C}\). Call it D. f. Construct g. Use the points you've already labeled to name one angle. ____________ Answer: The labeled angle is <ACD/<BAC.
Explanation: Here, we have to draw two points and then labeled them as A and B. And used a straightedge to draw \(\overline{A B}\). Then we have to draw a new point that is not on \(\overline{A B}\). And we will label it as C. Then we will draw \(\overline{A C}\). Then we will draw a point not on \(\overline{A B}\) or \(\overline{A C}\). And we will label it as D. And then we will construct We will use these points and we will label with one angle as <ACD/<BAC.
Question 2B C}\). e. Draw a new point that is not on \(\overline{A B}\) or \(\overline{A C}\) Label it D. f. Construct . g. Identify ∠DAB by drawing an arc to indicate the position of the angle. h. Identify another angle by referencing points that you have already drawn. _____________ Answer: The labeled angle is <ABC.
Explanation: Here, we have to draw two points and label them as A and B. And use a straightedge to draw \(\overline{A B}\). Then we have to draw a new point that is not on \(\overline{A B}\). Label it C. Then we will draw \(\overline{B C}\). And we will draw a new point that is not on \(\overline{A B}\) or \(\overline{A C}\) Label it D. So we will construct .
Then we have identified ∠DAB by drawing an arc to indicate the position of the angle. And we have Identified another angle by referencing points that we have already drawn show lines and rays. Extension: Draw a familiar figure. Label it with points, and then identify rays, lines, line segments, and angles as applicable. Answer: The ray of the house is marked as AB, The ray of the flash drive is marked as CD, The ray of the direction compass is marked as EF. The Line of the house is marked as AB, The Line of the flash drive is marked as CD, The Line of the direction compass is marked as EF. The Line segment of the house is marked as GH, The Line segment of the flash drive is marked as IJ, The Line segment of the direction compass is marked as EK. The Angle of the house is marked as <HGA, The Angle of the flash drive is marked as <CIJ, The Angle of the direction compass is marked as <KEF.
Explanation:
Ray: A Ray can be defined as a part of the line which has a fixed starting point but does not have any endpoint and it can be extended infinitely in one direction and a ray may pass through more than one point. The ray of the house is marked as AB, The ray of the flash drive is marked as CD, The ray of the direction compass is marked as EF. Line: A line can be defined as a long, straight and continuous path which is represented using arrowheads at both directions and the lines that do not have a fixed point it can be extended in two directions. The Line of the house is marked as AB, The Line of the flash drive is marked as CD, The Line of the compass rose is marked as EF. Line segment: A line segment is a straight line that passes through the two points and has fixed point and it can not be extended. The Line segment of the house is marked as GH, The Line segment of the flash drive is marked as IJ, The Line segment of the compass rose is marked as EK. Angle: A figure which is formed by two rays or lines that shares a common endpoint and is called an angle. So in the above, we can see the house, flash drive, and a compass rose. The Angle of the house is marked as <HGA, The Angle of the flash drive is marked as <CIJ, The Angle of the compass rose is marked as <KEF.
Eureka Math Grade 4 Module 4 Lesson 1 Exit Ticket Answer Key
Question 1. Draw a line segment to connect the word to its picture. Answer: Ray: A Ray can be defined as a part of the line which has a fixed starting point but does not have any endpoint and it can be extended infinitely in one direction and a ray may pass through more than one point. Line: A line can be defined as a long, straight and continuous path which is represented using arrowheads at both directions and the lines that do not have a fixed point it can be extended in two directions. Line segment: A line segment is a straight line that passes through the two points and has fixed point and it can not be extended. Point: A point is an exact location and it has no point only position and point can usually be named often with letters like A, B, etc. Angle: A figure which is formed by two rays or lines that shares a common endpoint and is called an angle.
Explanation:
Question 2. How is a line different from a line segment? Answer: A line can be defined as a long, straight and continuous path which is represented using arrowheads in both directions and the lines do not have a fixed point it can be extended in two directions, but the line segment is a straight line that passes through the two points and have a fixed point and it can not be extended.
Eureka Math Grade 4 Module 4 Lesson 1 Homework Answer Key
Question 1 point not on \(\overline{W X}\) or \(\overline{W Y}\). Call it Z. f. Construct g. Use the points you've already labeled to name one angle. ____________ Answer: The labeled angle is <XWY.
Explanation: Here, we will draw two points which are represented with W and X. And we will use a straightedge to draw \(\overline{W X}\). And then we will draw a new point that is not on \(\overline{W X}\). And we will label it as Y. Then we will draw \(\overline{W X}\). Next we will draw a point not on \(\overline{W X}\) or \(\overline{W Y}\). And we will call it Z. And then we will construct Then we will use these points which were already labeled and name the angle as <XWY.
Question 2 new point that is not on \(\overline{W Y}\) or on the line containing \(\overline{W X}\). Label it Z. f. Construct . g. Identify ∠ZWX by drawing an arc to indicate the position of the angle. h. Identify another angle by referencing points that you have already drawn. _____________ Answer: The identified angle is <XW.
Explanation: Here, we will draw two points and will label them as W and X. And we will use a straightedge to draw \(\overline{W X}\). Now we need to draw a new point that is not on \(\overline{W X}\). Label it Y. Then we will draw \(\overline{W Y}\). Next, we will draw a new point that is not on \(\overline{W Y}\) or on the line containing \(\overline{W X}\) and Label it as Z. Then we need to Construct . And then we will identify the angle ∠ZWX by drawing an arc to indicate the position of the angle. So we will Identify another angle by referencing points that we have already drawn and label the angle as <XW show lines and rays. Extension: Draw a familiar figure. Label it with points, and then identify rays, lines, line segments, and angles as applicable. Answer: The ray of the clock is marked as AB, The ray of the die is marked as EF, The ray of the number line is marked as AC. The Line of the clock is marked as BC, The Line of the die is marked as FG, The Line of the number line is marked as AB. The Line segment of the clock is marked as AC, The Line segment of the die is marked as GH, The Line segment of the number line is marked as BD. The Angle of the clock is marked as <CAB, The Angle of the die is marked as <FGH The Angle of the number line is marked as <CAB. | 677.169 | 1 |
Students will practice working with the Pythagorean Theorem with this Scavenger Hunt activity. This includes:
Find a missing leg or hypotenuse of a right triangle using the Pythagorean Theorem. Both basic and complex problems that require more than one step included.
Use the Pythagorean Theorem Converse to determine if a triangle is acute, right, or obtuse.
Solve word problems using the Pythagorean Theorem resource in a class that is behind in their math knowledge to failing an algebra 1 section. I added info to a few of the pictures due to lack of knowledge about trapezoids. The mix of easy and medium level was nice.
—TAMMY B.
Students loved moving around the room and working together! Impressive scaffolding of problems related to Pythagorean Theorem
—ROSHNI G.
I loved using this activity! It is very hard to find resources that include word-problems, basic right triangles, and challenge problems all in one activity; however, this scavenger hunt does that! I had students go in alphabetical order instead of following the "previous answer" in my 8th grade class and had students do the scavenger hunt in my Algebra I class, which I found very effective for their learning levels. | 677.169 | 1 |
Three of the edges of a cube are \overline{A B}, \overline{B C}, and \overline{C D}, and \overline{A D} is an interior diagonal. Points P, Q, and R are on \overline{A B}, \overline{B C}, and \overline{C D}, respectively, so that A P=5, P B=15, B Q=15, and C R=10. What is the area of the polygon that is the intersection of plane P Q R and the cube? | 677.169 | 1 |
Question 1.
Is it possible to have a triangle with the following sides?
(i) 2 cm, 3 cm, 5 cm
(ii) 3 cm, 6 cm, 7 cm
(iii) 6 cm, 3 cm, 2 cm
Solution:
In a triangle, the sum of the lengths of either two sides is always greater than the third side.
(i) Given that, the sides of the triangle are 2 cm, 3 cm, 5 cm.
It can be observed that,
(2 + 3) cm = 5 cm
However, 5 cm = 5 cm
Hence, this triangle is not possible.
Question 2.
Take any point 0 in the interior of a triangle PQR. Is
(i) OP + OQ > PQ?
(ii) OQ + OR > QR?
(iii) OR + OP > RP?
Solution:
If O is a point in the interior of a given triangle, then three triangles ∆OPQ, ∆OQR and ∆ORP can be constructed. In a triangle, the sum of the lengths of either two sides is always greater than the third side.
(i) Yes, as OQR is a triangle with sides OR, OQ and QR.
∴ OQ + OR> QR
(ii) Yes, as OQR is a triangle sides OR, OQ and QR.
∴ OQ + OR > QR
(iii) Yes, as A ORP is a triangle with sides OR, OP and PR.
∴ OR + OP > PR
Question 6.
The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?
Solution:
In a triangle, the Sum of the lengths of either two side is always greater than the third side.
Lengths of two sides of a triangle are 12 cm and 15 cm.
Let the third side be x cm.
∴ x + 12 > 15 ⇒ x > 3
x + 15 > 12 ⇒ x > – 3 but side length never be negative.
and 12 + 15 > x ⇒ 27 > x
Hence, third side can measure between 3 and 27. | 677.169 | 1 |
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Geometry Worksheets For Grade 4
In fourth grade, students deepen their understanding of geometry, exploring more complex shapes, angles, and spatial relationships. These comprehensive geometry worksheets challenge students to apply their knowledge to solve real-world problems, fostering critical thinking and problem-solving skills essential for future academic and career success.
Learn the Various Types of Triangles Worksheet
Learn to Draw Angles on a Straight Line Worksheet
Learning Geometrical Vocabulary Worksheet
By working through these engaging fourth-grade geometry worksheets, students will gain a robust foundation in geometric principles, empowering them to tackle increasingly sophisticated mathematical concepts and apply their knowledge in diverse academic and professional fields. | 677.169 | 1 |
two column triangle proofs worksheet
Double Triangle Proofs Worksheet – Triangles are one of the most fundamental forms in geometry. Understanding triangles is important for developing more advanced geometric ideas. In this blog post we will discuss the different types of triangles, triangle angles, how to determine the perimeter and area of a triangle, as well as provide an example of every. Types of Triangles There are three types in triangles, namely equilateral, isosceles, and scalene. Equilateral triangles contain three equal sides and … Read more | 677.169 | 1 |
For a given set of points in the 3D space or in a 2D space, can two different triangulation that conform to the Delanuay rules of empty circumcirle be created? If yes what would it depend on, the starting point? | 677.169 | 1 |
Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...
Prop. xxxI. This proposition is the general case of Prop. 47, Book 1, for any similar rectilineal figure described on the sides of a right-angled triangle. The demonstration, however, here given is wholly independent
of Euc. 1. 47.
Prop. XXXIII. In the demonstration of this important proposition, angles greater than two right angles are employed, in accordance with the criterion of proportionality laid down in Euc. v. def. 5.
This proposition forms the basis of the assumption of ares of circles for the measures of angles at their centers. One magnitude may be assumed as the measure of another magnitude of a different kind, when the two are so connected, that any variation in them takes place simultaneously, and in the same direct proportion. This being the case with angles at the center of a circle, and the arcs subtended by them; the arcs of circles can be assumed as the measures of the angles they subtend at the center of the circle.
Prop. B. The converse of this proposition does not hold good when the triangle is isosceles.
QUESTIONS ON BOOK VI.
1. DISTINGUISH between similar figures and equal figures.
2. What is the distinction between homologous sides, and equal sides in Geometrical figures?
3. What is the number of conditions requisite to determine similarity of figures? Is the number of conditions in Euclid's definition of similar figures greater than what is necessary? Propose a definition of similar figures which includes no superfluous condition.
4. Explain how Euclid makes use of the definition of proportion in Euc. vi. 1.
5. Prove that triangles on the same base are to one another as their altitudes.
6. If two triangles of the same altitude have their bases unequal, and if one of them be divided into m equal parts, and if the other contain of those parts; prove that the triangles have the same numerical relation as their bases. Why is this Proposition less general than Eue. vi. 1?
7. Are triangles which have one angle of one equal to one angle of another, and the sides about two other angles proportionals, necessarily
similar?
8. What are the conditions, considered by Euclid, under which two triangles are similar to each other?
9. Apply Euc. vI. 2, to trisect the diagonal of a parallelogram.
10. When are three lines said to be in harmonical proportion? If both the interior and exterior angles at the vertex of a triangle (Euc vi. 3, A.) be bisected by lines which meet the base, and the base produced, in D, G; the segments BG, GD, GC of the base shall be in Harmonical proportion.
11. If the angles at the base of the triangle in the figure Euc. vi. A, be equal to each other, how is the proposition modified?
12. Under what circumstances will the bisecting line in the fig. Euc. VI. A, meet the base on the side of the angle bisected? Shew that there is an indeterminate case.
13. State some of the uses to which Euc. vi. 4, may be applied. 14. Apply Euc. vi. 4, to prove that the rectangle contained by the segments of any chord passing through a given point within a circle is
constant.
15. Point out clearly the difference in the proofs of the two latter cases in Euc. vi. 7.
16. From the corollary of Euc. vr. 8, deduce a proof of Euc. 1. 47. 17. Shew how the last two properties stated in Euc. vi. 8. Cor. may be deduced from Euc. 1. 47; II. 2; vI. 17.
18. Given the nth part of a straight line, find by a Geometrical construction, the (n + 1)th part.
19. Define what is meant by a mean proportional between two given lines and find a mean proportional between the lines whose lengths are 4 and 9 units respectively. Is the method you employ suggested by any Propositions in any of the first four books?
20. Determine a third proportional to two lines of 5 and 7 units: and a fourth proportional to three lines of 5, 7, 9, units.
21. Find a straight line which shall have to a given straight line, the ratio of 1 to 5.
23. Give the corollary, Euc. vI. 8, and prove thence that the Arithmetic mean is greater than the Geometric between the same extremes. 24. If two equal triangles have two angles together equal to two right angles, the sides about those angles are reciprocally proportional. 25. Give Algebraical proofs of Prop. 16 and 17 of Book vi.
26. Enunciate and prove the converse of Euc. vi. 15.
27. Explain what is meant by saying, that "similar triangles are in the duplicate ratio of their homologous sides."
28. What are the data which determine triangles both in species and magnitude? How are those data expressed in Geometry?
29. If the ratio of the homologous sides of two triangles be as 1 to 4, what is the ratio of the triangles? And if the ratio of the triangles be as 1 to 4, what is the ratio of the homologous sides?
30. Shew that one of the triangles in the figure, Euc. Iv. 10, is a mean proportional between the other two.
31.
What is the algebraical interpretation of Euc. vi. 19?
32. From your definition of Proportion, prove that the diagonals of a square are in the same proportion as their sides.
33. What propositions does Euclid prove respecting similar polygons? 34. The parallelograms about the diameter of a parallelogram are similar to the whole and to one another. Shew when they are equal.
35. Prove Algebraically, that the areas (1) of similar triangles and (2) of similar parallelograms are proportional to the squares of their homologous sides.
36. How is it shewn that equiangular parallelograms have to one another the ratio which is compounded of the ratios of their bases and altitudes ?
37. To find two lines which shall have to each other, the ratio compounded of the ratios of the lines A to B, and C to D.
38. State the force of the condition "similarly described;" and shew that, on a given straight line, there may be described as many polygons of different magnitudes, similar to a given polygon, as there are sides of different lengths in the polygon.
39. Describe a triangle similar to a given triangle, and having its area double that of the given triangle.
40. The three sides of a triangle are 7, 8, 9 units respectively; determine the length of the lines which meeting the base, and the base produced, bisect the interior angle opposite to the greatest side of the triangle, and the adjacent exterior angle.
41. The three sides of a triangle are 3, 4, 5 inches respectively; find the lengths of the external segments of the sides determined by the lines which bisect the exterior angles of the triangle.
42. What are the segments into which the hypotenuse of a rightangled triangle is divided by a perpendicular drawn from the right angle, if the sides containing it are a and 3a units respectively?
43. If the three sides of a triangle be 3, 4, 5 units respectively: what are the parts into which they are divided by the lines which bisect the angles opposite to them?
44. If the homologous sides of two triangles be as 3 to 4, and the area of one triangle be known to contain 100 square units; how many square units are contained in the area of the other triangle?
45. Prove that if BD be taken in AB produced (fig. Euc. vi. 30) equal to the greater segment AC, then AD is divided in extreme and mean ratio in the point B.
Shew also, that in the series 1, 1, 2, 3, 5, 8, &c. in which each term is the sum of the two preceding terms, the last two terms perpetually approach to the proportion of the segments of a line divided in extreme and mean ratio. Find a general expression (free from surds) for the nth term of this series.
46. The parts of a line divided in extreme and mean ratio are incommensurable with each other.
47. Shew that in Euclid's figure (Euc. I. 11.) four other lines, besides the given line, are divided in the required manner.
48. Enunciate Euc. vi. 31. What theorem of a previous book is included in this proposition?
49. What is the superior limit, as to magnitude, of the angle at the circumference in Euc. vi. 33? Shew that the proof may be extended by withdrawing the usually supposed restriction as to angular magnitude; and then deduce, as a corollary, the proposition respecting the magnitudes of angles in segments greater than, equal to, or less than a semicircle. 50. The sides of a triangle inscribed in a circle are a, b, c, units respectively: find by Euc. vi. c, the radius of the circumscribing circle. 51.
Enunciate the converse of Euc. vi. D.
52. Shew independently that Euc. VI. D, is true when the quadrilateral figure is rectangular.
53. Shew that the rectangles contained by the opposite sides of a quadrilateral figure which does not admit of having a circle described about it, are together greater than the rectangle contained by the diagonals.
54. What different conditions may be stated as essential to the possibility of the inscription and circumscription of a circle in and about a quadrilateral figure?
55. Point out those propositions in the Sixth Book in which Euclid's definition of proportion is directly applied.
56. Explain briefly the advantages gained by the application of analysis to the solution of Geometrical Problems.
57. In what cases are triangles proved to be equal in Euclid, and in what cases are they proved to be similar?
To inscribe a square in a given triangle.
Analysis. Let ABC be the given triangle, of which the base B and the perpendicular AD are given.
Let FGHK be the required inscribed square. Then BHG, BDA are similar triangles, and GH is to GB, as AD is to AB, but GF is equal to GH;
therefore GF is to GB, as AD is to AB.
Let BF be joined and produced to meet a line drawn from A parallel to the base BC in the point E.
Synthesis.
Then the triangles BGF, BAE are similar,
and AE is to AB, as GF is to GB,
but GF is to GB, as AD is to AB;
wherefore AE is to AB, as AD is to AB;
hence AE is equal to AD.
Through the vertex A, draw AE parallel to BC the
base of the triangle,
make AE equal to AD,
join EB cutting AC in F,
through F, draw FG parallel to BC, and FK parallel to AD; also through G draw GH parallel to AD. Then GHKF is the square required.
The different cases may be considered when the triangle is equilateral, scalene, or isosceles, and when each side is taken as the base.
PROPOSITION II. THEOREM.
If from the extremities of any diameter of a given circle, perpendiculars be drawn to any chord of the circle, they shall meet the chord, or the chord produced in two points which are equidistant from the center.
First, let the chord CD intersect the diameter AB in Z, but not at right angles; and from A, B, let AE, BF be drawn perpendicular to CD. Then the points F, E are equidistant from the center of the chord CD.
Join EB, and from I the center of the circle, draw IG perpendicular to CD, and produce it to meet EB in H.
Then IG bisects CD in G; (III. 2.)
and IG, AE being both perpendicular to CD, are parallel. (1. 29.) Therefore BI is to BH, as IA is to HE; (VI. 2.)
and BH is to FG, as HE is to GE; therefore BI is to FG, as IA is to GE; but BI is equal to IA;
therefore FG is equal to GE.
It is also manifest that DE is equal to CF.
When the chord does not intersect the diameter, the perpendiculars intersect the chord produced.
PROPOSITION III. THEOREM.
If two diagonals of a regular pentagon be drawn to cut one another, the greater segments will be equal to the side of the pentagon, and the diagonals will cut one another in extreme and mean ratio.
Let the diagonals AC, BE be drawn from the extremities of the side AB of the regular pentagon ABCDE, and intersect each other in the point H.
Then BE and AC are cut in extreme and mean ratio in H, and the greater segment of each is equal to the side of the pentagon. Let the circle ABCDE be described about the pentagon. (IV. 14.) Because EA, AB are equal to AB, BC, and they contain equal angles;
therefore the base EB is equal to the base AC, (1. 4.)
and the triangle EAB is equal to the triangle CBA, and the remaining angles will be equal to the remaining angles, each to each, to which the equal sides are opposite.
Therefore the angle BAC is equal to the angle ABE; and the angle AHE is double of the angle BAH, (1. 32.) but the angle EAC is also double of the angle BAC, (vI. 33.) therefore the angle HAE is equal to AHE,
and consequently HF is equal to EA, (1. 6.) or to AB. And because BA is equal to AE, | 677.169 | 1 |
Interior & Exterior Angles of Polygons Boom Cards™
Description: This is a fun and interactive way to practice finding sums of interior and exterior angle measures of polygons, as well as the measure of a single interior/exterior angle measure in a regular polygon. Students will also have to solve for the number of sides in a polygon given the measure of the sum of its interior angles (or a single interior/exterior angle in a regular polygon). Some problems will require students to solve for x, given expressions for angle measures in polygons. Boom Cards are NO PREP required and SELF-CHECKING. I use them with all of my classes and my students LOVE them! | 677.169 | 1 |
ORANGE PUBLIC SCHOOLS OFFICE OF CURRICULUM AND INSTRUCTION OFFICE OF MATHEMATICS
GEOMETRY Pre - Assessment
School Year 2013-2014
Directions for Geometry Pre-Assessment The Geometry Pre-Assessment is made up of two sections. Section 1 is made up of 15 short response, 10 multiple choice, and 4 extended response questions. Section 2 is made up of one long task that is split up into multiple parts. Read each question carefully, including diagrams and graphs. Work as rapidly as you can without sacrificing accuracy. Do not spend too much time puzzling over a question that seems too difficult for you. Answer the easier questions first; then return to the harder ones. Try to answer every question, even if you have to guess. Where necessary, you may use scratch paper for your work. Do not use the margins of the test booklet to do scratch work. Record all answers in this test booklet. When necessary, be sure to provide all work and explanations in a clear and neat manner. You may use a calculator for this test.
Two quadrilaterals are shown in the coordinate plane below. Quadrilateral ABCD was dilated with a scale factor of 2 with the center at the origin and then rotated 180 about the origin to get the quadrilateral in Quadrant IV.
Part A
Label the image quadrilateral in Quadrant IV using W, X, Y, and Z.
Part B
Write a sentence that describes the relationship between the two quadrilaterals using the word "congruent" or the word "similar."
5
5.
A rectangle has a width of 3.5 inches and a length of x inches. If the diagonal of the rectangle is 12.5 inches, what is the value of x ? Give the exact solution or round your answer to the nearest tenth.
6.
Starting from the entrance of her school, Alyssa walked 400 feet due north, then 300 feet due east, and ended up at the entrance of a running track. Miki walked directly from the entrance of the school to the entrance of the running track. How many more feet did Alyssa walk than Miki?
7.
The figure above shows a movie screen with dimensions shown in feet. What is the length of the diagonal of the screen, in feet? Give the exact solution or round your answer to the nearest tenth of a foot.
6
8.
9.
A room is in the shape of a rectangular prism. The room has a width of 8 feet, a length of 12 feet, and a height of 9 feet. What is the greatest possible distance between two points on the walls in the room? Give the exact solution or round your answer to the nearest foot.
In the coordinate plane, what is the distance between the points 2, 3 and 4, 0 ? Give the exact solution or round your answer to the nearest tenth.
10.
In the coordinate plane above, what is the length of segment AB ? Give the exact solution or round your answer to the nearest tenth.
7
11.
In the coordinate plane above, what is the perimeter of parallelogram WXYZ ? Give the exact solution or round your answer to the nearest tenth.
12.
The cone in the figure above has a volume of 72 cubic centimeters and a height of 6 centimeters. What is the radius of the base of the cone, in centimeters?
13.
The diameter of a spherical basketball is 10 inches. What is the volume of the basketball? Give the exact solution or round your answer to the nearest cubic inch.
8
14.
Parallel lines and n (not shown) were each translated. Could lines t and u shown in the coordinate plane below be the image of lines and n after translation? Explain your reasoning.
15.
Describe a sequence of transformations that can be used to show that triangle ABC is congruent to triangle XYZ.
9
Multiple Choice Questions 16.
Angle ABC in the coordinate plane below will be rotated 90 degrees counterclockwise about the origin. What are the coordinates of the image of point A ?
Which of the following transformations shows that triangle ABC is congruent to triangle XYZ ? a. Triangle ABC is translated 7 units to the right and 2 units down. b. Triangle ABC is translated 14 units to the right and 2 units down. c. Triangle ABC is reflected over the x-axis and translated 2 units down. d. Triangle ABC is reflected over the y-axis and translated 2 units down. 10
The circle shown in the coordinate plane below is the preimage under a dilation centered at the 12
origin with scale factor 2. Which of the following points is NOT on the image of the dilation?
a. b. c. d.
21.
6, 6 0, 0 0, 6 6, 6
Quadrilateral ABCD, shown in the coordinate plane below, is dilated with the center at the origin 13
to form quadrilateral EFGH. What is the scale factor of the dilation?
a. b.
22.
1 4 1 3
c.
3
d.
4
Quadrilaterals JKLM and WXYZ are shown in the coordinate plane below. Quadrilateral WXYZ is the image of quadrilateral JKLM under a transformation. Which of the following best 14
describes the transformation?
a. b.
23.
2 . 3 3 A dilation with center 0, 0 with a scale factor of . 2
A dilation with center 0, 0 with a scale factor of
c.
A translation 2 units to the left and 3 units down.
d.
A translation 2 units to the right and 3 units up.
Triangles ABC, DEF , JKL, and PQR can be placed in the coordinate plane below and are related to each other in the following manner. 15
Triangle ABC is reflected over the x-axis to get triangle DEF. Triangle DEF is translated 6 units to the right and 4 units down to get triangle JKL. A transformation is applied to triangle JKL to get triangle PQR. Triangle PQR is similar to triangle ABC but NOT congruent to triangle ABC.
Which of the following could describe the transformation applied to triangle JKL to get triangle PQR ?
a. b.
Triangle JKL is rotated 90 counterclockwise about the origin to get PQR. Triangle JKL is rotated 180 about the origin to get PQR.
c.
Triangle JKL is dilated with a scale factor of 1 with the center at the origin to get PQR.
d.
24.
Triangle JKL is dilated with a scale factor of 5 with the center at 3, 4 to get PQR.
Which of the following expressions can be used to represent the distance between the points 2, 4 and 5, 3 in the coordinate plane? 16
a.
2 4
5 3
b.
2 4
2
5 3
c.
2 5
2
4 3
d.
2 5
4 3
2
2
2
2
2
2
25.
The cylindrical jar above has a height of 12 centimeters and a radius of 4 centimeters. The jar contains 20 spherical marbles. Each marble has a radius of 0.8 centimeters. Which of the following represents the amount of space in the jar that is NOT occupied by marbles? a. b. c. d.
Are triangles ABC and XYZ similar? Justify your answer using one or more transformations.
18
27.
Based on the figure below, determine whether each given statement must be true, and briefly explain why.
Statement
Must the statement be true? (Yes or No)
Explain why.
Line is parallel to line m.
Line t is parallel to line u.
x 110
y 70
19
28.
In the figure below, does a
120 ? Explain your answer.
29.
Jared drew the figure above in order to prove the Pythagorean theorem. The figure consists of 9 congruent right triangles that do not overlap. Part A: Explain why Jared's drawing as labeled CANNOT be used to prove the Pythagorean theorem.
Part B: What equation can you conclude to be true based on Jared's drawing?
In triangle PQR above, w2 x2 y 2. Triangle STU is a right triangle, as shown. The table below shows a partially completed proof that triangle PQR is a right triangle. Complete the table to describe the reasons for each statement in the proof.
Statement
Reason
1) w2 x2 y 2
1) Given.
2) w2 x2 z2
2)
3) y 2 z 2
3)
4) y z
4)
5) Triangle PQR is congruent with triangle STU.
5)
6) Angle Q is congruent with angle T.
6)
7) mQ 90.
7)
8) Triangle PQR is a right triangle.
8)
23
Part C: A student made this conjecture about reflections on an xy-coordinate plane.
When a polygon is reflected over the y-axis, the x-coordinates of the corresponding vertices of the polygon and its image are opposite, but the ycoordinates are the same. Develop a chain of reasoning to justify or disprove the conjecture. You must demonstrate that the conjecture is always true or that there is at least one counterexample in which the conjecture is not true. You may include one or more graphs in your response. | 677.169 | 1 |
Apollonius's Theorem
Apollonius's theorem is an elementary geometry theorem relating the length of a median of a triangle to the lengths of its sides. While most of the world refers to it as it is, in East Asia, the theorem is usually referred to as Pappus's theorem or midpoint theorem. It can be proved by Pythagorean theorem from the cosine rule as well as by vectors. The theorem is named after a Greek mathematician Apollonius of Perga.
Relations with Other Theorems
Substitute \(\overline{BP}=\overline{CP}\) into Stewart's theorem, \(\overline{CP}\cdot\overline{AB}^2+\overline{BP}\cdot\overline{AC}^2=\left(\overline{BP}+\overline{CP}\right)\left(\overline{AP}^2+\overline{BP}\cdot\overline{CP}\right)\), and you get
Examples
Triangle \(ABC\) has side lengths \(\overline{AB}=2\sqrt{3}\) and \(\overline{BC}=2,\) as shown. \(D\) is the midpoint of \(\overline{BC},\) satisfying \(\overline{AD}=\sqrt{7}.\)
Let \(E\) be the point of intersection between \(\overline{AB}\) and the bisector of \(\angle ACB.\) \(\overline{CE}\) meets with \(\overline{AD}\) at point \(P,\) and the bisector of \(\angle APE\) meets with \(\overline{AB}\) at point \(R.\) An extension of \(\overline{PR}\) meets with \(\overline{BC}\) at point \(Q.\)
Given that the area of \(\triangle PQC\) is \(a+b\sqrt{7}\) times larger than that of \(\triangle PRE\) for some rational numbers \(a\) and \(b,\) find the value of \(ab.\)
This problem is a part of <Grade 10 CSAT Mock Test> series.
Two sides of a triangle are \(10\) and \(5\) units in length, and the length of the median to the third side is \(6.5\) units. The area of the triangle is \(6\sqrt{x}\) units squared. Determine the value of \(x\).
This question has appeared in NMTC.
Triangle \(ABC\) with its centroid at \(G\) has side lengths \(AB=15, BC=18,AC=25\). \(D\) is the midpoint of \(BC\).
The length of \(GD\) can be expressed as \( \frac{ a \sqrt{d} } { b} \), where \(a\) and \(b\) are coprime positive integers and \(d\) is a square-free positive integer. | 677.169 | 1 |
One of the angle of a parallelogram is 45 of its adjacent angle. Find the measure of both angles.
Video Solution
Text Solution
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Step by step video, text & image solution for One of the angle of a parallelogram is 4/(5) of its adjacent angle. Find the measure of both angles. by Maths experts to help you in doubts & scoring excellent marks in Class 14 exams. | 677.169 | 1 |
Euclid's Elements [book 1-6] with corrections, by J.R. Young
the circle, as AC; and draw DC to the point C, where it meets the circumference. And because DA is equalt to DC, the angle DAC is equal* to the angle ACD: but DAC is a right angle; therefore ACD is a right angle; and therefore the angles DAC, ACD are equal
+Hyp.
*17. 1.
to two right angles; which is impossible: therefore the straight line drawn from A at right angles to BA does not fall within the circle. In the same manner it may be demonstrated, that it does not fall upon the circumference; therefore it must fall *See fig. 2. without the circle, as AE.*
*12. 1.
*17. 1.
Also, between the straight line AE, and the circumference, no straight line can be drawn from the point A which does not cut the circle. For, if possible, let AF be between them: from the point D draw* DG perpendicular to AF, and let it meet the circumference in H. And because AGD is a right angle, and DAG less than a right angle, DA is greater* +12 Def. than DG: but DA is equal to DH; therefore DH is greater than DG, a part than the whole, which is impossible. Therefore no straight line can be drawn from the point A, between AE and the circumference, which does not cut the circle.
*19. 1.
FE
H
A
COR. From this it is manifest, that the straight line which is drawn at right angles to the diameter of a circle from the t1 Def. 3. extremity of it, touchest the circle; and that it touches it only in one point, because, if it did meet
the circle in two, it would fall within it. Also, it *2. 3. is evident, that there can be but one straight line
which touches the circle in the same point.
PROP. XVII. PROB.
To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.
First, let A be a given point without the given circle BCD; it is required to draw a straight line from A which shall touch the circle.
*1. 3.
#11. 1
Find the centre E of the circle, and draw AE cutting the circle in D; and from the centre E, at the distance EA, describe the circle AFG; from the point D draw* DF at right angles to EA, and draw EBF, AB. AB shall touch the circle BCD. Because E is the centre of
the circles BCD, AFG, EA is +12 Def. 1. equal to EF, and ED to EB; therefore the two sides AE, EB, are equal to the two FE, ED, each to each; and they contain the angle at E common to the two triangles AEB,
#4. 1.
(CE
FED; therefore the base DF is equal to the base AB, and the triangle FED to the triangle AEB, and the other angles to the other angles: therefore the angle EBA is equal to the angle +Const. EDF: but EDF is a right† angle, wherefore EBA +1 Ax. is a right angle: and EB is drawn from the centre: but a straight line drawn from the extremity of a *Cor. 16. 3. diameter, at right angles to it, touches the circle: therefore AB touches the circle; and it is drawn from the given point A. Which was to be done.
But if the given point be in the circumference of the circle, as the point D, draw DE to the centre *Cor. 16. 3. E, and DF at right angles to DE, DF touches the circle.
It is evident that two touching lines may be drawn froin A, as the point F may fall on either side of AE.
If a straight line touch a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle.
Let the straight line DE touch the circle ABC in the point C; taket the centre F, and draw the straight line FC: FC shall be perpendicular to DE.
+1. 3.
For, if it be not, from the point F draw FBG perpendicular to DE: and because FGC is a right *17. 1. angle, GCF is* an acute angle; and to the greater *19. 1. angle the greater* side is opposite: therefore FC is +12 Def. 1. greater than FG: but FC is equalt
to FB; therefore FB is greater than FG, the less than the greater, which is impossible: therefore FG is not perpendicular to DE. In the same manner it may be shown, that no other is perpendicular to it besides FC, that is, FC is perpendicular to DE. Therefore, if a straight line, &c. D
Q. E. D.
F
B
GF
This proposition is unnecessary, and may therefore be omitted. It is obviously the converse of the first part of prop. xvi.;—that first part affirming that if a straight line from the extremity of a diameter be perpendicular to that diameter it shall touch the circle, and this 18th proposition states, conversely, that if a straight line touch a circle it will be perpendicular to the diameter from whose extremity it is drawn. But the second part of the 16th is in reality the converse of the first part; since it declares that if a straight line from the extremity of the diameter touch the circle, it can be no other than the perpendicular before mentioned.❤
* The ingenious Williamson, in his excellent edition of the Elements, is the only one, as far as I know, who has adverted to the uselessness of this proposition. He remarks, "There is something rather singular in the 18th proposition, for it seems to me to be nothing but the corollary to the 16th." The corollary, however, is, as Austin justly observes, a superfluous addition; as it is merely a re-statement of the proposition itself in other words. But this, like the other corollaries in the Elements, is supposed to have been subjoined, not by Euclid, but by some one of the early editors.
PROP. XIX. THEOR.
If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.
Let the straight line DE touch the circle ABC in C, and from C let CA be drawn at right angles to DE: the centre of the circle shall be in CA.
For, if not, let F be the centre, if possible, and join CF. Because DE touches the circle ABC, and FC is drawn from the centre to the point of contact, FC is per- B pendicular to DE: therefore FCE is a right angle: but ACE is also a right angle; therefore D
#18.3.
+Hyp.
+1 Ax.
C
E
the angle FCE is equal to the angle ACE, the less to the greater, which is impossible: therefore F is not the centre of the circle ABC. In the same manner it may be shown, that no other point which is not in CA, is the centre; that is, the centre is in CA. Therefore, if a straight line, &c. Q. E. D.
This proposition, like the preceding, is quite superfluous, and ought to be expunged from the elements, although we believe it has hitherto escaped all objection. The only thing that it proves is this, viz. that the line which is perpendicular to the diameter must have the diameter perpendicular to it.
PROP. XX. THEOR.
The angle at the centre of a circle is double of the angle at the circumference upon the same arc, that is, upon the same part of the circumference.
Let ABC be a circle, and BEC an angle at the centre, and BAC an angle at the circumference, which have the same arc BC for their base: the angle BEC shall be double of the angle BAC.
Join AE, and produce it to F. First, let the centre of the circle be within the angle BAC. Because EA is equal to EB, the angle EAB is equal to the angle EBA; therefore the angles EAB, EBA are together double of the angle EAB; but the *32. 1. angle BEF is equal to the angles EAB, EBA; therefore also the
*5. 1.
F
angle BEF is double of the angle EAB: for a like reason the angle FEC is double of the angle EAC: therefore the whole angle BEC is double of the whole angle BAC. Again, let the centre of the circle be without the angle BAC. It may be demonstrated, as in the first case, that the angle FEC is double of the angle FAC; and that FEB, a part of the first, is double of FAB, a part of the other; therefore the remaining angle BEC is double of the remaining angle BAC. Therefore the angle at the centre, &c. Q. E. D.
NOTE. It must not be inferred from this that to every angle at the circumference there corresponds an angle at the centre upon the same arc, and which is double of the former angle. Whenever the angle at the centre really stands upon the same arc, it is necessarily double that at the circumference; but if the angle at the circumference stand upon a semi-circle, then there can be no corresponding angle at the centre; and if the angle at the circumference stand on an arc still greater, then the angle at the centre, corresponding, stands not upon the same, but upon the opposite part of the circumference.
PROP. XXI. THEOR.
The angles in the same segment of a circle are equal to one another.
Let ABCD be a circle, and BAD, BED angles in the same segment BAED; these angles shall be equal to one another. | 677.169 | 1 |
4th Grade Geometry Quizzes, Questions & Answers
Are you ready to take your geometric knowledge to the next level? Dive into the world of shapes, angles, and patterns with our exciting 4th Grade Geometry Quizzes and trivia! Geometry is a fascinating branch of mathematics that helps us understand the world around us. In these quizzes, designed specifically for 4th graders, you'll embark on a journey of discovery as you explore the secrets of polygons, angles, symmetry, and more.
Each quiz is crafted to be engaging and educational, making learning geometry a fun adventure. Test your skills by identifying different types of triangles, solving puzzles involving symmetry and congruence, and even tackling geometry riddles that will put your problem-solving abilities to the test. Whether you're a student looking to ace your geometry class or a parent wanting to help your child excel in math, these quizzes are a fantastic resource. They cover key 4th-grade geometry topics in an interactive and enjoyable way, making learning a breeze.
Join us on this geometric quest, where you'll not only enhance your understanding of shapes and measurements but also develop critical thinking skills. Are you up for the challenge? Take our 4th Grade Geometry Quizzes, and let the geometric fun begin!
Top Trending Quizzes
During this year of the battle of the books seven books we have covered during the period between 2011 and 2012. How well do you remember each book we covered then? Take up the practice test below and refresh your mind by... | 677.169 | 1 |
Area of a triangle
4+
Designed for iPad
Screenshots
Description
The triangle area calculator will show you how to find and what is the area of a sided, equilateral, right-angled and isosceles triangle using different formulas.
Area of a triangle, equilateral isosceles triangle area formula calculator allows you to find an area of different types of triangles, such as equilateral, isosceles, right or scalene triangle, by different calculation formulas, like Geron's formula, length of triangle sides and angles, incircle or circumcircle radius.
What's New
26 Aug 2023
Version 1.0.2
Minor bug fixes
App Privacy
The developer, Roman Konasov | 677.169 | 1 |
Point-line distance in taxicab geometry
In the previous post I showed how ellipses are octagonal under taxicab geometry and how they degenerate into hexagons and "circular" diamonds. However, before we can talk about other shapes in taxicab geometry, we need to understand how to measure distance between a point and a line.
In the above example, the point is 4 units from the line horizontally, 3 units from the line vertically, and the closest point by Euclidean distance (perpendicular to the line) is somewhere between 3 and 4 units. The distance between the point and the line, then—the shortest distance—is 3 units. In general in taxicab geometry, the distance between a point and a line is always along the horizontal or vertical, and to see why, consider a circle growing from the given point:
The first point of the growing circle to touch the line is one of the corners, and the corners are always on the horizontal or vertical. (The exception, of course, is if the line is at a 45° angle to the axes, in which case an entire side of the circle will touch the line, but the shortest distance between the point and line will still be found at a corner touching the line.)
The illustration above is the start of a visual proof. If you want something more numerical, consider how the distance from a point to a line changes under the Manhattan distance as you slide along the rise and run of a line's slope. What I like about the visual proof is that it works for either taxicab or Euclidean geometry. A Euclidean circle grown from a point will also first touch a line at the spot closest to the circle's center. The shapes are different between Euclidean and taxicab geometry, but the visual logic is the same. I haven't worked out a rigorous numerical proof, but I suspect the logic needed for the taxicab geometry proof differs from the Euclidean proof. If you have suggestions for such a proof, or evidence that they're not really so different, feel free to email me. | 677.169 | 1 |
Incenter Investigation Benchmark
This shows the incenter, the point of concurrency of the 3 angle bisectors for a triangle. Use the distance tool to measure the distance from the point to each side by clicking the point, then the middle of the line segment.
Use the arrow tool to click and drag one vertex of the triangle so it changes the size and shape of the triangle. | 677.169 | 1 |
Question Video: Using the Perpendicular Bisector Theorem
Mathematics • Second Year of Preparatory School
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In the diagram, the line 𝐴𝐷 is the perpendicular bisector of the line 𝐵. Find the value of 𝑥.
02:19
Video Transcript
In the diagram, the line 𝐴𝐷 is the perpendicular bisector of the line 𝐵𝐶. Find the value of 𝑥.
So to solve this problem, what we're gonna use is the perpendicular bisector theorem. And this tells us that if a point is on the perpendicular bisector of a line, then it's equidistant from the ends of the segment. So if we look at that point being our vertical line in the middle of our triangle, then we can say that the distances to the left and right of this, they're gonna be equal. So as we know that the right-hand side of the point is equal to two 𝑥 plus four. And, therefore, the left-hand side from the point must also be equal to two 𝑥 plus four.
Well, now, if we take a look at our triangle, we've got two right angled triangles, if we split it into two. And we've got one on the left-hand side. And we've got one on the right-hand side. And we can see they've both got the bottom side that's the same because we've been told that in the question. Also, they share the vertical side. So this is the same. So, therefore, the diagonal sides must also be the same because our triangle is going to be congruent. And we can see that as we've shown here. But it also means that if we look to the bigger triangle, so both of them put together, that it would be an isosceles triangle.
So now that we've shown that the diagonals are equal. And we've done that because triangle 𝐴𝐵𝐷 is congruent to triangle 𝐴𝐶𝐷. And we showed that using side-angle-side because we had the bottom side, the right angle, and then the vertical shared side. What we can do is we can equate our diagonal lengths to each other to find 𝑥. So when we do that, we get three 𝑥 plus one equals five 𝑥 minus 12. So then we look at the side that's got the most 𝑥s. It's the right-hand side. So what we're gonna do is subtract three 𝑥 from each side of our equation.
And when we do that, we get one is equal to two 𝑥 minus 12. And then what we're gonna do is add 12 to each side of the equation, which gives us 13 is equal to two 𝑥. And then, finally, divide by two. And we get 6.5 is equal to 𝑥. So, therefore, we can say that the value of 𝑥 in our diagram is 6.5. | 677.169 | 1 |
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HEIGHT and DISTANCE – TRIGONOMETRY Quantitative Aptitude
Height and Distance is one the important part of Trigonometry segment in Quantitative Aptitude or Arithmetic Aptitude. Upjob.in now publishing the fully solved Question paper, Problem and Numerical with answer key for the preparation of banking examination. Today we are covering the important topic of Math for SSC, bank, and railway test. Hope following MCQ will be helpful in the preparation of your Banking examination.
Q2. The angle of elevation of a tower from a distance 50m, from its foot is 30 degree. The height of the tower is
(a) 50 root 3m (b) 50/ root 3m (c) 25 root 3m (d) 100 root 3m
Ans. (b)
Q3. A 10m long ladder is placed against a wall. It is inclined at an angle of 30 degree to the ground. The distance of the foot of the ladder from the wall is
(a) 7.32m (b) 8.26m (c) 8.66m (d) 8.16m
Ans. (c)
Q4. A kite is flying at a height of 75 m from the level ground, attach to a string inclined at 60 degree to the horizontal. The length of the string is:
(a) 50/ root 3m (b) 50 root 3m (c) 25 root 3m (d) 100 root 3m
Ans. (b)
Q5. From the top of a cliff 50m high, the angles of depression of the top and bottom of a tower are observed to be 30 degree and 60 degree respectively. What is the height of the tower?
(a) 30 m (b) 45 m (c) 60 m (d) 75 m
Ans. (c)
Q6. The angles of depression of two ships from the top of the light house are 45 and 30 degree towards east. If the ships are 200 m apart, the height of the light house is: (take root 3= 1.73)
(a) 100 m (b) 173 m (c) 200 m (d) 273 m
Ans. (d)
Q7. The height of a tree is 10 m. It is bend by the wind in such a way that its top touches the ground and makes an angle of 60 degree with the ground. At what height from the bottom did the tree get bent? (Take root 3= 1.732)
(a) 4.6 m (b) 4.8 m (c) 5.2 m (d) 5.4 m
Ans. (a)
Q8. From a point on a bridge across the river, the angles of depression of the bank on opposite sides of the river are 30 and 45 degree respectively. If the bridge is at a height of 2.5 m from the banks, find the width of the river. (Take root 3= 1.732)
(a) 5.78 m (b) 6.83 m (c) 7.24 m (d) 6.7 m
Ans. (b)
Q9. On the horizontal plane, there is a vertical tower with a flagpole on its top. At a point 9 m away from the foot of the tower, the angles of elevation of the top and bottom of the flagpole are 60 and 30 degree respectively. The height of the flagpole is:
(a) 6 root 3m (b) 5 root 3m (c) 25 m (d) 4 m
Ans. (a)
TIME AND WORK
Q1. A and B can do a piece of work in 8 hours while B alone can do it in 12 hours. Both A and B working together can finish the work in:
(a) 10 hours (b) 4 hours (c) 5 1/4 hours (d) 4 4/5 hours
Q2. To complete a work, A takes 50% more time than B. If together they take 18 days to complete the work, how much time shall B take to do it?
(a) 30 days (b) 35 days (c) 40 days (d) 45 days
Q3. Kamal can do a work in 15 days. Bimal is 50 % more efficient than Kamal. The number of days, Bimal will take to do the same piece of work, is (a) 10 days (b) 10 1/2 days (c) 12 days (d) 14 days
Q4. A and B together can complete a piece of work in 12 days, B and C can do it in 20 days and C and A can do it | 677.169 | 1 |
A Treatise on Trigonometry, Plane and Spherical: With Its Application to ...
substituting these for their equals in the preceding equation it becomes
As the angles have each the same relations to the corresponding sides of a triangle, the same formula by a proper modification will furnish the values of the angles A and c.
It may be expressed in ordinary language thus, the sine of half either angle of a triangle is equal to radius into the square root of half the sum of the three sides minus one of the adjacent sides, into half the sum minus the other adjacent side, divided by the rectangle of the adjacent sides.
To apply this to an
EXAMPLE.
Let there be three places at distances from each other respectively of 50, 60, and 70 miles. Required the angle under which two roads must depart from that which is 60 and 70 distant from the other two, in the direction of these last. 60 and 70 will be the sides of a triangle adjacent the required angle, and 50 the side opposite; then
Radius must be understood as a factor of the second member for the sake of homogeneity, since the quantity under the radical is the ratio of a surface to a surface, and therefore an abstract number.
From the tables we find the angle to be 22° 12′ 28′′. The whole angle required will be double this or
44° 24' 56'
The other two angles may be found in a similar manner; the one is 67° 07′ 18′′, the other 78° 27′ 40′′.
If R in the above formula should be made to pass (by squaring it) under the radical sign, it would be necessary to add twice 10 in order to effect the multiplication by this factor R2 before taking the square root. But as on the other hand twice 10 must be rejected for the arithmetical complements used, these two operations exactly counterbalance each other, and neither of them need be performed. By adding together the four logarithms, therefore, and dividing by 2, the same result will be obtained. The operation in the above example would be as follows:
The best mode of proceeding in the solution of a plane triangle when three sides are given, is to prepare a blank form similar to that on p. 58, by ruling four columns, the first for the arguments, and each of the other three for the trigonometrical functions of those arguments necessary to be employed in the calculation of one of the three angles. Thus,
There are other forms which have some advantages over the above, and which may be derived in an analogous manner. They are as follows.
The student may write the blank forms for these formulas as an exercise.
EXAMPLES.
1. Given in a plane triangle the three sides 120, 112 65, and 112, to find the three angles.
57° 27' Ans. 57° 58' 39' 64° 34' 21''
* Derived from (5) and (6) of Art. 72, and (1) of Art 73. A convenient form.
By dividing the formula sin A√cos A by the formula cos A=
cosa is found tan or at once dividing (3) by
=
sin A
from which (4) above may be derived; (4). In a similar manner may be found cota= 2 sec a sec a +1
2 sec a
and coseca =
sec a
-1
74. Before treating of the only remaining case in the solution of triangles, it will be convenient to demonstrate some additional general formulas which shall present certain important relations of the trigonometrical lines of two different arcs; which formulas are of frequent use in the higher analysis, are employed in the subsequent parts of the work, and will be immediately of service in deriving a formula for the last case of plane trigonometry which we have to consider.
Add together equations (3) and (7) of (Art. 70) which express the values of sin (a+b) and sin (ab) and the resulting equation, cancelling the second terms of the second members which are similar with contrary signs, is
subtracting the same equation, the second from the first,
substituting in equation (1) the values of a+b, ab, a and b in terms of p and q, that equation becomes
sin p+ sin q2 sin (p+q) cos (p − q) * (2)
Which may be translated into ordinary language thus: the sum of the sines of two arcs is equal to twice the sine of half the sum into the cosine of half the difference of those arcs.
By subtracting the latter of the same equations (3) and (7) of (Art. 69) from the former, and reducing similar terms, there results
sin (a + b) — sin (a - b) — 2 sin b cos a
(3)
* R must be understood either as a divisor of the second member or multiplier of the first, because the sum of two lines cannot be equal to a rectangle.
making the same substitutions as above in equation (1) this last equation
or, the difference of the sines of two arcs is equal to twice the sine of half their difference into the cosine of half their sum.*
which may be expressed in a proportion thus: the sum of the sines of any two arcs is to the difference of their sines as the tangent of half their sum is to the tangent of half their difference. | 677.169 | 1 |
Shapes Of Animals
Shapes Of Animals - Circle, oval, square, rhombus, trapezoid. Web learn about the different types of animals, including invertebrates and vertebrates, with examples and facts. Web animal bodies come in a variety of sizes and shapes. Limits on animal size and shape include impacts to their. Web cats in the form of geometric shapes. Web learn about the different types of body plans, symmetry, and limits on animal size and shape. Vector set of farm animal in simple geometric shape. Web collection of geometric polygon animals, horse, lion, giraffe, butterfly, elephant, leopard, wolf, eagle, deer, buffalo, shark, vector illustration.
Shape Animal Crafts
Web learn about the different types of animals, including invertebrates and vertebrates, with examples and facts. Web learn about the different types of body plans, symmetry, and limits on animal size and shape. Limits on animal size and shape include impacts to their. Vector set of farm animal in simple geometric shape. Circle, oval, square, rhombus, trapezoid.
Web learn about the different types of body plans, symmetry, and limits on animal size and shape. Vector set of farm animal in simple geometric shape. Web learn about the different types of animals, including invertebrates and vertebrates, with examples and facts. Web cats in the form of geometric shapes. Limits on animal size and shape include impacts to their.
28+ Animals With Shape Pictures
Web cats in the form of geometric shapes. Web animal bodies come in a variety of sizes and shapes. Limits on animal size and shape include impacts to their. Web learn about the different types of animals, including invertebrates and vertebrates, with examples and facts. Web learn about the different types of body plans, symmetry, and limits on animal size.
Web collection of geometric polygon animals, horse, lion, giraffe, butterfly, elephant, leopard, wolf, eagle, deer, buffalo, shark, vector illustration. Circle, oval, square, rhombus, trapezoid. Web animal bodies come in a variety of sizes and shapes. Web learn about the different types of body plans, symmetry, and limits on animal size and shape. Vector set of farm animal in simple geometric shape. Web cats in the form of geometric shapes. Limits on animal size and shape include impacts to their. Web learn about the different types of animals, including invertebrates and vertebrates, with examples and facts.
Web cats in the form of geometric shapes. Web animal bodies come in a variety of sizes and shapes. Web learn about the different types of body plans, symmetry, and limits on animal size and shape. Circle, oval, square, rhombus, trapezoid.
Web Learn About The Different Types Of Animals, Including Invertebrates And Vertebrates, With Examples And Facts.
Limits on animal size and shape include impacts to their. Vector set of farm animal in simple geometric shape. | 677.169 | 1 |
angles of a triangle sum | 677.169 | 1 |
Worksheet Generator
S.T.W.
4th Grade Common Core: 4.G.3
Common Core Identifier: 4.G.3 / Grade: 4
Curriculum: Geometry: Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles.
Detail: Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. | 677.169 | 1 |
Standard - 5.G.1: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in
the plane located by using an ordered pair of numbers, called its
coordinates. Understand that the first number indicates how far to
travel from the origin in the direction of one axis, and the second
number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the
coordinates correspond (e.g., x-axis and x-coordinate, y-axis and
y-coordinate).
Standard - 5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Standard - 5.G.3: Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. | 677.169 | 1 |
Euclidean Distance
In the field of data analysis, measuring distances between data points is essential for understanding patterns and relationships within datasets. Distance metrics provide a quantitative measure of similarity or dissimilarity between objects, enabling the application of various analytical techniques. One such fundamental metric is Euclidean Distance. Euclidean Distance, named after the Greek mathematician Euclid, measures the straight-line distance between two points in Euclidean space. This widely-used metric serves as a cornerstone in numerous applications, ranging from image processing and machine learning to geospatial analysis and robotics. In this article, we delve into the concept of Euclidean Distance, exploring its definition, applications, properties, and its role in machine learning and real-world examples.
Definition and explanation of Euclidean Distance
Euclidean Distance is a fundamental metric used in data analysis to quantify the spatial separation between two points in Euclidean space. It measures the length of the straight-line segment connecting the two points. Calculated using the Pythagorean theorem, Euclidean Distance takes into account the coordinates of the points and determines their proximity or dissimilarity. In a two-dimensional plane, Euclidean Distance can be visualized as the direct path between two points on a Cartesian plane. In three-dimensional space, it represents the distance between points in a three-dimensional coordinate system. Overall, Euclidean Distance provides a quantitative measure that facilitates various applications in fields such as image processing, machine learning, and geospatial analysis.
Importance of Euclidean Distance in data analysis
Euclidean Distance holds immense importance in the field of data analysis. By measuring distances between data points, it enables us to understand the relationships and similarities within datasets. With a simple and intuitive calculation method, Euclidean Distance allows for effective comparisons and clustering of data. In fields like image processing, machine learning, and geospatial analysis, it serves as a fundamental metric for tasks such as image similarity detection, clustering algorithms, and route optimization. Its versatility and wide range of applications make Euclidean Distance an indispensable tool in the analysis and interpretation of complex datasets.
Preview of topics to be covered in the essay
In this essay, we will delve into the concept of Euclidean Distance and its significance in data analysis. We will begin by providing a definition and explanation of Euclidean Distance, highlighting how it is calculated in Euclidean space. Moreover, we will explore the diverse applications of Euclidean Distance in various fields such as image processing, machine learning, geospatial analysis, recommender systems, and robotics. Additionally, we will discuss the key properties of Euclidean Distance, its usage in machine learning algorithms, and compare it with other distance metrics. Furthermore, we will examine the challenges of applying Euclidean Distance in high-dimensional spaces and present real-world examples where Euclidean Distance is utilized. Lastly, we will address the limitations and future research directions concerning Euclidean Distance.
One important application of Euclidean Distance in machine learning is in the field of nearest neighbor classifiers. In this technique, Euclidean Distance is used to determine the similarity between a query point and the training samples. By calculating the Euclidean Distance between the query point and each training sample, the nearest neighbor can be identified. This approach is particularly useful in classification problems, where the class label of the nearest neighbor is assigned to the query point. The Euclidean Distance metric enables the identification of the most similar data points, aiding in accurate and efficient classification tasks.
Understanding Euclidean Distance
Euclidean Distance is a fundamental concept in data analysis that measures the distance between two points in Euclidean space. It is calculated using the Pythagorean theorem, taking the square root of the sum of squared differences in each coordinate. Visual representations make it easier to understand as a straight line connecting the two points. Euclidean Distance finds wide applications in various fields, such as image processing, clustering, geospatial analysis, and robotics. Its simplicity and intuitiveness make it a popular choice for measuring similarity and distance in machine learning algorithms, including K-Means clustering, nearest neighbor classifiers, and principal component analysis.
Definition and calculation of Euclidean Distance in Euclidean space
Euclidean Distance is a fundamental metric used in data analysis to measure the similarity or dissimilarity between data points in Euclidean space. It is defined as the straight-line distance between two points, calculated using the Pythagorean theorem. In a two-dimensional Euclidean space, the distance between two points (x1, y1) and (x2, y2) can be computed as the square root of the sum of the squared differences in the x and y coordinates. This straightforward calculation allows for the comparison of points in a geometric manner, providing essential insights for various applications in data analysis.
Visual representation of Euclidean Distance in 2D and 3D spaces
The concept of Euclidean Distance can be easily visualized in 2D and 3D spaces. In 2D space, Euclidean Distance represents the straight-line distance between two points and can be calculated using the Pythagorean theorem. The distance is represented by the length of the line connecting the two points. Similarly, in 3D space, Euclidean Distance is the shortest distance between two points in three-dimensional Cartesian coordinates, forming a line segment. Visual representations of Euclidean Distance allow for a clear understanding of the geometric relationship between data points, aiding in data analysis and decision-making processes.
Comparison with other distance metrics
When compared to other distance metrics, Euclidean Distance has its own strengths and weaknesses. One commonly used alternative is Manhattan Distance, which calculates the distance as the sum of the absolute differences between the coordinates. While Euclidean Distance considers the actual distance between points, Manhattan Distance is more suitable for grid-like structures. Another metric is Minkowski Distance, which generalizes both Euclidean and Manhattan distances. Additionally, Cosine Similarity measures the angle between vectors, making it useful for comparing documents or text data. Lastly, Mahalanobis Distance takes into account the covariance between variables, making it effective for data with different scales and distributions. The choice of distance metric depends on the specific problem and data characteristics, highlighting the versatility and context-dependent nature of distance calculations in data analysis.
In comparing Euclidean Distance with other distance metrics, it is important to consider their respective strengths and weaknesses. One such alternative is the Manhattan Distance, also known as the City Block distance. While Euclidean Distance calculates the shortest distance between two points in a straight line, Manhattan Distance considers only horizontal and vertical movements, which can be more suitable when movements are restricted to specific directions. Minkowski Distance, on the other hand, generalizes both Euclidean and Manhattan distances by introducing a parameter that allows for different levels of emphasis on each dimension. Cosine Similarity, often used in text analysis and recommendation systems, measures the cosine of the angle between two vectors, thus evaluating the similarity of their orientations rather than their magnitudes. Another important distance metric is Mahalanobis Distance, which takes into account the correlation structure of the data, making it useful in cases where variables are interdependent. In the end, the choice of distance metric depends on the specific task and nature of the data being analyzed.
Applications of Euclidean Distance
Euclidean Distance finds extensive applications across various fields due to its ability to measure similarity and dissimilarity between data points. In image processing and computer vision, it is utilized for image matching, object detection, and feature extraction. In machine learning, Euclidean Distance serves as a fundamental metric in clustering and classification algorithms such as K-Means and nearest neighbor classifiers. Geospatial analysis and GIS employ Euclidean Distance to calculate distances between locations and determine optimal routes. Recommender systems leverage Euclidean Distance for collaborative filtering, while robotics benefits from path planning using Euclidean Distance. These applications highlight the versatility and importance of Euclidean Distance in data analysis.
Image processing and computer vision
In the field of image processing and computer vision, Euclidean Distance plays a crucial role in various applications. One such application is image similarity and retrieval, where Euclidean Distance is used to measure the similarity between images based on their pixel values. This enables tasks such as image search, content-based image retrieval, and image clustering. Euclidean Distance also aids in image registration, which involves aligning and comparing multiple images. By calculating the distance between corresponding pixels, it helps identify and quantify spatial transformations, essential for tasks such as image stitching and object tracking in videos. Overall, Euclidean Distance serves as a fundamental metric in analyzing and processing images to extract meaningful information and facilitate computer vision tasks.
Clustering and classification in machine learning
Clustering and classification are fundamental tasks in machine learning, and Euclidean Distance plays a crucial role in both. In the context of clustering, Euclidean Distance is used to determine the similarity or dissimilarity between data points, enabling the grouping of similar instances into clusters. This allows for the identification of hidden patterns and structures within datasets. In classification, Euclidean Distance is utilized to determine the distance between an unlabeled datapoint and labeled training instances, aiding in the assignment of the datapoint to the most similar class. Ultimately, Euclidean Distance facilitates the effective organization and prediction of data in machine learning algorithms.
Geospatial analysis and GIS
Geospatial analysis and Geographic Information Systems (GIS) heavily rely on Euclidean Distance as a key metric. By calculating distances between points in a geographic space, GIS enables various applications such as route optimization, spatial clustering, and proximity analysis. Euclidean Distance plays a vital role in determining the nearest facilities, identifying patterns of spatial distributions, and measuring spatial relationships between different locations. Moreover, the accuracy of GIS-based spatial analysis heavily relies on the precise calculation of distances, making Euclidean Distance an essential tool for geospatial professionals and researchers.
Recommender systems and collaborative filtering
Recommender systems and collaborative filtering are areas where Euclidean Distance plays a crucial role. These systems aim to provide personalized recommendations by analyzing the similarity between users or items. Euclidean Distance is used to measure the distance between users or items based on their feature vectors, allowing for the identification of similar profiles or preferences. By utilizing Euclidean Distance, recommender systems can effectively match users with items they are likely to be interested in, improving user satisfaction and engagement. This application of Euclidean Distance highlights its versatility and importance in enhancing user experiences in various domains.
Robotics and path planning
In the field of robotics and path planning, Euclidean Distance is a crucial metric used to determine the distance between two points in a given environment. By calculating the Euclidean Distance, robots can navigate efficiently from their current position to a desired destination. This metric aids in determining the shortest and most optimal path, allowing robots to avoid obstacles and reach their goals effectively. Euclidean Distance plays a vital role in various robotic applications, such as autonomous vehicles, warehouse automation, and industrial robotics, ensuring safe and efficient movement in complex environments.
In the realm of machine learning, Euclidean Distance plays a crucial role as a similarity metric. One popular application is the K-Means clustering algorithm, which groups data points based on their distance from cluster centroids. Euclidean Distance is also essential in nearest neighbor classifiers, where the distance between test samples and training samples is calculated to identify the closest neighbors and make predictions. Moreover, Euclidean Distance aids in Principal Component Analysis (PCA), a technique that reduces the dimensionality of data by finding the directions of maximum variance. Its versatility and effectiveness make Euclidean Distance a fundamental tool for various machine learning tasks.
Properties of Euclidean Distance
The properties of Euclidean Distance play a crucial role in its application and interpretation. Firstly, Euclidean Distance exhibits symmetry, meaning the distance between point A and point B is the same as the distance between point B and point A. Additionally, Euclidean Distance is non-negative, ensuring that distances are always greater than or equal to zero. Another important property is the triangle inequality, which states that the distance between two points is always shorter than or equal to the sum of the distances between those points and a third point. Finally, the Euclidean Distance can also be interpreted as the Euclidean norm or Euclidean length, representing the magnitude of a vector in Euclidean space. These properties underscore the reliability and usefulness of Euclidean Distance in various data analysis tasks.
Symmetry and non-negativity
One of the key properties of Euclidean Distance is its symmetry and non-negativity. Symmetry means that the distance between two points A and B is the same as the distance between B and A. This property is essential in many applications, as it ensures consistent and fair comparisons between data points. Additionally, Euclidean Distance is always non-negative, meaning that it is never less than zero. This property aligns with our intuitive understanding of distance as a positive measure and allows for meaningful interpretations in various analytical contexts.
Triangle inequality
The triangle inequality is a fundamental property of Euclidean Distance that holds true in any dimensional space. It states that the distance between two points is always less than or equal to the sum of the distances between those points and a third point. Mathematically, for three points A, B, and C, the inequality is represented as: d(A,C) ≤ d(A,B) + d(B,C). This property is valuable in data analysis as it allows for the establishment of upper bounds on distances and aids in the optimization of algorithms and decision-making processes.
The Euclidean norm or Euclidean length
The Euclidean norm, also known as Euclidean length, is a mathematical concept closely related to Euclidean Distance. It quantifies the magnitude or length of a vector in Euclidean space. The Euclidean norm is calculated by taking the square root of the sum of the squares of the vector's components. It provides a measure of the vector's "distance" from the origin or the point it represents. The Euclidean norm is a fundamental tool in various mathematical and statistical applications, enabling the computation of distances and similarities between vectors in multidimensional spaces.
In the realm of high-dimensional data analysis, Euclidean Distance encounters challenges and limitations. One major issue is its sensitivity to scale and units. When data attributes have different units or scales, Euclidean Distance may be distorted, leading to inaccurate results. Additionally, the curse of dimensionality poses a significant challenge. As the number of dimensions increases, the space becomes sparser, and the distances between points become less informative. This can hinder the effectiveness of using Euclidean Distance as a metric in high-dimensional spaces. Moreover, handling missing data can be problematic since Euclidean Distance assumes complete information for all dimensions. These limitations highlight the need for alternative distance metrics and techniques in complex data analysis scenarios.
Euclidean Distance in Machine Learning
Euclidean Distance is widely utilized as a similarity metric in machine learning algorithms. For instance, in K-Means clustering, the distance between data points is measured using Euclidean Distance, allowing for the identification of clusters with minimal intra-cluster distances. Similarly, nearest neighbor classifiers employ Euclidean Distance to determine the proximity between an unlabeled instance and existing labeled instances for classification. Furthermore, Euclidean Distance plays a vital role in Principal Component Analysis (PCA), where it is used to calculate the covariance matrix and identify the most significant dimensions. Overall, Euclidean Distance is a critical tool in machine learning for pattern recognition, classification, and dimensionality reduction.
Similarity metric in machine learning
In machine learning, Euclidean Distance serves as a crucial similarity metric that enables various algorithms to make informed decisions. For instance, the K-Means clustering algorithm relies on Euclidean Distance to group similar data points together. Similarly, nearest neighbor classifiers utilize Euclidean Distance to determine the similarity between a test instance and labeled training data. Euclidean Distance also plays a significant role in Principal Component Analysis (PCA) by quantifying the variance between data points and aiding in the identification of meaningful features. These examples highlight the essentiality of Euclidean Distance in the realm of machine learning.
Examples of algorithms and techniques using Euclidean Distance
Several algorithms and techniques in machine learning heavily rely on Euclidean Distance as a similarity metric. One such example is the K-Means clustering algorithm, which partitions data points into clusters based on their proximity determined by Euclidean Distance. Nearest neighbor classifiers also employ Euclidean Distance to determine the closest neighbors and make predictions. Additionally, Principal Component Analysis (PCA), a dimensionality reduction technique, uses Euclidean Distance to find the axes that capture the maximum variance in the data. These examples highlight the wide-ranging applications of Euclidean Distance in data analysis and its fundamental role in various machine learning algorithms.
Advantages and limitations of using Euclidean Distance in machine learning
Euclidean Distance has several advantages when used in machine learning. Firstly, it is a simple and intuitive metric that is easy to understand and interpret. It can effectively measure the similarity or dissimilarity between data points, making it useful for tasks such as clustering and classification. Additionally, Euclidean Distance can be computed efficiently, making it suitable for large-scale datasets. However, it is important to acknowledge the limitations of Euclidean Distance. It assumes that all features or dimensions are equally important, which may not always be the case. Euclidean Distance is also sensitive to differences in scale and can be affected by outliers. Despite these limitations, Euclidean Distance remains a valuable tool in many machine learning applications.
Euclidean Distance is widely applied in various real-world examples to analyze and solve practical problems. For instance, in image processing and computer vision, Euclidean Distance is utilized to calculate similarities between images, enabling image retrieval and classification. In geospatial analysis and GIS, Euclidean Distance measures the distances between locations, aiding in route planning and spatial analysis. Furthermore, Euclidean Distance plays a crucial role in recommender systems, where it gauges the similarity between users or items to provide personalized recommendations. In robotics, Euclidean Distance assists in path planning and obstacle avoidance. These examples highlight the practical significance and versatility of Euclidean Distance in diverse fields.
Euclidean Distance in Multidimensional Spaces
In the context of multidimensional spaces, Euclidean Distance extends its applicability to higher dimensions, encompassing datasets with numerous variables. However, as the dimensionality increases, challenges arise. The curse of dimensionality highlights the sparsity of data points as the number of dimensions grows, potentially diminishing the effectiveness of Euclidean Distance in capturing meaningful differences. To address this issue, dimensionality reduction techniques such as Principal Component Analysis (PCA) can be employed, which transform the original data into a lower-dimensional subspace while retaining as much variance as possible. By leveraging such techniques, Euclidean Distance can continue to play a crucial role in analyzing complex, multidimensional datasets.
Extending Euclidean Distance to high-dimensional spaces
Extending Euclidean Distance to high-dimensional spaces poses unique challenges and considerations in data analysis. As the number of dimensions increases, the curse of dimensionality becomes evident, leading to sparsity and increased computational complexity. Techniques such as dimensionality reduction and feature selection become crucial to mitigate these challenges and maintain the effectiveness of Euclidean Distance as a distance metric. By reducing the dimensionality of the data, the impact of irrelevant or redundant features can be minimized, allowing for more meaningful and accurate distance calculations in high-dimensional spaces. These techniques play a vital role in enabling the application of Euclidean Distance in real-world scenarios, where data often exhibits high dimensionality.
Challenges and considerations in high-dimensional data analysis
When dealing with high-dimensional data analysis, there are several challenges and considerations that need to be addressed. One major challenge is the curse of dimensionality, where the increase in the number of dimensions leads to sparsity of data points and a decrease in the effectiveness of distance metrics. Another consideration is the difficulty of visualizing and interpreting high-dimensional data, making it crucial to employ dimensionality reduction techniques to reduce complexity and extract meaningful information. Additionally, feature selection becomes crucial to eliminate irrelevant or redundant features and improve the performance of models. Overall, high-dimensional data analysis requires careful handling to overcome these challenges and make accurate inferences.
Techniques for dimensionality reduction and feature selection
Techniques for dimensionality reduction and feature selection play a crucial role in dealing with high-dimensional data in the context of Euclidean Distance. As the number of features increase, the complexity and computational requirements of distance calculations also grow. Therefore, various methods have been developed to reduce the dimensionality of data while retaining the most relevant information. Principal Component Analysis (PCA) is one commonly used technique that transforms the data into a lower-dimensional space by identifying the principal components that explain the maximum variance. This allows for more efficient Euclidean Distance calculations and facilitates data analysis and visualization. Additionally, feature selection techniques such as Mutual Information and Recursive Feature Elimination aim to identify the subset of features that are most informative and discriminative for the Euclidean Distance computations, further improving the accuracy and efficiency of the analysis.
One area where Euclidean Distance plays a crucial role is in robotics and path planning. In robotics, determining the shortest path between two points is essential for efficient and accurate navigation. By utilizing Euclidean Distance, robots can calculate the distance between their current location and the target destination, helping them identify the optimal path to reach it. This distance metric is particularly valuable in obstacle avoidance and collision detection, allowing robots to navigate through complex environments while maintaining safety and efficiency. Additionally, Euclidean Distance is used in path planning algorithms, such as the A* algorithm, to find the most efficient route between multiple waypoints. By leveraging Euclidean Distance, robotics researchers and engineers can design intelligent and adaptable systems that can effectively navigate real-world environments.
Real-world Examples of Euclidean Distance
In real-world examples, Euclidean Distance is applied to a wide range of problems across various disciplines. For instance, in image processing and computer vision, Euclidean Distance is employed to measure similarity between images, aiding in tasks such as image matching and object recognition. In the field of geospatial analysis and GIS, Euclidean Distance is utilized to calculate the shortest paths between locations, facilitating route planning and optimization. Additionally, Euclidean Distance plays a crucial role in recommender systems and collaborative filtering, where it measures the similarity between users or items to make personalized recommendations. Moreover, in robotics and path planning, Euclidean Distance is used to determine the optimal trajectory for navigating robotic systems. These real-world examples illustrate the versatility and practicality of Euclidean Distance as a fundamental metric in solving a multitude of data analysis problems.
Case studies demonstrating the application of Euclidean Distance
Case studies demonstrate the extensive application of Euclidean Distance in solving practical problems across various domains. For instance, in image processing and computer vision, Euclidean Distance is used to measure the similarity between images, enabling tasks like image retrieval and object recognition. In geospatial analysis and GIS, Euclidean Distance helps calculate the shortest path between two points on a map, aiding in route planning and navigation systems. Euclidean Distance also plays a crucial role in recommender systems and collaborative filtering, recommending items or users based on their proximity in feature space. These case studies exemplify the versatility and effectiveness of Euclidean Distance in real-world scenarios.
Practical problems solved using Euclidean Distance
Euclidean Distance is a versatile metric that finds practical applications in various fields. For instance, in image processing and computer vision, it is used to compare similarity between images and detect patterns. In the field of geospatial analysis and GIS, Euclidean Distance helps determine the shortest path between two locations on a map. It is also utilized in recommender systems and collaborative filtering to recommend products or services based on similarity. Furthermore, Euclidean Distance plays a crucial role in robotics and path planning by calculating the distance between robot positions and obstacles, enabling efficient navigation.
In the realm of machine learning, Euclidean Distance serves as a crucial similarity metric. It is utilized in various algorithms and techniques, such as K-means clustering, nearest neighbor classifiers, and Principal Component Analysis (PCA). Euclidean Distance allows for the quantification of distances between data points, enabling the identification of patterns and similarities within datasets. By calculating the distance in Euclidean space, it becomes possible to uncover clusters, make predictions, and perform data-driven decision making. The versatility and reliability of Euclidean Distance make it an indispensable tool in machine learning workflows, ultimately contributing to the advancement of various domains and industries.
Limitations and Challenges of Euclidean Distance
One of the limitations of Euclidean Distance is its sensitivity to scale and units. Since Euclidean Distance calculates the straight-line distance between two points, it assumes that all dimensions or variables have equal importance and are measured on the same scale. However, in real-world data analysis, the scales and units of different variables can vary significantly. This can lead to biased results or inaccurate distance calculations. Another challenge is the curse of dimensionality, which refers to the increase in computational complexity and decreased effectiveness of Euclidean Distance as the number of dimensions or variables increases. High-dimensional data can result in sparse and noisy distance calculations, making it difficult to interpret and analyze the results accurately. Additionally, Euclidean Distance struggles with missing data. If a value is missing for one or more dimensions, it becomes challenging to compute the distance accurately. These limitations and challenges highlight the need for alternative distance metrics and techniques in certain data analysis scenarios.
Sensitivity to scale and units
A notable limitation of Euclidean Distance is its sensitivity to scale and units. This means that the values and units of measurement used in the dataset can greatly impact the calculated distances. If the variables in the dataset have different scales or units, the Euclidean Distance may be dominated by the variables with larger values. This can lead to misleading results and inaccurate comparisons. To mitigate this issue, it is crucial to normalize or standardize the data before applying Euclidean Distance, ensuring that all variables are on a comparable scale.
Curse of dimensionality
The curse of dimensionality refers to the challenges and limitations that arise when dealing with high-dimensional data using Euclidean Distance. As the number of dimensions increases, the amount of data required to maintain a certain level of density grows exponentially. Consequently, this leads to sparsity in the data distribution, making it difficult to accurately measure distances and identify meaningful patterns. The curse of dimensionality poses a significant hurdle in data analysis, necessitating the use of dimensionality reduction techniques and careful consideration of feature selection to mitigate its effects and ensure reliable results.
Handling missing data
Handling missing data is a crucial aspect in utilizing Euclidean Distance for data analysis. When dealing with datasets that contain missing values, it is essential to employ appropriate strategies to address this issue. One common approach is to impute the missing data using methods such as mean imputation, regression imputation, or multiple imputation. Another technique involves considering missing values as a separate category, allowing them to be included in the distance calculations. Careful handling of missing data ensures accurate distance measurements and minimizes the potential bias introduced by incomplete information.
Euclidean Distance, a fundamental metric in data analysis, plays a crucial role in various fields. In image processing and computer vision, it aids in measuring the similarity between images for object recognition. In machine learning, Euclidean Distance is used in algorithms like K-Means clustering and nearest neighbor classifiers to identify patterns and make predictions. Geospatial analysis and GIS employ Euclidean Distance to calculate distances between locations for optimal routing. Recommender systems utilize it to suggest similar items to users based on their preferences. Furthermore, Euclidean Distance facilitates path planning in robotics by determining the shortest distance between points. Its versatility and widespread application make Euclidean Distance an indispensable tool in data analysis.
Future Trends and Research Directions
In looking towards the future, there are several promising trends and research directions associated with Euclidean Distance. One area of interest is in the healthcare field, where Euclidean Distance could potentially be utilized for personalized medicine and disease prediction models. Additionally, in the realm of autonomous vehicles, Euclidean Distance can play a crucial role in improving path planning algorithms and obstacle avoidance systems. Furthermore, ongoing research is exploring the integration of Euclidean Distance into social network analysis and recommendation systems to enhance personalized content suggestions and social connections. These emerging trends and research directions highlight the continued relevance and applicability of Euclidean Distance in advancing various domains.
Emerging trends and research areas related to Euclidean Distance
Emerging trends and research areas related to Euclidean Distance are continuously expanding as the field of data analysis evolves. One notable area of interest is the application of Euclidean Distance in healthcare, where it is being used to analyze medical data and patient profiles for personalized treatment and diagnosis. Additionally, in the field of autonomous vehicles, Euclidean Distance is being explored for path planning and obstacle avoidance, enabling safer and more efficient navigation. As data analysis techniques advance, there is immense potential for Euclidean Distance to be further utilized in diverse domains, paving the way for exciting advancements in multiple industries.
Potential advancements and applications in various fields
Potential advancements and applications of Euclidean Distance span across various fields, offering new possibilities for data analysis. In healthcare, Euclidean Distance can be utilized in medical imaging for identifying patterns and anomalies. Autonomous vehicles can leverage this metric for path planning and obstacle avoidance. Furthermore, Euclidean Distance has the potential to enhance recommendation systems by measuring similarity between user preferences. As technology advances, the applications of Euclidean Distance are expected to expand, encouraging further research and innovation in fields such as artificial intelligence, robotics, and personalized medicine.
Furthermore, Euclidean Distance is not without its limitations and challenges. One such limitation is its sensitivity to scale and units. In other words, the measurement of distance can be significantly influenced by the choice of measurement units and the range of values. Additionally, when dealing with high-dimensional data, known as the "curse of dimensionality", Euclidean Distance may become less reliable due to the increased sparsity and complexity of the data. Lastly, the presence of missing data can pose a challenge since Euclidean Distance requires complete data points for accurate calculations. As data analysis continues to evolve, addressing these limitations and challenges will be crucial for further advancements in utilizing Euclidean Distance effectively.
Conclusion
In conclusion, Euclidean Distance emerges as a fundamental metric in data analysis due to its versatile applications and intuitive interpretation. It provides a reliable measure of similarity and dissimilarity between data points in various domains such as image processing, machine learning, geospatial analysis, and robotics. Despite its limitations in handling scale, high-dimensional data, and missing values, Euclidean Distance remains a popular choice due to its simplicity and effectiveness. Furthermore, as data analysis continues to evolve, Euclidean Distance is expected to play an increasingly crucial role in emerging fields and contribute to advancements in healthcare, autonomous vehicles, and other domains.
Summary of key points discussed in the essay
In summary, this essay has provided an in-depth understanding of Euclidean Distance as a fundamental metric in data analysis. The concept of measuring distances between data points was introduced, emphasizing the importance of such measurements in various fields. The definition and calculation of Euclidean Distance in Euclidean space were explained, along with its visual representation in 2D and 3D spaces. The diverse applications of Euclidean Distance, ranging from image processing to robotics, were discussed. Furthermore, the properties of Euclidean Distance, such as symmetry and the Euclidean norm, were explored. The use of Euclidean Distance in machine learning algorithms and its comparison with other distance metrics were highlighted. The challenges and considerations in high-dimensional data analysis were presented, along with real-world examples showcasing the application of Euclidean Distance. Finally, the limitations, future trends, and research directions pertaining to Euclidean Distance were examined. Throughout the essay, it became evident that Euclidean Distance is a crucial tool in data analysis, enabling researchers and practitioners to make informed decisions and solve complex problems.
Reinforcement of the significance of Euclidean Distance in data analysis
The significance of Euclidean Distance in data analysis cannot be understated. As a fundamental metric, Euclidean Distance plays a pivotal role in measuring the similarity and dissimilarity between data points. Its calculation in Euclidean space provides a straightforward and intuitive measure of distance, making it widely used in various fields including image processing, machine learning, geospatial analysis, and recommender systems. Furthermore, the properties of Euclidean Distance, such as symmetry, non-negativity, and the triangle inequality, ensure its validity and applicability in diverse analytical scenarios. Thus, Euclidean Distance remains a critical tool for data analysts seeking to understand and explore complex datasets | 677.169 | 1 |
So we can now make all lines through the point \( e_{12}\), but we only have one point. We were promised origin independence! So where are the other points? To find them, we need another line orthogonal to \(e_1\) and \(e_2 \).
Martin Roelfs
An Algebra for Geometry
{GA}
Martin Roelfs
An Algebra for Geometry
{GA}
The line at infinity, \(e_0\), is orthogonal to \(e_1\) and \(e_2 \). Besides finding a new line, we also find two new points: \(e_{01}, e_{02}\). Because these lie at infinity, they are called directions.
Recall that lines are specified by the linear equation
$$ ax + by + \delta = 0 $$
We can thus represent arbitrary lines in 2D as
$$ u = ae_1 + be_2 + \delta e_0, $$
where \(\delta\) is the distance from the origin. Similarly, arbitrary directions (points on the horizon)
$$ p_\infty = y e_{01} + x e_{20} $$
whereas arbitrary points are given by
$$ p = e_{12} + p_\infty = e_{12} + y e_{01} + x e_{20} $$
What is the square of \(e_0\)? Well, you cannot reflect in the horizon and live to tell about it!
e_0^2 = 0
\begin{pmatrix}x \\ y \\ 0\end{pmatrix}
\begin{pmatrix}x \\ y \\ 1\end{pmatrix}
\mathbb{R}_{2,0,1}
Martin Roelfs
An Algebra for Geometry
{GA}
Question: how would we now rotate around a given point \((x, y)\) by an angle \( \theta \)?
p = e_{12} + y e_{01} + x e_{20}
Since \(e_0^2 = 0\), all points are created equal:
$$ p^2 = e_{12}^2 = -1$$
Thus all points generate their own local Euler's formula under exponentiation:
$$ R = e^{\theta p} = \cos\theta + p \sin\theta$$
Directions are even easier, because they square to \(p_\infty^2 = 0\): | 677.169 | 1 |
how do you label a right triangle
Label Sides Of Right Triangle Worksheet – Triangles are among the most fundamental forms in geometry. Understanding the concept of triangles is essential for understanding more advanced geometric principles. In this blog post we will look at the various kinds of triangles and triangle angles, as well as how to calculate the length and width of a triangle, and give instances of each. Types of Triangles There are three types for triangles: Equal, isosceles, as well … Read more | 677.169 | 1 |
Angles x and y are inscribed angles and in any given circle,an arc with a larger inscribed angle is longer than an arc with a smaller inscribed angle. So as x>y => Arc PQ > Arc QR But arcs listed in Quantities A and B are different,each of them is a full circle with one section removed | 677.169 | 1 |
Estimating Angles
Why play this game?
Estimating Angles is an engaging game that enables students to improve their familiarity with angles of different sizes. By setting this activity up as a game with a target to beat, students are likely to persevere and engage for longer than they might with a more traditional angles exercise.
The interactive provides instant feedback and in 2 player mode students can challenge a friend to beat their score.
Possible approach
Start the lesson by asking students what they know about angles, and collect together ideas such as names of angles (acute, obtuse, reflex, right-angled) and landmark angles (90 degrees, 180 degrees, 270 degrees, 360 degrees).
Draw acute, obtuse and reflex angles on the board and ask students to estimate their size. Encourage justification for estimates.
Draw another five angles on the board and challenge the class to make better estimates than you can. Invite students to estimate each angle, and then estimate the angles yourself. Then measure each angle with a protractor. The closest estimate gains a point.
You may wish to use the GeoGebra applet below - move the dot to change the angle, and tick/untick the box to show/hide its measurement.
Demonstrate the interactivity at Level 1 or 2 before setting the group off to work in pairs. The challenge is to score more than 50 points in 10 turns.
Keep a record of the highest score on the board. How close to 100 points can any pair get? Can anyone get an average of more than 8 after more than 10 turns? These could be long-term challenges, with students taking a screenshot when they get their average above a certain target.
Pairs could move on to Level 3 or 4 when appropriate.
Key questions
Which angles are easy to estimate?
What strategies can you use to improve your estimates | 677.169 | 1 |
Cos 20 Degrees
The value of cos 20 degrees is 0.9396926. . .. Cos 20 degrees in radians is written as cos (20° × π/180°), i.e., cos (π/9) or cos (0.349065. . .). In this article, we will discuss the methods to find the value of cos 20 degrees with examples.
Cos 20°: 0.9396926. . .
Cos (-20 degrees): 0.9396926. . .
Cos 20° in radians: cos (π/9) or cos (0.3490658 . . .)
What is the Value of Cos 20 Degrees?
The value of cos 20 degrees in decimal is 0.939692620. . .. Cos 20 degrees can also be expressed using the equivalent of the given angle (20 degrees) in radians (0.34906 . . .)
FAQs on Cos 20 Degrees
What is Cos 20 Degrees?
Cos 20 degrees is the value of cosine trigonometric function for an angle equal to 20 degrees. The value of cos 20° is 0.9397 (approx)
What is the Value of Cos 20 Degrees in Terms of Cot 20°?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 20° can be written as cot 20°/√(1 + cot²(20°)). Here, the value of cot 20° is equal to 2.74747.
What is the Exact Value of cos 20 Degrees?
The exact value of cos 20 degrees can be given accurately up to 8 decimal places as 0.93969262.
How to Find the Value of Cos 20 Degrees?
The value of cos 20 degrees can be calculated by constructing an angle of 20° with the x-axis, and then finding the coordinates of the corresponding point (0.9397, 0.342) on the unit circle. The value of cos 20° is equal to the x-coordinate (0.9397). ∴ cos 20° = 0.9397.
How to Find Cos 20° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 20° can be given in terms of other trigonometric functions as: | 677.169 | 1 |
TANGENT to Sine Wave Orthographic UHD1970-01-01T00:00:00+00:00TANGENT to Sine Wave Perspective UHD
and the route taken by a red line. All these items have been given a thickness to make the model easier to understand.
TANGENTS just touch a surface or line. They represent the slope of the curve at that point. In a circle, the tangent is perpendicular to the radius. For example, the ground is tangential to a ball that sits on it.
TANGENT to Sine Wave Perspective UHD2020-09-09T02:48:31ZVolume of Sphere Derivation1970-01-01T00:00:00+00:00The volume of a sphere is mapped into an equivalent pyramid up to illustrate how the formula sphere volume = 4/3 π r3 can be understood.
The animation starts with a translucent sphere (pale orange). A dark red wedge is shown running from the centre of...PT34S | 677.169 | 1 |
The upper $$(\frac{3}{4})$$ th portion of a vertical pole subtends an angel $$\tan ^{-1}\left(\frac{3}{5}\right)$$ at a point in the horizontal plane through its foot and at a distance $$40 \mathrm{~m}$$ from the foot. A possible height of the vertical is
A
80 m
B
20 m
C
40 m
D
60 m
2
BITSAT 2023
MCQ (Single Correct Answer)
+3
-1
A tower $$T_1$$ of the height $$60 \mathrm{~m}$$ is located exactly opposite to a tower $$T_2$$ of height $$80 \mathrm{~m}$$ on a straight road. From the top of $$T_1$$, if the angle of depression of the foot of $$T_2$$ is twice the angle of elevation of the top of $$T_2$$, then the width (in $$\mathrm{m}$$) of the road between the feet of the towers $$T_1$$ and $$T_2$$ is
A
$$20 \sqrt{3}$$
B
$$10 \sqrt{3}$$
C
$$10 \sqrt{2}$$
D
$$20 \sqrt{2}$$
3
BITSAT 2023
MCQ (Single Correct Answer)
+3
-1
If $$A, B, C \in[0, \pi]$$ and if $$A, B, C$$ are in $$\mathrm{AP}$$, then $$\frac{\sin A+\sin C}{\cos A+\cos C}$$ is equal to
A
$$\sin B$$
B
$$\cos B$$
C
$$\cot B$$
D
$$\tan B$$
4
BITSAT 2022
MCQ (Single Correct Answer)
+3
-1
If $$\alpha,\beta,\gamma \in[0,\pi]$$ and if $$\alpha,\beta,\gamma$$ are in AP, then $${{\sin \alpha - \sin \gamma } \over {\cos \gamma - \cos \alpha }}$$ is equal to | 677.169 | 1 |
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Parabola
Parabola
Parabola is conic section defined as a plane curve obtained by intersection of a cone and a plane in which the plane misses the vertex and the plane is parallel to the generator
Definition 2Definition 3Thus
Parabola is a plane curve defined a locus of a point in which the distance from a fixed point (or
focus) and distance from a fixed line (or directrix) is always equal.
Parabola is a plane curve defined a locus of a point which is always equidistant from a fixed point (or
focus) to a fixed line (or directrix)Standard Equ | 677.169 | 1 |
Hypotenuse Leg Theorem – Explanation & Examples
In this article, we'll learn about the hypotenuse leg (HL) theorem. Like, SAS, SSS, ASA, and AAS, it is also one of the congruency postulates of a triangle.
The difference is that the other 4 postulates apply to all triangles. Simultaneously, the Hypotenuse Leg Theorem is true for the right triangles only because, obviously, the hypotenuse is one of the right-angled triangle legs.
What is Hypotenuse Leg Theorem?
The hypotenuse leg theorem is a criterion used to prove whether a given set of right triangles are congruent.
The hypotenuse leg (HL) theorem states that; a given set of triangles are congruent if the corresponding lengths of their hypotenuse and one leg are equal.
Unlike other congruency postulates such as; SSS, SAS, ASA, and AAS, three quantities are tested, with hypotenuse leg (HL) theorem, two sides of a right triangle are only considered.
Illustration:
Proof of Hypotenuse Leg Theorem
In the diagram above, triangles ABC and PQR are right triangles with AB = RQ, AC = PQ.
By Pythagorean Theorem,
AC2 = AB2 + BC2 and PQ2 = RQ2 + RP2
Since AC = PQ, substitute to get;
AB2 + BC2 = RQ2 + RP2
But, AB = RQ,
By substitution;
RQ2 + BC2 = RQ2 + RP2
Collect like terms to get;
BC2 =RP2
Hence, △ABC ≅△ PQR
Example 1
If PR ⊥ QS, prove that PQR and PRS are congruent
Solution
Triangle PQR and PRS are right triangles because they both have a 90-degree angle at point R.
Given;
PQ = PS (Hypotenuse)
PR = PR (Common side)
Therefore, by Hypotenuse – Leg (HL) theorem, △ PQR ≅△ PR.
Example 2
If FB = DB,BA = BC, FB ⊥ AE and DB ⊥ CE, show that AE = CE.
Solution
By Hypotenuse Leg rule,
BA = BC (hypotenuse)
FB = DB (equal side)
Since, ∆ AFB≅ ∆ BDC, then ∠A = ∠ Therefore, AE = CE
Hence proved.
Example 3
Given that ∆ABC is an isosceles triangle and ∠ BAM = ∠MAD. Prove that M is the midpoint of BD.
Solution
Given ∠ BAM = ∠MAD, then line AM is the bisector of ∠ BAD.
AB = AD (hypotenuse)
AM = AM (common leg)
∠ AMB = ∠AMD (right angle)
Therefore, BM = MD.
Example 4
Check whether ∆XYZ and ∆STR are congruent.
Solution
Both ∆XYZ and ∆STR are right triangles (presence of a 90 – degree angle)
XZ = TR (equal hypotenuse).
XY = SR (Equal leg)
Hence, by Hypotenuse-Leg (HL) theorem, ∆XYZ ≅∆STR.
Example 5
Given: ∠A=∠C = 90 degrees, AD= BC. Show that △ABD ≅△DBC.
Solution
Given,
AD = BC (equal leg)
∠A=∠C (right angle)
BD = DB (common side, hypotenuse)
By, by Hypotenuse-Leg (HL) theorem, △ABD ≅△DBC
Example 6
Suppose ∠W = ∠ Z = 90 degrees and M is the midpoint of WZ and XY. Show that the two triangles WMX and YMZ are congruent.
Solution
△WMX and △YMZ are right triangles because they both have an angle of 900 (right angles)
WM = MZ (leg)
XM = MY (Hypotenuse)
Therefore, by Hypotenuse-Leg (HL) theorem, △WMX≅ △YMZ.
Example 7
Calculate the value of x in the following congruent triangles.
Solution
Given the two triangles are congruent, then;
⇒2x + 2 = 5x – 19
⇒2x – 5x = -19 – 2
⇒ -3x = – 21
x =- 21/-3
x = 7.
Therefore, the value of x = 7
Proof:
⇒ 2x + 2 = 2(7) + 2
⇒14 + 2 = 16
⇒ 5x -19 = 5(7) – 19
⇒ 35 – 19 = 16
Yes, it worked!
Example 8
If ∠ A = ∠ C = 90 degrees and AD = BC. Find the value of x and y that will make the two triangles ABD and DBC congruent.
Solution
Given,
△ABD ≅△DBC
Calculate the value of x
⇒ 6x – 7 = 4x + 2
⇒ 6x – 4x = 2 + 7
⇒ 2x = 9
⇒ x = 9/2
x = 4.5
Calculate the value of y.
⇒ 4y + 25 = 7y – 5
⇒ 4y – 7y = – 5 – 25
⇒ -3y = -30
y = -30/-3 =10
Therefore, △ABD ≅△DBC, when x = 4.5 and y = 2.72.
Practice Questions
1. Suppose that $\Delta ABC \cong \Delta LMN$, which of the following shows the value of $a$?
$a = 36$
$a = 48$
$a = 72$
$a = 96$
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2. Suppose that $\Delta ABC \cong \Delta LMN$, which of the following shows the value of $c$?
$c = 72$
$c = 92$
$c = 96$
$c = 120$
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3.
Which of the following shows the value of $x$?
$x = -30$
$x = -15$
$x = 15$
$x = 30$
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4.
Which of the following shows the value of $y$?
$y = -20$
$x = -10$
$y = 10$
$y = 20$
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5. True or False: We can use the Hypotenuse Leg theorem to show that the two triangles shown below are similar to each other. | 677.169 | 1 |
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