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Translations
The coordinate plane, a foundational concept in mathematics, is more than just a grid of points. It serves as a playground for various geometric transformations, among which, translations are fundamental. A translation moves a shape to a new position without changing its size, orientation, or shape. But how do we systematically carry out translations on […] | 677.169 | 1 |
Geometry
Geometry: Since the beginning of human society, the importance of determining directions and angles of the components of the environment that surrounds us has been very high. Geometry is one of the oldest branches of mathematics which deals with the size, shapes, angles, and dimensions of things. One of the earliest uses of geometry was seen when the early humans devised tools out of stone or traced the direction of their travel by referring to the positions and angles of the stars and the sun. Geometry deals with the things which are used in daily life. Geometry includes 2D as well as 3D shapes i.e 2 dimensional and 3-dimensional shapes.
In-plane geometry, 2d shapes such as triangles, squares, rectangles, and circles are also called flat shapes. In solid geometry, 3d shapes such as a cube, cuboid, cone, etc. are also called solids. The basic geometry is based on points, lines, and planes which come under coordinate geometry.
In this article, we are providing you the detailed information about geometry, geometry shapes, and geometry formulas. Understanding geometry will help candidates to solve the problems related to that and asked in the competitive exams.
Geometry Definition
Geometry word is derived from Ancient Greek words – 'Geo' means 'Earth' and 'metron' means 'measurement'. It can also be traced back to the Sanskrit word 'Jyamiti'. Geometry is concerned with properties of space that are related to distance, shape, size, and relative position of figures. The basics of geometry depend majorly on points, lines, angles, and planes.
What are the Branches of Geometry?
There are various uses of Geometry in daily life. It ranges from choosing the right size of any item to planning highly sophisticated space missions. Depending upon these applications, geometry can be classified in various branches. The branches of geometry are as follows:
Algebraic geometry
Discrete geometry
Differential geometry
Euclidean geometry
Coordinate geometry
Convex geometry
Topology
Plane Geometry (2D Geometry)
Plane Geometry means flat shapes that can be drawn on a piece of paper. These include lines, circles & triangles of two dimensions. Plane geometry is also known as two-dimensional geometry which utilizes two coordinates in any space i.e. X and Y axis. Examples of 2D Geometry is square, triangle, rectangle, circle, lines, etc. Here we are providing you with the properties of the 2D shapes below.
Point
A point is a location or place on a plane. A dot usually represents them. It is important to understand that a point is not a thing, but a place. The point has no dimension and it has the only position. This position is also known as coordinates.
Line
The line is straight with no curves), has no thickness, and extends in both directions without end infinitely. It can be seen as a collection of points adjacent to each other. Almost all the shapes in geometry make use of lines and points. For example, shapes such as angles, triangles, rectangles, pentagons, and so on are made up using a collection of lines joined with each other.
Angles in Geometry
Angles are formed by the intersection of two lines called rays at the same point. which is called the vertex of the angle. It is utilized in the formation of a triangle, rectangle or more complex figures. These angles are of various types such as acute, right, obtuse and reflex. In the next section let us look at these in detail.
Types of Angle
Acute Angle – An Acute angle is an angle smaller than a right angle ie. it can be from 0 – 90 degrees.
Obtuse Angle – An Obtuse angle is more than 90 degrees but is less than 180 degrees.
Right Angle – An angle of 90 degrees.
Straight Angle – An angle of 180 degrees is a straight angle, i.e. the angle formed by a straight line.
Reflex Angle – These angles are greater than 180 degrees but less than 360 degrees.
Polygons
A polygon is an unrestrictedshapefigure that has a minimum of three sides and three vertices. The term' poly' means' numerous' and' gon' means' angle'. Therefore, polygons havenumerousangles. The border and area of a polygon depend upon its type. The distribution of polygons is described as grounded on the figures of sides and vertices.
Types of Polygons
The types of polygons depending upon the number of sides and vertices are:
Triangles
Quadrilaterals
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
We have discussed the property as well as examples of the polygons with their properties in the below table. These figures will help candidates prepare for geometry for various competitive exams.
A 4-sided polygon with four edges and four vertices with a sum of internal angles is 360 degrees
Square has 4 equal sides and vertices which are at right angles.
Rectangle has equal opposite sides and all angles are at right angles.
A parallelogram has two pairs of parallel sides. The opposite sides & opposite angles are equal in measure.
The rhombus has all four sides to be of equal length. However, they do not have their internal angle to be 90 degrees
Trapezium has one pair of opposite sides to be parallel.
Pentagon
A plane figure with five straight sides and five angles
Sum of all the interior angles is 540 degrees
In a regular pentagon, each interior angle is of 108 degrees and each exterior angle is of 72 degrees
Hexagon
A plane figure with six straight sides and six angles
Sum of all the interior angles is 720 degrees
In a regular pentagon, each interior angle is of 120 degrees and each exterior angle is of 60 degrees
Heptagon
A plane figure with seven sides and seven angles
Sum of all the interior angles is 900 degrees
In a regular pentagon, each interior angle is of 128.57 degrees and each exterior angle is of 51.43 degrees
Octagon
A plane figure with eight straight sides and eight angles.
Sum of all the interior angles is 1080 degrees
In a regular pentagon, each interior angle is of 135 degrees and each exterior angle is of 45 degrees
Nonagon
A plane figure with nine straight sides and nine angles.
Sum of all the interior angles is 1260 degrees
In a regular pentagon, each interior angle is of 140 degrees and each exterior angle is of 40 degrees
Decagon
A plane figure with ten straight sides and ten angles.
Sum of all the interior angles is 1440 degrees
In a regular pentagon, each interior angle is of 144 degrees and each exterior angle is of 36 degrees
Geometry Formulas
Every geometry shape and figure has its own formula for the area and perimeter. This is important for the candidates to solve various questions asked in the geometry section in the Quantitative Aptitude section of Competitive exams. These formulas are also used while solving many questions in the Reasoning section as well. Some of the most important Geometry formulas are tabulated below.
Shape
Area
Perimeter
Rectangle (l= Length and b= breadth)
(l*b)
2(l+b)
Square (a is the side of the square)
a2
4a
Triangle (a,b and c are sides of the triangle)
1/2 (b × h) [where b is the length of the base and h is the length of the height of the triangle]
What is Geometry | 677.169 | 1 |
Ch 5.6
We will look at the properties of the Hinge
Theorem.
Learn a new kind of proof.
ACB
In an indirect, you assume that whatever
you want to prove is false and then find a
counterexample.
Yeah, we are
learning another
kind of proof.
Prove: All even numbers are divisible by 2.
Prove: All squares are quadrilaterals.
Prove: All perpendicular bisectors do not
intersect inside a triangle.
Given: x is a perfect square.
Prove: The square root of x is a whole number. | 677.169 | 1 |
If the weight of an angle A ( = 48°22'40'') is 4, then the weight of the angle (A2=24°11′20′′) will be ________
16
Open in App
Solution
The correct option is A 16 If a quantity of given weight is divided by a factor, the weight of the result is obtained by multiplying the given weight by the square of that factor.
Angle 'A' has weight = 4 ∴ Weight of A2=4×(2)2=16 | 677.169 | 1 |
Stretching Geometry
A square shaped garden is marked with integer coordinates. The area of the garden is infinite. The lower left corner of the garden is O (0,0) and the coordinate system is like normal Cartesian coordinate system. A mad problem setter has planted (15001*15001) trees in 15001 rows and 15001 columns. So the area where trees are planted is also a square. Trees are planted in a coordinate (p,q) iff the following two conditions are true:
a) d|p and d|q
b) p ≤ 15000d and q ≤ 15000d.
Here d is an arbitrary integer (0 < d < 11). In figure 1 below the value of d is 4.
2/2
The mad problem setter has drawn a 2D image of this garden, which looks somewhat like Figure 1 but a lot larger. But the problem is that while manipulating the image with a high precision imaging software he has stretched the image b (b > 0) times along x-axis and a (a > 0) times along y-axis. For example Figure 2 is found by stretching Figure-1, 3 times along x-axis and 2 times along y-axis. The trees in the pictures can be considered point objects. So they remain point objects after any sort of stretching. Before or after this stretching not all trees are visible from the origin O (0, 0) due to some other trees, which are on the same line with respect to O. In the above pictures the visible trees are shown with red. You will be given the coordinate of one of the visible trees in this stretched figure A (x1, y1). There will always be a tree at position B (x2, y2) in this stretched figure that is visible from O and whose angular distance φ from A with respect to O is minimum (I mean angle AOB is minimum) and in the positive direction (Counter clockwise). Your job is to determine the coordinate of B (x2, y2). Of course, the mad problemsetter is looking for a very efficient solution.
Input
The input file contains less than 10001 lines of input. Each line contains four integers x1 (0 ≤ x1 ≤ 15000∗d∗b), y1 (0 ≤ y1 ≤ 15000∗d∗a), d (0 < d < 11), ab (0 < ab ≤ 100000000, ab means a∗b) and φ (expressed in degree). The meanings of these symbols are described in the problem statement. Input is terminated by a line where the value of x1 is '-1'. This line should not be processed. φ will always be in the format D.DDDDDDDDDDDDDDDeN, here D denotes any decimal digit and N is an integer (−4 > N > −16). (If you are intelligent and use floating-point number with reasonable precision (such as double in C/C++) then you should not have any trouble.)
Output
For each line of input except the last one produce one line of output which contains two integers which are the values of x2 and y2.
Sample Input
152896 108 4 36 1.058827657765851e-6
43492 24 2 6 5.815846810938229e-7
-1 -1 -1 -1 1.000000000000000e-8
Sample Output
203856 144
54364 30 | 677.169 | 1 |
my-frugal-lifestyle
3 TRUE OR FALSE QUESTIONS!!! PLEASE HELP!Isometry is any transformation that results in an image tha...
4 months ago
Q:
3 TRUE OR FALSE QUESTIONS!!! PLEASE HELP!Isometry is any transformation that results in an image that is twice as large as the original image.Question 11 options: True FalseQuestion 12 (1 point) Question 12 UnsavedCorresponding points must be an equal distance from the Line of Reflection.Question 12 options: True FalseQuestion 15 (1 point) Question 15 UnsavedAngle of rotation is the measure of degrees that a figure is rotated about a fixed point.Question 15 options: True False
Accepted Solution
A:
question 11 is false. isometry is to be geomatrically congruent. question 15 is true. | 677.169 | 1 |
Radian/Square Millisecond Converter
Category
Angular Acceleration
Category
From
Radian/Square Millisecond
From
To
Select Unit
To
What Unit of Measure is Radian/Square Millisecond?
Radian per square millisecond is a unit of measurement for angular acceleration. By definition, if an object accelerates at one radian per square millisecond, its angular velocity is increasing by one radian per millisecond every millisecond.
What is the Symbol of Radian/Square Millisecond?
The symbol of Radian/Square Millisecond is rad/ms2. This means you can also write one Radian/Square Millisecond as 1 rad/ms2.
Manually converting Radian/Square Millisecond Radian/Square Millisecond converter tool to get the job done as soon as possible.
We have so many online tools available to convert Radian/Square Millisecond to other Angular Acceleration units, but not every online tool gives an accurate result and that is why we have created this online Radian/Square Millisecond converter tool. It is a very simple and easy-to-use tool. Most important thing is that it is beginner-friendly.
How to Use Radian/Square Millisecond Converter Tool
As you can see, we have 2 input fields and 2 dropdowns. For instance, you want to convert Radian/Square Millisecond to Radian/Square Microsecond.
From the first dropdown, select Radian/Square Millisecond and in the first input field, enter a value.
From the second dropdown, select Radian/Square Microsecond.
Instantly, the tool will convert the value from Radian/Square Millisecond to Radian/Square Microsecond and display the result in the second input field.
Example of Radian/Square Millisecond Converter Tool
Radian/Square Millisecond
1
Radian/Square Microsecond
0.000001
Radian/Square Millisecond to Other Units Conversion Table
Conversion
Description
1 Radian/Square Millisecond = 57295779.51 Degree/Square Second
1 Radian/Square Millisecond in Degree/Square Second is equal to 57295779.51
1 Radian/Square Millisecond = 57.3 Degree/Square Millisecond
1 Radian/Square Millisecond in Degree/Square Millisecond is equal to 57.3 | 677.169 | 1 |
gc aesthetics implantate
There's multiple ways to do this and correspondingly multiple solutions, but the simplest is just to connect each triangle to the square along an edge (since that then shares two vertices and an edge, twice). Counting Faces, Vertices and Edges. | 677.169 | 1 |
If s is arc length, establish the Frenet equation T′s=κN.
Solution
By definition, the curvature is κ=T′s. Since T is a unit vector, T·T=1, and T·T′=2T·T′=0, so T′ is orthogonal to T. The direction of T′ defines the direction of the principal normal, the unit vector N. Hence, T′/κ=N, or T′=κN | 677.169 | 1 |
... these more:
Trace angle 3 on a piece of patty paper. Then trace angle 5 so that it is adjacent to angle 3. What
does it mean for angles to be adjacent? [they have the same vertex, share a side, and do not share any
interior points.]
...
... Thus the desired congruence for the two triangles ABC and A0B0C0 in this particular
picture is the composition of a translation, a rotation, and a reflection.
It remains to address the other possibilities and how they affect the above argument. If A
= A0 to begin with, then the initial translation w ...
... Write the contrapositive of this statement. If you
live in Boston, then you live in Massachusetts.
A. If you do not live in
Massachusetts, then you do
not live in Boston.
B. If you live in Massachusetts,
then you do not live in Boston.
C. If you do not live in
Massachusetts, then you live
in Boston. ...
... a supplier in the Yellow Pages under "Restaurant Equipment and Supplies" or order them directly from
Key Curriculum Press.
Patty paper geometry evolved rather quickly after I saw a teacher use patty papers in one paper folding
activity. It was exciting to see, and each time I worked with patty paper ...
... class have won a competition for a playground
makeover and can choose the equipment they
would like. Ask the children to discuss what it
means that the area is 72 m2.
- Remind the children that the perimeter is the
distance around the outside of an area. Ask them
to work out the perimeter of the 3 m ...
... Constructing the Inscribed and Circumscribed Circles of a Triangle
1. The Inscribed Circle: On a separate piece of paper draw a scalene triangle. The side lengths
should be at least 2 inches each. The incenter of the triangle is found by bisecting two of the angles
and finding the intersection of th ...
... • Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite
angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles
are parallelograms with congruent diagonals.
BIG IDEA (Why is this included in the curriculum?)
• A nat ...
... that two large redwood trees visible from where they
are sitting are 180 feet apart, and Ladie says they are
275 feet apart.
The problem is, they can't measure the distance to see
whose estimate is better, because their cabin is located
between the trees. All of a sudden, Ladie recalls her geo ...
... Label the angles as shown. Place a second piece of patty paper over the
first and copy the trapezoid and its mid-segment.
Compare the trapezoid's base angles with the corresponding angles at the
mid-segment by sliding the copy up over the original.
Are the corresponding angles congruent? What ...
... 8. Fold and crease the paper so that the two horizontal lines are perfectly aligned on top of
each other.
9. Mark the intersection of the horizontal crease and segment PQ. (If you were careful, you
will notice that the vertical crease and the horizontal crease intersect on segment PQ in
the same spo ...
... 3. On a 5 × 5 dot paper draw
Two parallel lines
Two parallel line segments that touch a total of 9 dots
Two perpendicular lines
Two perpendicular line segments that touch a total of 9 dots
Two intersecting line segments that touch a total of 9 dots, but are not perpendicular
4. On a 5 × 5 dot paper
...
... Step 2 Align one side of the equilateral triangle with one side of the equilateral
triangle on the second sheet. Trace the triangle so that the two copies of the
triangle are now on one of the patty papers.
Step 3 Continue tracing the triangles in this manner filing the paper with tessellations
of e ...
... points X, Y, and Z) to the incenter are equal. Hence, the
line segments IX, IY, and IZ are all equal. From Calculus,
for any point not on a line, there exists another line
containing that point that is perpendicular to the line. Thus,
we have that IX is perpendicular to AB, IY perpendicular to
AC, a ...
Toilet paper orientation
Toilet paper when used with a toilet roll holder with a horizontal axle parallel to the floor and also parallel to the wall has two possible orientations: the toilet paper may hang over (in front of) or under (behind) the roll; if perpendicular to the wall, the two orientations are right-left or near-away. The choice is largely a matter of personal preference, dictated by habit. In surveys of US consumers and of bath and kitchen specialists, 60–70 percent of respondents prefer over.While many people consider this topic unimportant, some hold strong opinions on the matter. Advice columnist Ann Landers said that the subject was the most responded to (15,000 letters in 1986) and controversial issue in her column's history. Defenders of either position cite advantages ranging from aesthetics, hospitality, and cleanliness to paper conservation, the ease of detaching individual squares, and compatibility with a recreational vehicle or a cat. Some writers have proposed connections to age, sex, or political philosophy, and survey evidence has shown a correlation with socioeconomic status. A generic answer is that it should hang the way the person doing the roll changing prefers.Solutions range from compromise, to using separate dispensers or separate bathrooms entirely, or simply ignoring the issue altogether. One man advocates a plan under which his country will standardize on a single forced orientation, and at least one inventor hopes to popularize a new kind of toilet roll holder which swivels from one orientation to the other. | 677.169 | 1 |
The sum of the squares of the diameters of any parallelogram is equal to the sum of the squares of the sides of the parallelogram.
Let ABCD be a parallelogram, of which the diameters are AC and BD; the sum of the squares of AC and BD is equal to the sum of the squares of AB, BC, CD, DA. Let AC and BD intersect one another in E: and because the vertical angles AED, CEB are equal 2, a and also the alternate angles EAD, ECB, the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each: but the sides AD and BC, which are opposite to equal angles in these triangles, are also equale; therefore the other sides which are op
COR. From this demonstration, it is manifest that the diameters of every parallelogram bisect one another.
२८ हूँ
OF
GEOMETRY.
BOOK III.
DEFINITIONS.
A
THE
HE radius of a circle is the straight line drawn from Book III. the centre to the circumference.
I.
A straight line is said to
touch a circle, when it meets the circle, and being produced, does not cut it.
II.
Circles are said to touch
one another, which meet, but do not cut one another.
III.
Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal. IV.
And the straight line on which the greater perpendicular falls, is said to be farther from the centre.
An arch of a circle is any part of the circumference.
To find the centre of a given circle.
Let ABC be the given circle; it is required to find its
centre.
Draw within it any straight line AB, and bisecta it in Book III. D; from the point Ď drawb BC at right angles to AB, and produce it to E, and bisect CE in F: the point F is the centre of the circle ABC.
For, if it be not, let, if possible, G be the centre, and join GA, GD, GB: Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG are equal to the two BD, DG, each to each; but the base GA is also equal to the base GB, because they are radii of the same circle: therefore the angle ADG is equal to the angle GDB: But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of the angles is a right angled. Therefore the angle
c
F
a 10. 1. b 11. 1.
c 8. 1.
D
B
GDB is a right angle: But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible. Therefore G is not the centre of the circle ABC: In the same manner, it can be shown, that no other point but F is the centre; that is, F is the centre of the circle ABC: Which was to be found.
COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.
d 7. def. 1.
PROP. II. THEOR.
If any two points be taken in the circumference of a circle, the straight line which joins them will fall within the circle.
Book III.
a 5. 1.
b 16. 1.
c 19. 1.
a 1. 3.
C
Let ABC be a circle, and A, B any two points in the circumference: the straight line drawn from A to B will fall within the circle.
a
A
b
D
B
F
Take any point in AB as E; find D the centre of the circle ABC; join AD, DB and DE, and let DE meet the circumference in F. Then because DA is equal to DB, the angle DAB is equal to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater than the angle DAE; but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: Now to the greater angle the greater side is opposite; DB is therefore greater than DE: but DB is equal to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle. Wherefore, if any two points, &c. Q. E. D.
PROP. III. THEOR.
If a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and, "if it cut it at right angles, it will bisect it.
Let ABC be a circle, and let CD, a straight line drawn through the centre, bisect any straight line AB, which does not pass through the centre, in the point F: It also cuts it at right angles.
Take a E the centre of the circle, and join EA, EB. Then, because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two sides in the | 677.169 | 1 |
AC.
Definition of AC, such as calculating the area of a hyperbolic sector or determining the length of a hyperbolic arc.
Syntax of ACOSH Function
The syntax of the ACOSH function in Google Sheets is as follows:
ACOSH(number)
Where number is the number for which you want to calculate the inverse hyperbolic cosine. This argument must be a number greater than or equal to 1.
You can use the ACOSH function in a cell of a Google Sheets spreadsheet by typing the function and its argument, surrounded by = signs, as follows:
=ACOSH(number)
where number is the number for which you want to calculate the inverse hyperbolic cosine. The function will then return the angle, in radians, that has a hyperbolic cosine equal to the specified number.
Examples of ACOSH Function
Here are three examples of how to use the ACOSH function in Google Sheets:
To calculate the angle, in radians, whose hyperbolic cosine is 1.5, you could use the following formula: =ACOSH(1.5). This formula would return the value 0.962424, which is the angle, in radians, that has a hyperbolic cosine of 1.5.
To calculate the angle, in radians, whose hyperbolic cosine is 2.5, you could use the following formula: =ACOSH(2.5). This formula would return the value 1.316957, which is the angle, in radians, that has a hyperbolic cosine of 2.5.
To calculate the angle, in radians, whose hyperbolic cosine is 3.5, you could use the following formula: =ACOSH(3.5). This formula would return the value 1.762747, which is the angle, in radians, that has a hyperbolic cosine of 3.5.
In all of these examples, the ACOSH function is used to calculate the inverse hyperbolic cosine of a specified number, and returns the angle, in radians, that has a hyperbolic cosine equal to the specified number. This can be useful for performing complex mathematical calculations in a Google Sheets spreadsheet.
Use Case of ACOSH Function
Here are three examples of how the ACOSH function could be used in real-life scenarios in Google Sheets:
An astronomer is studying the properties of a distant galaxy and wants to calculate the area of a hyperbolic sector in the galaxy. The astronomer measures the radius and height of the sector and determines that the hyperbolic cosine of the sector's angle is 2.5. To calculate the area of the hyperbolic sector, the astronomer could use the following formula: =ACOSH(2.5). This would return the value 1.316957, which is the angle, in radians, that has a hyperbolic cosine of 2.5. The astronomer could then use this value, along with the radius and height of the sector, to calculate the area of the hyperbolic sector.
A geographer is studying the shape of the Earth and wants to use Google Sheets to calculate the lengths of different arcs on the Earth's surface. The geographer creates a spreadsheet with a column for each arc and a row for each angle, in degrees. To calculate the hyperbolic cosine of each angle, the geographer uses the COSH function, and to calculate the inverse hyperbolic cosine of each angle, the geographer uses the ACOSH function. This allows the geographer to determine the lengths of the different arcs on the Earth's surface.
A physicist is studying the behavior of subatomic particles and wants to calculate the angle, in radians, of a particle's trajectory. The physicist measures the velocity and acceleration of the particle and determines that the hyperbolic cosine of the angle of the particle's trajectory is 3.5. To calculate the angle, in radians, of the particle's trajectory, the physicist could use the following formula: =ACOSH(3.5). This would return the value 1.762747, which is the angle, in radians, that has a hyperbolic cosine of 3.5. This allows the physicist to study the behavior of the particle and understand its trajectory.
Limitations of ACOSH Function
The ACOSH function in Google Sheets has some limitations that users should be aware of:
The function only works with numbers greater than or equal to 1. If the number passed as an argument to the function is less than 1, the function will return the #NUM! error.
The function only returns the angle, in radians, that has a hyperbolic cosine equal to the specified number. If you need to convert the result to degrees, you must use the DEGREE function to convert the value from radians to degrees.
The function is not capable of performing other complex mathematical calculations, such as calculating the area of a hyperbolic sector or determining the length of a hyperbolic arc. If you need to perform these calculations, you must use the appropriate formulas or functions.
Overall, while the ACOSH function is useful for performing inverse hyperbolic cosine calculations in Google Sheets, it has some limitations that users should be aware of.
Commonly Used Functions Along With ACOSH
In Google Sheets, the ACOSH function is commonly used along with other mathematical functions, such as the SIN, COS, TAN, and ASIN functions, which are used to calculate the sine, cosine, tangent, and inverse sine of a given value, respectively. Other commonly used functions in Google Sheets include the SUM, AVERAGE, MAX, and MIN functions, which are used to calculate the sum, average, maximum, and minimum of a given range of values, respectively.
Summary
The ACOSH function is a useful mathematical tool that allows users to calculate the inverse hyperbolic cosine of a given value. It is a part of the standard library of many programming languages and is commonly used in fields that involve complex calculations, such as computer programming and engineering. In Google Sheets, the ACOSH function can be used in conjunction with other mathematical functions to perform a wide variety of calculations. If you are interested in trying out the ACOSH function for yourself, you can easily do so by using Google Sheets, which is a free online tool that allows users to create and edit spreadsheets. Give the ACOSH function a try and see how it can help you with your calculations!
Video: ACOSH Function
In this video, you will see how to use ACOSH function. Be sure to watch the video to understand the usage of ACOSH formula. | 677.169 | 1 |
Measure the Earth's Circumference with a Shadow
Summary
Key Concepts
Circumference, angles, parallel lines, Earth's equator, latitude
Credits
Ben Finio, PhD, Science Buddies
Introduction
How long of a tape measure would you need to measure the circumference of the Earth? Would you need to walk the whole way around the Earth to measure it? Do you think you can do it with just a meter stick in one location? Try this project to find out!
Important: this project will only work within about 2 weeks of the spring or fall equinox (usually about March 20th and September 23rd).What is the circumference of the Earth? In the age of modern technology, this might seem like an easy question for scientists to answer with tools like satellites and GPS, and it might be even easier for you to look up the answer online. It might seem like it would be impossible for you to measure the circumference of the Earth using just a meter stick. However, the Greek mathematician Eratosthenes was able to estimate the circumference of the Earth over two thousand years ago, without the aid of any modern technology. How? Using a little knowledge about geometry!
At the time, Eratosthenes was in the city of Alexandria in Egypt. He read that in a city named Syene south of Alexandria, on a particular day of the year at noon, the reflection of the Sun was visible at the bottom of a deep well. This meant that the Sun had to be directly overhead (another way to think about this is that perfectly vertical objects would cast no shadow). On that same day in Alexandria, a vertical object did cast a shadow. Using geometry*, this allowed him to calculate the circumference of the Earth using this equation:
[Please enable JavaScript to view equation]
He knew there were 360 degrees in a circle.
He could measure the angle of the shadow cast by a tall object in Alexandria.
He knew the overland distance between Alexandria and Syene (the two cities were close enough that at the time, this distance could be measured on foot).
The only unknown in the equation is the circumference of the Earth!
In this project, you will do this calculation yourself by measuring the angle formed by a meter stick's shadow at your location. You will need to do the experiment near the fall or spring equinox, when the Sun is directly overhead at the Earth's equator. Then, you can look up the distance between your city and the equator, and use the same equation Eratosthenes did to calculate the circumference of the Earth. How close do you think your result will be to the "real" value?
* There is a geometric rule about the angles formed by a line that intersects two parallel lines. Eratosthenes assumed that the Sun was far enough away from the Earth that its rays were effectively parallel when they arrived at the Earth. This told him that the angle of the shadow he measured in Alexandria was equal to the angle between Alexandria and Syene, measured at the center of the Earth. If this sounds confusing, don't worry! It is much easier to visualize with a picture. See the references in the More to Explore section for some helpful diagrams and a more detailed explanation of the geometry involved.
Materials
Sunny day on or near the spring or fall equinox (about March 20th or September 23rd)
Flat, level ground that will be in direct sunlight around noon
Meter stick
Volunteer to help hold the meter stick while you take measurements. If you are doing the experiment alone, you will need a bucket of sand or dirt to insert one end of the meter stick to hold it upright.
Optional: plumb bob (you can make one yourself by tying a small weight to the end of a string) or post level to make sure the meter stick is vertical
Calculator
Two options to measure the angle formed by the shadow:
A protractor and a long piece of string
A calculator with trigonometric functions
Preparation
Look at your local weather forecast a few days in advance, and pick a day where it looks like it will be mostly sunny around noon. You have a window of several weeks to do this project, so don't get discouraged if it turns out to be cloudy! You can try again.
Look up the sunrise and sunrise times for that day in your local paper or on a weather website. You will need to calculate "solar noon," the time exactly halfway between sunrise and sunset when the sun is directly overhead. This will probably not be exactly 12:00 noon.
Go outside and set up for your experiment about 10 minutes before solar noon so you have everything ready.
Instructions
Set up your meter stick vertically, outside in a sunny spot at solar noon.
If you have a volunteer to help, have them hold the meterstick. Otherwise, bury one end of the meter stick in a bucket of sand or dirt so it stays upright.
If you have a post level or plumb bob, use it to make sure the meter stick is perfectly vertical. Otherwise, do your best to eyeball it.
Mark the end of the meter stick's shadow on the ground with a stick or a rock.
Draw an imaginary line between the top of the meter stick and the tip of its shadow. Your goal is to measure the angle between this line and the meter stick. There are two different ways to do this:
Method 1: use a protractor
Have your volunteer stretch a piece of string between the top of the meter stick and the end of its shadow.
Use a protractor to measure the angle between the string and the meter stick in degrees. Write this angle down.
Method 2: use a calculator and trigonometry
Mark the exact spot where the meter stick touches the ground (or where it would touch the ground if it wasn't inside a bucket).
Use the meter stick to measure the length of the shadow (the distance between the bottom of the meter stick and the tip of the shadow that you marked previously). Write this distance down (in meters).
Use a calculator with trigonometric functions to calculate the angle, using the atan (or tan-1) button. Make sure the calculator is set to degrees and not radians. Then enter atan(distance), where "distance" is the length of the shadow you measured in the previous step. Write the resulting angle down.
Note: the previous step assumes you are using a meter stick and not a yard stick. If you are using a yard stick, measure the shadow length in inches, and then calculate atan(distance/36).
Look up the distance between your city and the equator. There are several options to do this:
If you have internet access, there are websites like that will automatically calculate the distance between a city and other locations (other cities, the north and south poles, the equator, etc.).
Use the scale on a world map to measure the distance between your city and the equator. Note that since flat maps always require some sort of distortion, this method may not be very accurate for cities that are far away from the equator.
Here is a list of some major US cities and their distance from the equator. If you are at the same latitude as one of these cities, you can just use these values. Or, you can look on a map of the United States to determine how far north or south you are from one of these cities, and add (or subtract) that value from the distance:
Miami, FL: 2,852 km (1,772 mi)
Orlando, FL: 3,159 km (1,963 mi)
Houston, TX: 3,293 km (2,046 mi)
San Diego, CA: 3,621 km (2,250 mi)
Dallas, TX: 3,629 km (2,255 mi)
Phoenix, AZ: 3,703 km (2,301 mi)
Atlanta, GA: 3,737 km (2,322 mi)
Los Angeles, CA: 3,769 (2,342 mi)
Charlotte, NC: 3,899 km (2,423 mi)
Nashville, TN: 4,004 km (2,488 mi)
San Francisco, CA: 4,183 km (2,599 mi)
Washington, DC: 4,308 km (2,677 mi)
Kansas City, MO: 4,329 km (2,690 mi)
Baltimore, MD: 4,350 km (2,703 mi)
Denver, CO: 4,402 km (2,735 mi)
Philadelphia, PA: 4,424 km (2,749 mi)
New York, NY: 4,508 km (2,801 mi)
Chicago, IN: 4,637 km (2,881 mi)
Detroit, MI: 4,688 km (2,913 mi)
Boston, MA: 4,691 km (2,915 mi)
Portland, ME: 4,836 km (3,005 mi)
Minneapolis, MN: 4,983 km (3,096 mi)
Portland, OR: 5,042 km (3,133 mi)
Seattle, WA: 5,277 km (3,279 mi)
Calculate the circumference of the Earth using this equation:
[Please enable JavaScript to view equation]
What value do you get? How close is your answer to the true circumference of the Earth (see Observations and results section)?
Extra: try repeating your experiment on different days before, on, and after the equinox, or at different times before, at, and after solar noon. How much does the accuracy of your answer change?
Extra: ask a friend or family member in a different city to try the experiment on the same day and compare your results. Do you get the same answer?
Observations and Results
In 200 B.C.E., Erastothenes estimated the circumference of the Earth to be about 46,250 km (at the time he used a different unit for distance, the stadia). Today we know that the Earth's circumference is roughly 40,000 km (24,854 miles). Not bad for a two-thousand year old estimate with no modern technology! Depending on the error in your measurements, like the exact day and time you did the experiment, how accurately you measured the angle or length of the shadow, and how accurately you measured the distance between your city and the equator, you should be able to calculate a value fairly close to 40,000 km (within a few hundred, or maybe a few thousand, km). All without leaving your own back yard!
Careers
Have you ever been in a new city and needed to figure out how to get from point A to point B? Have you ever tried to figure out the best time of the year to go on vacation so that you have good weather? Many people in these situations turn to a map. Maps are important sources of information, and geographic information systems (GIS) technicians are the professionals who gather data from a variety of sources, store it in databases, and use those databases to make accurate maps. Because maps are…
Read more
Maps can give us much more information than ways to get from A to B. Maps can give us topographic, climate, and even political information. Cartographers and photogrammetrists collect a vast amount of data, such as aerial data and survey data to produce accurate maps and models. For example, by collecting rainfall data, a cartographer can make an accurate model of how rainfall can affect an area's watershed. The maps and models can then be used by policy makers to make informed decisions.
Read more
Essential members of any construction team include mapping and surveying technicians—the "instrument people"—who set up and operate special equipment that measures distances, curves, elevations, and angles between points on Earth's surface. These technicians then take the data gathered by the instruments and create maps and charts on a computer. About half of their work is spent in hands-on, high-technology data collection in the field, while the other half is spent in an…
Read more
Did you know three of the four United States presidents on Mount Rushmore had the proud distinction of being surveyors? Surveying is an unusual mix of law and civil (construction) engineering. Surveyors protect the interests and rights of property owners. They create original legal documents describing property boundaries in land and water, and can act as expert witnesses in property or criminal cases | 677.169 | 1 |
Name
Synopsis
설명
Return an offset line at a given distance and side from an input line. All points of the returned geometries are not further than the given distance from the input geometry. Useful for computing parallel lines about a center line.
For positive distance the offset is on the left side of the input line and retains the same direction. For a negative distance it is on the right side and in the opposite direction.
거리의 단위는 공간 참조 시스템의 단위로 측정됩니다.
Note that output may be a MULTILINESTRING or EMPTY for some jigsaw-shaped input geometries. | 677.169 | 1 |
19. Point in a polygon
One way to determine whether a point lies inside a polygon is to add up the angles that it makes with the polygon's vertices. If the test point is P and A and B are two adjacent polygon points, then you look at the angle ∠APB.
If you add up all of the angles between the test point and each of the polygon's edges, the result will be either 0, 2π, or –2π. If the total is 2π or –2π, then the point is inside the polygon. If the total is 0, then the point lies outside of the polygon. You can probably convince yourself that this works if you draw a few examples. For example, try placing points inside and outside of a square, draw the angles, and estimate their values.
The idea is straightforward. The hard part is calculating ...
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O'Reilly members experience books, live events, courses curated by job role, and more from O'Reilly and nearly 200 top publishers. | 677.169 | 1 |
Here are solutions for section 8 of the first practice test in The Official SAT Study Guide, second edition, found on pages 413–418. The following solutions illustrate faster, less formal methods that may work better than formal methods on a fast-paced test such as the SAT. To learn more about these methods, see my e-book Succeeding in SAT Math or the SAT math tips page.
Estimate the answer: Looking at the graph, it should be fairly obvious that the answer is at least 10, but well under 25.
Look at the answer choices: Based on our estimate, we can eliminate (A), (B), and (E), leaving (C) 11 and (D) 13.
Draw a diagram: A diagram is already drawn, but you can fill in what you know on it. Since AD = 6, and AD is the radius of the circle, BA, CA, and AE are also 6. Since the two triangles are congruent, DE = 4. Fill in these numbers on the diagram.
Look at the answer choices: Comparing the answer choices with the numbers we filled in on the diagram, answers (A) through (D) are all incorrect, and (E) ED = 4 does match the diagram. Select (E) ED = 4.
Read the question and determine what it is asking: These types of questions can look intimidating, but really it's just asking you to plug a = 5, b = 2, c = 6 into ab − ac + c.
Estimate the answer: The coordinate axes divide the big square into four little squares. Each little square has area 2 × 2 = 4, so the answer should be 4 × 4 = 16.
Look at the answer choices: The only choice near our estimate is (C) 16. Select that answer.
Read the question and understand what it is asking: Read the question through a first time so you know what to look for. Then read the question through a second time, looking for the relevant information in each sentence:
Sentence 1: One of Owen, Chadd, Steph, and Daria is the oldest.
Sentence 2: Chadd is not the oldest.
Sentence 3: Daria might be the oldest.
Sentence 4: Steph is not the oldest.
Sentence 5: Owen is not the oldest.
Since Chadd, Steph, and Owen are not the oldest, Daria must be the oldest. Select (B) Daria.
Solution 1:
Estimate the answer: We can assume the diagram to be drawn to scale. y appears to be somewhat more than 90, and x appears to be somewhat less than 90. So, x + y is about 180, and 2(x + y) is around 360.
Look at the answer choices: (E) 360 is the only answer in range of our estimate ((D) 270 is too small, as it means that x must be less than 45, which isn't what the diagram shows). Select (E) 360.
Solution 2:
Draw a diagram: A diagram is given, but extend all of the lines in the diagram.
Since QR || PS, we know, from what we know about a line intersecting a parallel line that the angle to the left of the one marked y° must equal x°. So, x and y are supplemental. So, x and y = 180 and 2(x + y) = 360. Select (E) 360.
Convert the sentences into equations. The first sentence can be written as:
⅓(x + y + z) = 12
Since z is the largest of the three numbers, the second sentence can be written as:
(x + y) − z = 4
Look at the answer choices: If we manipulate the above equations slightly, they are the same equations as given in (A). Select (A) x + y + z = 36, x + y − z = 4.
Estimate the answer: Since 81 = 34, the answer has to be less than 4, maybe around 2.
Look at the answer choices: We can eliminate (C), (D), and (E), leaving (A) 3/2 and (B) 2.
Based on the laws of exponents,
32x·32y = 32x + 2y
This equals 81, which is 34. So,
32x + 2y = 34
and so
2x + 2y = 4 x + y = 2
Therefore, the answer is (B) 2.
Estimate the answer: f is tallest somewhere around halfway between 0 and 8, say around 4.
Look at the answer choices: We can eliminate (A) 2, (D) 6, and (E) 8.
To be on the safe side, it's a good idea to count the dashes on the graph to see whether the answer is (B) 4 or (C) 5. Counting, the function is tallest at x = 4. Select (B) 4.
Solution 1: Multiply both sides of the equation by 9:
9k = 3x
Therefore, the answer is (B) 9k.
Solution 2:
Estimate the answer: To make our estimate a bit easier, assume that k and x are positive. Now, if k = x/3, then x must always be greater than k (as an example, if k = 1, then x = 3). Therefore, 3x must be greater than 3k.
Look at the answers: The only answer that is always > 3k is (B) 9k (if necessary, you may want to play around with a few values for (C) to convince yourself that (C) can't be the answer). Therefore, the answer is (B) 9k.
Draw a diagram: If you find it difficult to visualize a cube, it may be helpful to draw a diagram.
A cube has six faces. If two are black, then 6 − 2 = 4 are white. If the total area of the four white faces is 64 square inches, the area of each white face is ¼(64) = 16 square inches. Since each face is a square, the side length must be √16 = 4 inches. Now, the area of the cube is 4 × 4 × 4 = 64. Select (A) 64.
Solution 1:
Draw a diagram: The diagram is already given. You may find it helpful to multiply everything on the number line by 4. So, 1 becomes 4, −1 becomes −4, and now x becomes 1, y becomes 3, w becomes −2, and v becomes −3.
Look at the answer choices:
(A) is equal to 0.
(B) is equal to −2.
(C) is equal to −1.
(D) is equal to −1.
(E) is equal to 2.
(B) has the least value. Select (B) v + x.
Solution 2:
Look at the answer choices:
(A) is equal to 0.
(B) is somewhat negative.
(C) is somewhat negative, but not as small as (B).
(D) is somewhat negative, but not as small as (B).
(E) is positive.
(B) has the least value. Select (B) v + x.
Read the question and understand what it is asking: You are asked to find the median of a set of 7 integers. The median is the middle value. Since there are an odd number of integers, there is only one middle value. Therefore, the median must be an integer. This insight allows us to eliminate II. right off the bat.
Try a special case: Try a small number, like 1, and see what the median is:
The median of 1,3,4,6,7,10,12 is 6.
Try a large number, like 20, and see what the median is:
The median of 3,4,6,7,10,12,20 is 7.
So, both I. and III. are possible. Therefore, the answer is (D) I and III only.
Solution 1: We can select any one of 5 colors for color 1. Having done that, we can select any one of 4 colors for color 2. The answer is 4 × 5 = 20. Select (B) 20.
Solution 2: While this solution takes more time, you could list out all of the possibilities. Say that the colors are Red (R), Orange (O), Yellow (Y), Green (G), and Blue (B). The possibilities are:
Estimate the answer: If one side of a rectangle gets 30% longer, the area increases by 30%. If one side gets 30% shorter, the area decreases by 30%. The change has to be between +30% and −30%. Probably the area isn't going to change too much.
Look at the answer choices: "(C) It is unchanged." Looks promising. There are also two answers with small decreases. We can definitely eliminate (A) and (B).
Take advantage of question order: This is the second-last question in the section, so it's a hard question. Therefore, the obvious answer is probably wrong. Eliminate (C) It is unchanged.
Try a special case:
Say that the length of the rectangle is 10 and the width is 10. Then the area is 100.
If the length of the rectangle is increased by 30%, it becomes 13.
If the width is decreased by 30%, it becomes 7.
The new area of the rectangle is 13 × 7 = 91. This represents a (100 − 91)⁄100 × 100% = 9% decrease.
Therefore, the answer is (E) It is decreased by 9%.
Try a special case: To make calculations easier, say that k = 0.
Determine how many bees in the hive were on day number 10:
n(10) = 10²⁄2 − 20(10) + 0 n(10) = 50 − 200 n(10) = −150
Therefore, there are −150 bees in the hive on day 10 (this seems nonsensical, but this is because we decided to let k = 0 to make the calculations easier, so we'll accept it. If you wanted to go back and let k = 150 or 200 or something, you could do that too).
Guess and check: Try each answer out:
On day 20, there are ½(20)² −20(20) = −200 bees in the hive. Nope.
On day 30, there are ½(30)² −20(30) = −150 bees in the hive. This is the answer we're looking for.
Therefore, the answer is (B) 30.
Last updated April 14, 2014.
URL:
For questions or comments, e-mail James Yolkowski ([email protected]). | 677.169 | 1 |
Is a triangle with two equal sides always isosceles?
An isosceles triangle therefore has both two equal sides and two equal angles. A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle.
What is the sum of an isosceles triangle?
180°
The sum of the internal angles of all triangles is always 180°. Therefore, the angles of an isosceles triangle add up to 180°.
How do you prove that two sides of an isosceles triangle are equal?
Hence proved. Theorem 2: Sides opposite to the equal angles of a triangle are equal. Proof: In a triangle ABC, base angles are equal and we need to prove that AC = BC or ∆ABC is an isosceles triangle….Isosceles Triangle Theorems and Proofs.
MATHS Related Links
Types Of Polygon
Parts Of A Circle
Reflection Symmetry
Surface Area Of A Cone
What does an isosceles triangle sides add up to?
All the three angles situated within the isosceles triangle are acute, which signifies that the angles are less than 90°. The sum of three angles of an isosceles triangle is always 180°, which means we can find out the third angle of a triangle if the two angles of an isosceles triangle are known.
How many sides are equal on a isosceles triangle?
two sides
In geometry, an isosceles triangle is a triangle that has two sides of equal length.
What is true about an isosceles triangle?
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.
How do you justify an isosceles triangle?
If a triangle has two congruent angles, then the sides opposite those angles are congruent. In other words, iso-angular triangles are iso-lateral (isosceles). This is the converse of the isosceles triangle theorem.
How many equal sides are there in an isosceles triangle?
two equal sides
An isosceles triangle can be drawn in many different ways. It can be drawn to have two equal sides and two equal angles or with two acute angles and one obtuse | 677.169 | 1 |
I understand what cosine similarity is and how to calculate it, specifically in the context of text mining (i.e. comparing tf-idf document vectors to find similar documents). What I'm looking for is some better intuition for interpreting the results/similarity scores I come up with.
My question: If I have a cosine similarity of less than 0.707 (i.e. an angle greater than 45 degrees), is is fair to say that those respective documents/vectors are more "different" (less "similar") since the angle between them is more orthogonal? My initial thought was 'yes,' but in practice for me so far it doesn't seem like that's the right way to read into the numbers.
$\begingroup$Cosine similarity is like Pearson correlation, only done on not centered data. For binary data, it is known also as Ochiai coefficient. Yes, cosine similarity bw two vectors is, like r, is the cosine of the angle between them.$\endgroup$
$\begingroup$You are asking specifically about using it in text mining. I'm not text analyst, I just suppose the compared documents are formed as binary data, yes? Then probably the meaning is the meaning of Ochiai? (see its formula).$\endgroup$
3 Answers
3
I believe another difference between cosine similarity and TF-IDF is that cosine similarity is done in an embedding space, such as one created by doc2vec.
Such an embedding puts words that are used in similar contexts near to each other, so you could use clustering to find similar documents. But cosine distance probably makes more sense for a couple of reasons:
An embedding like doc2vec encodes information in direction and distance. Look at the examples of king - man + woman yielding queen. I'd guess that direction dominates this comparison.
In high-dimensional spaces, "nearby" (distance) can begin to lose its meaning, so directional measures -- which are also by definition finite and determined a priori -- might make more sense if the "inner product space" supports it. (I threw the last part in there not totally understanding what an "inner product space" is, but it sounds cool and it is related... I just couldn't explain how.)
So, given that, I'd say that the idea of "orthogonality" isn't meaningful here. Two documents are either together in a smaller wedge of the space or a larger wedge of the space and that's that: 100 degrees apart is farther apart than 90 degrees, and 80 degrees apart is closer.
The answer mentioned here is correct. The measure of cosine distance as a measure of similarity only makes sense under some specific assumptions.
That it is possible to represent multiple complex objects as commensurable entities.
We can use quantitative methods to find qualitative answers. For example, we can measure the similarity using some numbers.
One can argue that these assumptions can be used as a guiding principle for a wide range of measures. Indeed they are. But cosine similarity is more popular because they match well with our spatial intuition and common sense. If you ask someone totally unrelated, to state the similarity of two documents between -1 and 1, then the answer would probably close to what cosine similarity would give as well.
There are other factors that also make cosine similarity a good choice. When thinking about the similarity of two documents, we do not care about the order of the words or other specific grammatical constructs. Hence the cosine distance, you will notice, does not take into account these factors and hence kind of captures that essence.
Lastly another advantage of cosine similarity is that, in high dimensional spaces such as text word embedding, are essentially non-intuitive to humans due to our inability to grasp non-euclidean spaces. Hence to understand meaning in a non euclidean space we need some way of mapping this high dimension to a low dimension space. Thus cosine distance helps the researcher to visualise the vector space to a large extent.
So to answer your question, the absolute value of cosine distance does not make sense by itself. It only makes sense if you are comparing between multiple choices. If you started with "king", then chances are that the text is talking about a "man" rather than a "woman" or a "bird" based on the cosine distances. Just the cosine distance between "man" and "king" has not value by itself.
Cosine similarity is computed all the text documents after pre processing , like removals of stop words, stemming and applying Term frequency. Let us say A,B,C,D are four documents, I need to find out the similarity I can apply cosine similarity after going through all the preprocessing and calculating the weigths. Finally determine which are similarity based on the Angel, Now I would like to see the key tokens which contribution, (like man of the match in Cricket) and what semantic it is occurring weather is object or thing or Noun or Verb and predict the tokens which is relevant , For instance ,I live in Java and place is good. Here the context is JAVA ia Island , not bike nor oops. These way CS(cosine similarity is applied and predictive approach is dtermeined) | 677.169 | 1 |
From @mail.uunet.ca:[email protected] Wed Dec 8 14:21:29 1993
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Subject: Antipodal Edge Position
From: [email protected] (Mark Longridge)
Message-Id: <[email protected]>
Date: Wed, 8 Dec 1993 12:33:00 -0500
Organization: CRS Online (Toronto, Ontario)
>>It's got to be all edges flipped in place.
Oops. Well I figured if all edges flipped was one of the hardest
know cube states that in the case of edges-only it would be the
antipode. I'm now sure (I think) that it is really:
all edges flipped + 4 X
(with the 4 X on sides F, R, B, L which should match Dan's diagram)
Hmmmm, I don't know if this is a standard form of representation,
but this picture looks like a folded out cube:
+ T + + F +
T T R L
+ T + + B +
+ L + + F + + R + + D + + D + + D +
L L F F R R => F B R L B F
+ L + + F + + R + + T + + T + + T +
+ D + + B +
D D R L
+ D + + F +
+ B + + T +
B B ---------> R L
+ B + | + D +
|
+ D +
In my program I would have L R on the screen for the bottom face.
+ T +
The idea is you are always looking at a cube face head-on (just to
clarify the difference in diagrams).
More quotes for Jerry Bryan:
>The "edges of the 3x3x3 without centers" is a little tougher. Early
>in the project, I generated a data base for the first few levels
>(six or seven, I think), and I have a "Solver program" that will
>work up to that level. However, the full "edges of the 3x3x3 without
>centers" is a 4.2 gigabyte file on tape, so it is hard to process.
>Also, the size of the equivalence classes is not in the data base,
>only the level. I have to calculate the size of each equivalence
>class, and it is an expensive calculation.
>
>I made a pass at the
>file and calculated the number of equivalence classes (took
>125 hours on a mainframe), but I only saved a summary. I did not
>save the number of equivalence classes for each state. I found
>the antipodal by looking for level 15, since I knew there was
>only one occurrence, and since the level was in the data base.
>
>I am not yet for sure what they look like, but of the other two states
>with order-24 equivalence classes, one is at level 9 and the other
>is at level 12. Since the only one at an even level is at level 12,
>I am assuming that will be the one which is Start with the edges all
>flipped. The one at level 9 will probably be the mirror image of
Start.
I'd still like to see the process for all-edges-flipped (not
caring about the centres or corners). So "level" is the number of moves
required to solve the position? That means edges flipped in place
can be done in 12 qtw. | 677.169 | 1 |
18 ... Q. E. D. PROP . VIII . THEOR . 1 If two triangles have two sides of the one equal to two sides of the other , each to each , and have likewise their bases equal ; the angle which is contained by the two sides of the one shall be equal ...
Page 19 ... Q. E. D. PROP . IX . PROB . To bisect a given rectilineal angle , that is , to divide it into two equal angles . it . Let BAC be the given rectilineal angle , it is required to bisect A Take any point D in AB , and from AC cut ( 3. 1 ...
Page 22 ... Q. E. D. PROP . XIV . THEOR . 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 ...
Page 23 ... Q. E. D. PROP . XV . THEOR . IF two straight lines cut one another , the vertical or opposite angles shall be equal . Let the two straight lines AB , CD cut one another in the point E ; the angle AEC shall be equal to the angle DEB ...
Page 24 ... Q. E. D. PROP . XVII . THEOR . ANY two angles of a triangle are together less than two right angles . Let ABC be any triangle ; any two of its angles together are less than two right angles . Produce BC to D ; and because ACD is the | 677.169 | 1 |
Equatian of the Ecliptic
The equation of the ecliptic is the relationship between two equatorial coordinates (right ascension and declination) of any point on the zodiac circle. Astrologers use this equation to calculate primary directions. In this article, we will derive the ecliptic formula, which is essential for an astrologer's work. | 677.169 | 1 |
Find a vector orthogonal to the plane of (subspace spanned by) the vectors u and v. Show work.
Help: R^3: is Euclidean 3-space
Please see the attached file for the complete solution.
Thanks for using BrainMass.
44705 Vectors, Dot Products and Orthogonality 3. Two vectors Xand Y are said to be orthogonal (perpendicular) if the angle between them is r/2.
(a) Prove that X and V are orthogonal if and only if X . Y = 0.
Thus they are mutually orthogonal to each other.
(b) Now we want to find such that is a set of mutually orthogonalvectors. Then we have
Then we solve the above system of equations and find one solution as follows.
199559 Orthogonal unit vectors Find two unit vectorsorthogonal to both i + j and i - j + k To find the vector perpendicular to both i+j and i-j+k, we need to calculate
(i+j)x(i-j+k) = i-j-2k
Now the two unit vectors are
u = (i-j-2k)/|i-j-2k| = (1/
49716 Area Measure andOrthogonalVectors Let m be an area measure on {z in C:|z| < 1}.
Show that 1, z, z^2,... are orthogonalvectors in L^2(m).
Find ||z^n||, n >= 0.
If e_n=(z^n)/||z^n||, n >= 0, is {e_0, e_1,...} a basis for L^2(m)?
52711 Vectors : Area of Parallelogram, Perpendicular Vectors , Angles Between Vectors, OrthogonalVectorsand Determinants 1. For vectors v and w in , show that v - w and v + w are perpendicular if and only if .
2 | 677.169 | 1 |
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Rhombus and its Properties
Apr 13, 2022
Do you wonder what shape will form if you tilt the square by 45°? Nevertheless, a rhombus is different from this. Let's learn more about Rhombus and its properties of Rhombuses.
A In a rhombus, look for symmetry lines for its proper identification.
The Rhombus resembles the shape of a quadrilateral. It has four vertices and four sides enclosing four angles, exactly like most other quadrilaterals such as rectangles, squares, and so on. It, however, is not all. There is a lot more to learn about this fascinating 2D shape which is an important component of mathematics.
It is also one of the key courses that follow us from school to university. So, let's learn everything there is to know about Rhombus, including its qualities, angles, sides, and two diagonals. But first, let's have an understanding of quadrilaterals.
What do you understand by a Quadrilateral?
As the name indicates, quad means four. Before we go into the Rhombus and its attributes, do you know what a quadrilateral is? It is a polygon with four sides and four vertices with four angles. The total sum of a quadrilateral's internal angles equals 360 degrees.
What is a Rhombus?
A rhombus is a four-sided figure (quadrilateral) that is a particular case of a parallelogram. The opposite sides of a rhombus are parallel, and the interior angles are equal.
Furthermore, all of a rhombus's sides are the same length, and the diagonals intersect each other at angles perpendicular to each other. A rhombus can alternatively be referred to as a diamond or a rhombus diamond because of its unique shape. Rhombi or rhombuses are the plural representation of Rhombus.
Is a square a Rhombus?
A square, like a rhombus, has all of its sides equal. In addition, the square's diagonals are perpendicular to one another and bisect the angles opposite to each other. As a result, a square is a kind of Rhombus.
What are the facts regarding the Angles of a Rhombus?
One might argue whether Rhombus is a square or not. However, although a rhombus is not a square, a square is a type of Rhombus. It makes it slightly different from the square shape.
Following are some important facts regarding the angles in a rhombus.
There are four interior angles in a rhombus
The sum of all the interior angles in a rhombus is 360 degrees
Interior opposite angles are equal to each other
In a rhombus, adjacent angles are supplementary to each other
Diagonals inside a rhombus intersect each other perpendicularly at right angles
The diagonals bisect each angle formed between them
Angles of Rhombus
Any rhombus has four angles, with the opposite ones being equal. Furthermore, this shape is made up of diagonals that intersect each other at right angles. To put it another way, each diagonal of a rhombus divides the other into two equal pieces, and the angle generated at their intersection points is 90 degrees.
The Rhombus has four interior angles, and as the sum of two alternate sides is 180 degrees, the overall sum of the Rhombus' four interior angles is 360 degrees. The diagonals also bisect the Rhombus's opposite angles, dividing the Rhombus into two separate triangles that are congruent to each other.
State the Properties of Rhombus
Because it has all of the qualities of a parallelogram, a rhombus is considered one of the exceptional parallelograms.
The symmetrical lines of a rhombus are its two diagonals. A line that separates an object into two identical halves is known as an axis of symmetry. Both sides of the object are reflected in a mirror-like reflection. Over both of its diagonals, a rhombus is said to exhibit reflection symmetry. Some of the parallelogram's general qualities are as follows:
Opposite angles inside a parallelogram are equal or congruent
In a parallelogram, the opposing sides are parallel to each other
The opposite sides are equal in length
Each diagonal bisects the other one
The sum of two consecutive angles in a parallelogram is 180 degrees
The diagonal of a rhombus should be remembered because, in addition to bisecting each other at right angles, the two diagonals bisected will also be the same length.
Like other geometric shapes, the properties of rhombus are particular to it. The properties of rhombus are shown in the table below.
Properties of Rhombuses
References
A rhombus has congruent sides (equal).
In this case, AB = CD = DA = BC
The perpendicular bisector of each of the two diagonals in a rhombus is the diagonal that it bisects at 90 degrees.
Here, diagonals DB and CA form a 90° angle with one another.
The opposing sides are parallel, and the opposing angles are equal.
CD || AB and BC || AD
Angles adjacent to one another add up to 180 degrees.
∠A + ∠B = 180° ∠B + ∠C = 180° ∠C + ∠D = 180° ∠A + ∠D = 180°
For example, if a diagonal is 12 cm long and is bisected by another diagonal, it is divided into two 6 cm pieces. One can calculate the length of the diagonal if one knows the side of the Rhombus and the values of certain angles.
How to identify a Rhombus?
A rhombus is a quadrilateral. It is shaped like a diamond and has equal sides on all sides. In our daily lives, we see rhombus-shaped figures. A diamond, a kite, and other real-life examples.
How do find Rhombus in daily life?
Even though one might not notice it, at all times, the Rhombus shape is present in front of our eyes. From the kite shape to the diamond shape. The application of Rhombic shape may be seen everywhere, from the design of a kite to the shape of jewellery.
The shape of a Rhombus is so common that it can be found in shop signs, road signs, key chains, tiles, baseball fields, and so on. This shape can also be seen in several well-known architectural structures around the world.
The Rhombic shape is so popular because it is symmetrical and has a very attractive and pleasing shape. Because all four sides of the Rhombus are equal, the figure is also geometrically viable.
Rhombus Formulas
The formula of a rhombus deals with two main parameters. These are – the perimeter and area.
The expression gives the formula for the area of a rhombus-
A = ½ × d1 × d2
Here, d1 and d2 are the diagonals of the Rhombus.
The formula for the perimeter of the Rhombus is given by-.
P = side + side + side + side
Since the four adjacent sides of a rhombus are equal,
P = 4 × side
What is the Perimeter of a Rhombus, and how to calculate it?
A perimeter of a shape refers to the total length covered by all the sides of that shape. In other words, it is better defined as a boundary that restricts a figure and confines its outlines.
For a rhombus, its perimeter will be the sum of all its four sides that confines its countries. So, the perimeter will be :
P = side + side + side + side
P = 4 × side
For example, as it is known, all the sides of a rhombus are equal. So, if one side = 7 cm
Then, the perimeter will be:
P = 7 + 7 + 7 + 7
P = 28 cm
What is the area of a Rhombus, and how to calculate it?
An area of an object is defined as the total space occupied by that particular object. In this case, the area of a rhombus is the entire space occupied by the four sides and everything between them.
The area of a rhombus is given by the following equation:
A = ½ × d1 × d2
Here, d1 and d2 refer to the diagonals of a rhombus.
For example: if the diagonals of a rhombus are 12 cm and 6 cm. Then, is the area will be:
A = ½ × d1 × d2
A = ½ × 12 × 6
A = 36 cm²
What are the various other Properties of Rhombus?
For a quick review of the properties of rhombus, read the following points.
All four sides are the same length.
The opposite sides are parallel.
Opposite interior angles are the same.
At right angles or 90 degrees, diagonals bisect each other perpendicularly.
The diagonals of a rhombus intersect its opposite angles.
The total of two neighboring angles, i.e. 180°, is supplementary.
A rectangle is formed by uniting the midpoints of the sides of a rhombus.
There is no possibility of an inscribing circle within a rhombus.
Two congruent equilateral triangles are created when the shortest diagonal side of a rhombus equals one of its sides.
Summary
To summarize, the Rhombus' shape is symmetric along its diagonals, which means that the area on both sides of the diagonals is equal. That is, dividing the Rhombus from one of its diagonals yields symmetric objects with the same area and perimeter. The Rhombus' symmetric feature is mostly due to the presence of the two diagonals that are equal and bisect each other. The properties of rhombus are an important concept in mathematics and geometry.
Solved Rhombus Examples
Example 1: David has drawn a rhombus with two diagonals that are, respectively, 5 units and 10 units long. He asks Linda, his sister, to guide him to the location. Can you help Linda to find the solution?
Solution:
Given:
The diagonal values are d1 = 5 units and d2 = 10 units.
A = (d1 × d2)/2
A = (5×10)/2
A = 25 square meters
The rhombus has a surface area of 25 square units.
Example 2: Sam and Victor were playing a game of hopscotch when they noticed a tile at the playground that was shaped like a rhombus. The tile was 15 units long on either side. Can you help Sam and Victor in finding the tile's perimeter?
Solution:
Given:
The tile is 15 units long.
All four sides of a rhombus are equal to 15 units since all its sides are equal.
4 units for the perimeter, 4 units for each side, and 15 units total.
The tile's perimeter is 60 units, thus that's the answer.
Frequently Asked Questions
1. What are the qualities of a rhombus?
Ans. A rhombus has four equal sides, and two of those sides are perpendicular to the other two.
Rhombuses are named for their shape: a rhombus can be thought of as a diamond cut in half.
2. What is a real-life example of a rhombus?
Ans. A rhombus is a four-sided shape with all sides equal in length and all angles equal. The most common example of a rhombus is a diamond. The shape of a diamond is a rhombus because all of its edges are the same length and all of its angles are equal.
3. What do all rhombuses have in common?
Ans. Rhombuses are all about corners. They have four of them, and they're all 90 degrees. Rhombuses are also parallelograms, which means that they have two pairs of parallel sides.
4. What are the facts about rhombus?
Ans. -A rhombus is a quadrilateral with four equal sides and four equal angles.
-Rhombuses have many different names, including parallelograms and kite shapes.
-A rhombus can be a square if all of its sides are square.
5. What are the properties of rhombus?
Ans. A rhombus is a parallelogram that has four equal sides. The angles of a rhombus are always equal to one another. It is also a special case of a parallelogram, as it has all sides equal. In addition, the diagonals of a rhombus are perpendicular to each other | 677.169 | 1 |
3rd Grade: Geometry (3.G)
Geometry (3.G)
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3.G.A.1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.A.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. | 677.169 | 1 |
Term 4 - Shape, Fractions, length
Shape Objectives
identify and describe the properties of 2-D shapes, including the number of sides and line symmetry in a vertical line § identify and describe the properties of 3-D shapes, including the number of edges, vertices and faces § identify 2-D shapes on the surface of 3-D shapes, [for example, a circle on a cylinder and a triangle on a pyramid] § compare and sort common 2-D and 3-D shapes and everyday objects | 677.169 | 1 |
Two mirrors A and B are placed at right angles to each other
Two mirrors A and B are placed at right angles to each other as shown in Figure.
A ray of light incident on mirror A at an angle of 25° falls on mirror B after reflection. The angle of reflection for the ray reflected from mirror B would be
25°
50°
65°
115°
Answer
Angle of reflection for the ray reflected from mirror B will be 65° because figure below shows how reflected ray from mirror A forms incident ray on mirror B and then reflected back by an angle of 65°. | 677.169 | 1 |
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Trigonometry – Area of a Triangle
£2.50
Using the sine formula to find the area of any triangle no prep lesson.
Power point with proof of formula and 3 examples. Each example has one for teacher to do and then one for students to try,
Worksheet on finding areas, missing lengths and missing angles. | 677.169 | 1 |
Always, Sometimes or Never True - Set #2 center of a circle that circumscribes a triangle is inside the triangle.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
The center of the circle that circumscribes a triangle sometimes lies inside the triangle. The
statement is true when the triangle is acute. In the case of a right triangle the center of the
circumscribing circle lies on the triangle. In the case of an obtuse triangle, the center of the
circumscribing triangle lies outside the triangle.
2. An altitude subdivides a triangle into two similar triangles.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
It is sometimes true that an altitude subdivides a triangle into similar triangle. Two interesting case
are that of an altitude drawn in an isosceles triangle from the non-base angle, and that of an altitude
drawn in a right triangle from the right angle.
3. (a + b)2 = a2 + b2
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
This is only true when a = 0 or b = 0 or a = b = 0.
4. 3x2 = (3x)2
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
Again this is true only when x = 0
5. A shape with a finite area has a finite perimeter.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Sometimes true.
If the shape has fixed area A and dimensions x and A/x. The perimeter is thus 2(x + A/x). As x is
varied, this can be made as large or as small as we please. So theoretically, the perimeter may
become infinite - but then is an infinitely thin rectangle still a rectangle (it feels more like a line to
me!)? As an aside, it may also be noted that many fractals also have infinite boundaries but enclose
finite areas. Moving up a dimension, it is also interesting to note that the trumpet shape obtained
by rotating the curve y = 1/x around the x axis for x > 1 has a finite volume, but an infinite surface
area. (Does this suggest that we can fill the shape with a finite amount of paint, but that the paint
will never fully coat the surface?)
6. A shape with a finite perimeter has a finite area.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
Sample Solution: Always true.
It is always true that a shape with a finite perimeter has a finite area. If the dimensions of the
rectangle are x and y, then if 2(x + y) is finite and x and y are both positive, both x and y must be
finite, so the area must be finite. | 677.169 | 1 |
Circular sector
Circular sector
A circular sector is shaded in green
A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, r the radius of the circle, and L is the arc length of the minor sector.
A sector with the central angle of 180° is called a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants (90°), sextants (60°) and octants (45°).
The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.
Contents
Area
The total area of a circle is πr2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle and 2π (because the area of the sector is proportional to the angle, and 2π is the angle for the whole circle):
Another approach is to consider this area as the result of the following integral :
Perimeter
The length of the perimeter of a sector is the sum of the arc length and the two radii:
where θ is in radians.
Center of Mass
The distance from the center of the circle (that the sector is a part of) to the center of mass of the sector is two thirds of the corresponding distance for the center of mass of the arc of the sector. In particular, as the central angle approaches zero the center of mass of the arc is at distance r from the center of the circle, so that of the sector is at distance 2r/3. As the central angle approaches 2π (the whole circle), the center of mass of the arc converges to the center of the circle, whence so does that of the circular sector.
See also
Circular segment - the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
Sector — may refer to: * Sector, Devon, a location in the county of Devon in south western England * Sector, West Virginia, an unincorporated community in Hampshire County, West Virginia, United States of America * Sector (economic): one of several… … Wikipedia
Circular segment — In geometry, a circular segment is an area of a circle informally defined as an area which is cut off from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding the circleSector instrument — A five sector mass spectrometer A sector instrument is a general term for a class of mass spectrometer that uses a static electric or magnetic sector or some combination of the two (separately in space) as a mass analyzer.[1] A popular… … Wikipedia | 677.169 | 1 |
How do I get horizontal distance in AutoCAD?
Analyze tab of the Ribbon, find Inquiry Tool. Run the Point Inverse function. This will report horizontal distance.
How do you measure horizontal distance?
Rearrange the terms of equation: multiply both sides by run. Divide both sides by slope percent. run = (rise × 100 ) / slope % is a measure of horizontal distance.
How do you measure distance in AutoCAD?
You can measure the length of objects, the angle and distance between objects, and the radius of circles and arcs by tapping MEASURE, then tapping in the drawing area. … To measure distance:
Tap MEASURE.
Choose Distance.
Specify a first and a second point. Use object snaps for precision.
How do you find the distance between two lines in AutoCAD?
In the drawing area, double-click the dimension you want to edit. The Power Dimensioning Ribbon Contextual Tab displays. In the Power Dimensioning Edit Geometry dialog box, enter a numerical value in the Text offset from dimension line box. Click OK.
How do I Measure distance in AutoCAD 2020?
In AutoCAD 2020, measuring distances is faster than ever with the new Quick option of the MEASUREGEOM command, which measures 2D drawings simply by hovering your mouse. When this option is active, dimensions, distances, and angles are displayed dynamically as you move your mouse over and between objects.
How do you Measure distance in AutoCAD 2021?
How do you measure horizontal distance from a total station?
When calculating the horizontal distance between points, this difference also needs to be taken into consideration. Because the total station gives a true slope distance, the horizontal difference is simply H = s * cos α (see figure below).
What does horizontal distance mean?
Horizontal distance means the distance between two points measured at a zero percent slope | 677.169 | 1 |
...quadrilateral is a straight line joining its opposite angles. Circ1e. 30. A circle is a plane figure, enclosed by one line, which is called the circumference, and is such that all lines drawn from a certain point within the figure to the circumference are equal to one another. 31....
...is one which is less than a right angle. ^ XIII. A CIRCLE is a plane figure contained by one lino, which is called the CIRCUMFERENCE, and is such, that all straight lines drawn to the circumference from a certain point (called the CENTRE) within the figure are equal to one another....
...is the extremity of any thing. 14. A figure is that which is enclosed by one or more boundaries. 15. A circle is a plane figure contained by one line,...figure to the circumference are equal to one another : 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line...
...as above, and using the letters affixed to these lines instead of A, J3, C, D. ff Definitions. — A circle is a plane figure contained by one line,...within the figure to the circumference are equal. And that point is called the centre of the circle. General Enunciation. From the greater of two given...
...those which are called curvilinear, being bounded not by straight lines but by curved lines. Def.—A circle is a plane figure contained by one line, which...is called the circumference, and is such that all strait lines drawn from its centre to the circumference are equal to one another. The straight line...
...angles which are not adjacent to one another. XVII. A CIRCLE is a figure contained, on a flat surface, by one line which is called the CIRCUMFERENCE ; and is such that all straight lines drawn to the circumference from a certain point within the figure, called the CENTRE, are equal to each other....
...which is contained by one or more boundaries. 15. A CIRCLE is a plane figure contained by one line called the CIRCUMFERENCE, and is such that all straight lines drawn from a certain point within it, called the CENTRE, to the circumference are equal to each other. Each of such equal straight lines...
...VII. — A plane figure is a portion of a plane surface inclosed by a line or lines. "Def. VIII. — \circle is a plane figure contained by one line, which is called the circumference, &c. "Def. XXII. — A plane rectilineal figure is a portion of a plane surface inclosed by straight...
...figurca, u well u to plane figures XV. A circle is a plane figure contained [or, bounded] by •>ae line, which is called the circumference, and is such...figure to the circumference, are equal to one another. The circumference of a circle ie its boundary. The space contained within the boundary, \s called the...
...them lies wholly in that superficies. An obtuse angle is that which is greater than a right angle. A circle is a plane figure contained by one line which...circumference, and is such that all straight lines drawn from я certain point (called the centre) within the figure to the circumference are equal to one another.... | 677.169 | 1 |
Get Your First Solution Free
In a triangle, the sum of any two sides is greater than the other side. The Triangle class must adhere to this rule. Create the IllegalTriangleException class, and modify the constructor of the Triangle class to throw an Illegal Triangle Exception object
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Description
In a triangle, the sum of any two sides is greater than the other side. The Triangle class must adhere to this rule. Create the IllegalTriangleException class, and modify the constructor of the Triangle class to throw an IllegalTriangleException object if a triangle is created with sides that violate the rule, as follows:
/** Construct a triangle with the specified sides */
public Triangle(double side1, double side2, double side3) throws IllegalTriangleException {
// Implement it | 677.169 | 1 |
Advanced mathematics
Hexagon Cut Out
The diagram shows an irregular hexagon with interior angles all equal to 120 degrees made by cutting the corners off a piece of card in the shape of an equilateral triangle with sides of length 20 units.
An identical hexagon could also be made by cutting the corners off a different equilateral triangle | 677.169 | 1 |
How to Find Arc Length of a Circle: A Comprehensive Guide
Learn how to find arc length of a circle using the central angle method and the formula method. Discover the definition of arc length, relationship between arc length and circumference, and how radians and degrees differ. Explore examples and tips to simplify the concept of arc length in circles. Foster a deeper understanding of this vital mathematical concept and its practical applications in various fields.
I. Introduction
Circles are among the most basic and prevalent shapes in mathematics. One of the fundamental concepts associated with circles is the arc length. Understanding arc length is vital in various disciplines, including physical sciences, engineering, and architecture. Arc length can be used to determine the position of an object in motion, measure the perimeter of a sector, and identify the length of a curved line. Therefore, in this article, we will explore how to find arc length of a circle, emphasizing the formulas, approaches, and tips to master the concept.
II. Basics of Arc Length Calculation in a Circle
Arc length is the distance along a circle's edge between two points, measured in linear units. The arc length symbol is "s," and it is denoted in units of length, such as inches or meters. For a given circle, arc length s is related to the circle's circumference C and the measure of the central angle θ (in radians) as follows:
Where:
s: arc length
θ: central angle (in radians)
C: circumference of the circle
The central angle θ is the angle formed by two radii of a circle that intersect at the center. One full revolution, or 360°, is equal to 2π radians. Therefore, we can convert degrees to radians by multiplying by π/180 or use a calculator's degree-to-radian conversion function.
III. Finding Arc Length Using Central Angle
The formula for finding arc length using the central angle θ is:
Where:
r: radius of the circle
Therefore, the arc length s is equal to the central angle θ multiplied by the radius r. This equation is applicable when the angle θ is measured in radians.
Example problem: Find the arc length of a circle with a radius of 10 inches and a central angle of 60°.
First, convert 60° to radians by multiplying 60 by π/180, which gives θ = π/3 rad. Then, substitute the values of r and θ into the formula:
Therefore, the arc length of the circle is approximately 10.47 inches.
IV. Formula to Find Arc Length in a Circle
The arc length formula expressed as a function of the circle's diameter and central angle θ is:
Where:
d: diameter of the circle
This equation is more general than the previous formula and applies to both degrees and radians. To use this equation, the measure of the central angle must be in the same unit as the formula's denominator.
To obtain the arc length for a given measure of the central angle, the following formula can be used:
Where:
r: radius of the circle
θ: central angle (in radians)
This equation is useful when the diameter of the circle can be measured more accurately than the radius.
Example problem: Find the arc length of a circle with a diameter of 20 meters and a central angle of 135°.
First, convert 135° to radians by multiplying 135 by π/180, which gives θ = 3π/4 rad. Then, use the formula:
Therefore, the arc length of the circle is approximately 7.85 meters.
V. Mastering the Concept of Arc Length in a Circle- A Beginner's Guide
As with any mathematical concept, understanding arc length requires practice, patience, and knowledge of the basics. Here are some common misconceptions and tips to help you master the concept.
Common misconceptions and mistakes:
Confusing arc length with arc angle: arc angle is the measure of the central angle, while arc length is the distance traveled along the arc.
Assuming that arc length is equal to the chord length: a chord is a line drawn between two points on the circumference and is not equivalent to an arc length.
Tips for understanding and visualizing the concept:
Use a compass and ruler to draw circles and their properties.
Relate arc length to real-life scenarios, such as a vehicle traveling along a circular track.
VI. Simplifying the Process of Finding Arc Length in a Circle – Tips and Tricks
There are many useful shortcuts and tricks that can be helpful in calculating arc length.
Useful shortcuts for calculations:
When using the formula , it is sometimes easier to convert degrees to radians first, perform the multiplication, and then convert the answer back to degrees.
When using the formula , it is useful to reduce the fraction in the equation before plugging in the values.
For large central angles, it may be simpler to divide the circle into smaller sectors and add their arc lengths.
Practical applications of arc length:
Determining the distance traveled by a rotating object
Measuring the perimeter of a field or a circular painting
Calculating the circumference of a round playground, gyroscope or a tire
VII. Different Methods to Find Arc Length in a Circle – A Comparative Study
There are two primary methods to calculate arc length: using central angle and using the formula. Each method has its advantages and disadvantages.
Comparison between the formula method and central angle method:
The formula method is more general and applies to both degrees and radians, while the central angle method is only applicable to radians.
The central angle method is more straightforward when the angle is given in radians, while the formula method is simpler when using degrees.
Both approaches can yield the same result; the choice depends on the given situation and personal preference.
VIII. Conclusion
Arc length is a vital concept that is frequently utilized in various fields to measure distances and quantities. Understanding arc length requires knowledge of basic concepts such as the circle's circumference, radius, and central angle. The two primary methods to calculate arc length are using central angle or using the formula. Tips and tricks such as converting units, dividing sectors, and visualizing the concept can aid in mastering the concept. By practicing and developing proficiency in calculating arc length, individuals can apply this concept in numerous practical scenarios. | 677.169 | 1 |
The first three books of Euclid's Elements of geometry, with theorems and problems, by T. Tate
Dentro del libro
Resultados 1-5 de 10
Página 87 ... hypotenuse of a right - angled triangle , and the difference of the two acute angles , to construct the triangle . 13. To construct a right - angled triangle , having given the hypotenuse , and one of the acute angles equal to one third ...
Página 96 ... hypotenuse produced , they will cut off equal segments , and the perpendiculars will together be equal to the hypotenuse . 13. If a line PQ be drawn parallel to the base BC of a triangle ABC ( see fig . Theo . 1. No. 2. ) through the ...
Página 98 ... hypotenuse . 7. In any isosceles triangle , if a line be drawn from the vertex to any point in the base , the square upon this line , together with the rectangle contained by the segments of the base , is equal to the square upon either ...
Página 99 ... hypotenuse , to describe a right - angled triangle , such that the hypotenuse , F 2 EXERCISES ON BOOK II . 99.
Página 100 Euclid, Thomas Tate. right - angled triangle , such that the hypotenuse , together with the less of the two sides , shall be double the greater . 25. Show that the algebraical proposition ( a + b ) 2 + ( a − b ) 2 = 2a2 + 2b3 , | 677.169 | 1 |
Polar Graph. The Polar Graph is a graph in the polar coordinate system in which the each point on the plane is defined by two values - the polar angle and the polar radius. The certain equations have very complex graphs in the Cartesian coordinates, but the application of the polar coordinate system allows usually produce the simple Polar Graphs for these equations.
Create flowcharts, org charts, shipping flowcharts, floor plans, business diagrams and more with ConceptDraw. Includes 1000s of professional-looking business templates that you can modify and make your own | 677.169 | 1 |
The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...
angles, EAF, EBF, there are two angles in one equal to Book III. two angles in the other, and the side EF, which is opposite to one of the equal angles in each, is common to both; therefore the other sides are equale; AF therefore is equal é 26. 1. to FB. Wherefore, if a straight line, &c. Q. E. D.
PROP. IV. THEOR.
Ir in a circle two straight lines cut one another which do not both pass through the centre, they do not bisect each other.
Let ABCD be a circle, and AC, BD two straight lines in it which cut one another in the point E, and do not both pass through the centre: AC, BD, do not bisect one another.
For, if it is possible, let AE be equal to ÉC, and BE to ED: If one of the lines pass through the centre, it is plain that it cannot be bisected by the other which does not pass through the centre: But if neither of them pass through the centre, take a F the centre of the circle, and join A EF; and because FE, a straight line through the centre, bisects another AC which does not pass
E
C
1.3.
through the centre, it shall cut it at right angles; where- b 3. 3. fore FEA is a right angle: Again, because the straight line FE bisects the straight line BD, which does not pass through the centre, it shall cut it at right angles: wherefore FEB is a right angle: And FEA was shown to be a right angle; therefore FEA is equal to the angle FEB, the less to the greater, which is impossible: Therefore AC, BD do not bisect one another. Wherefore, if in a circle, &c. Q. E. D.
PROP. V. THEOR.
Ir two circles cut one another, they shall not have the same centre.
Let the two circles ABC, CDG cut one another in the points B, C; they have not the same centre.
BOOK III.
For, if it be possible, let E be their centre; join EC, and draw any straight line EFG meet
ing them in F and G; and be- cause E is the centre of the circle ABC, CE is equal to EF: Again, A/D because E is the centre of the cir- cle CDG, CE is equal to EG: But CE was shown to be equal to EF; therefore EF is equal to EG, the less to the greater, which is impossible: Therefore E is not
E
the centre of the circles ABC, CDG. Wherefore, if two circles, &c. Q. E. D.
PROP. VI. THEOR.
Ir two circles touch one another internally, they shall not have the same centre.
Let the two circles ABC, CDE, touch one another inter- nally in the point C: They have not the same centre. For, if they have, let it be F; join FC and draw any straight line FEBmeeting them in
E and B; And because F is the centre of the circle ABC, CF is equal to FB; Also, because Fis the centre of the circle CDE, CF is equal to FE: And CF was shown to be equal to FB; there- fore FE is equal to FB, the less to the greater, which is impossible: Wherefore F is not the centre of
A
the circles ABC, CDE. Therefore, if two circles, &c. Q. E. D.
BOOK III.
PROP. VII. THEOR.
If; and, of any others, that which is nearer to the line which passes through the centre is always greater than one more remote: And from the same point there can be drawn only two straight lines that are equal to one another, one upon each side of the shortest line.
Let ABCD be a circle, and AD its diameter, in which let any point F be taken which is not the centre: Let the centre be E; of all the straight lines FB, FC, FG, &c. that can be drawn from F to the circumference, FA is the greatest, and FD, the other part of the diameter, AD, is the least: And of the other, FB is greater than FC, and FC than FG.
B
a
Join BE, CE, GE; and because two sides of a triangle are greater than the third, BE, EF, are greater than BF; * 20. 1. but AE is equal to EB; therefore AE, EF, that is AF, is greater than BF: Again, because BE is equal to CE, and FE common to the triangles BEF, CEF, the two sides BE, EF are equal to the two CE, EF; but the angle BEF is greater than the angle CEF; therefore the base BF is greaterb than the base FC: For the
H
same reason, CF is greater than GF: Again, because GF, FE are greater than EG, and EG is equal to ED; GF, FE are greater than ED: Take away the common part FE, and the remainder GF is greater than the remainder FD: Therefore FA is the greatest, and FD the least of all the straight lines from F to the circumference; and BF is greater than CF, and CF than GF.
If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to that through the centre is always greater than the more remote: But of those which fall upon the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than the more remote: And only two equal straight lines can be drawn from the point unto the circumference, one upon each side of the least.
Let ABC be a circle, and D any point without it, from which let the straight lines DA, DE, DF, DC, be drawn to the circumference, whereof DA passes through the centre. Of those which fall upon the concave part of the circumference AEFC, the greatest is AD which passes through the centre; and the nearer to it is always greater than the more remote, viz. DE than DF, and DF than DC: But of those which fall upon the convex circumference HLKG, the least is DG between the point D and the diameter AG;
and the nearer to it is always less than the more remote, Book III. viz. DK than DL, and DL than DH.
€ 24.1.
Take a M the centre of the circle ABC, and join ME, MF, 1. 3. MC, MK, ML, MH: And because AM is equal to ME, add MD to each, therefore AD is equal to EM, MD; but EM, MD are greater than ED; therefore also AD is 20. 1. greater than ED. Again, because ME is equal to MF, and MD common to the triangles EMD, FMD; EM, MD, are equal to FM, MD: but the angle EMD is greater than the angle FMD; therefore the base ED is greater than the base FD. In like manner it may be shown that FD is greater than CD: Therefore DA is the greatest; and DE greater than DF, and DF than DC: And because MK, KD are greater than MD, andMK is equal to MG, the remain- C der KD is greater than the remainder GD, that is GD is less than KD: And because MK, F DK, are drawn to the point K
within the triangle MLD, from
H
GB
N
M
d4 Ax.
E
A
M, D, the extremities of its side MD, MK, KD, are less e 21. 1. than ML, LD, whereof MK is equal to ML; therefore the remainder DK is less than the remainder DL: In like manner it may be shown, that DL is less than DH: Therefore DG is the least, and DK less than DL, and DL than DH. Also there can be drawn only two equal straight lines from the point D to the circumference, one upon each side of the least. At the point M, in the straight line MP, make the angle DMB equal to the angle DMK, and join DB: And because MK is equal to MB, and MD common to the triangles KMD, BMD, the two sides KM, MD are equal to the two BM, MD; and the angle KMD is equal to the angle BMD; therefore the base DK is equal to the 4. 1. base DB: But, besides DB, there can be no straight line drawn from D to the circumference equal to DK: For, if there can, let it be DN; and because DK is equal to DN, and also to DB; therefore DB is equal to DN, that is, the nearer to the least equal to the more remote, which is impossible. If therefore any point, &c. Q.E.D. | 677.169 | 1 |
The figures of Euclid with the enunciations, as printed in Euclid's Elements of plane geometry [book 1-4, 6] by W.D. Cooley
Dentro del libro
Página 37 ... tangent to the circle : and through the same point there cannot be drawn another straight line also a tangent to the circle . A B E PROP . XVII . PROB . To draw a tangent BOOK III . 37.
Página 38 Euclides William Desborough Cooley. PROP . XVII . PROB . To draw a tangent to a given circle from a given point , either in or outside of its circum- ference ... tangent . B E PROP . XIX . THEOR . If a straight line 38 EUCLID'S ELEMENTS .
Página 39 ... tangent , passes through the centre of the circle . PROP . XX . THEOR . The angle at the centre of a circle , is double of the angle at the circumference , standing on the same arch . D B E G PROP . XXI . THEOR . All angles in the BOOK ...
Página 44 ... tangent are equal to the angles in the alternate segments of the circle . E PROP . XXXIII . PROB . Upon a given straight line to describe a segment of a circle containing an angle equal to a given angle . F B PROP . XXXIV . PROB . To ...
Página 46 ... tangent . B B E PROP . XXXVII . THEOR . If from a point outside of a circle two straight lines be drawn , the one cutting the circle , the other meeting it , and if the rectangle contained by the whole cutting line and its external | 677.169 | 1 |
Year 4 Identify Angles Game
Test your knowledge of acute, obtuse and right angles in this Year 4 Identify Angles Game. Can you answer all the questions correctly?
Teacher Specific Information
Year 4 Identify Angles Game includes five questions designed to check pupils' understanding of acute, obtuse and right angles. Pupils will label angles, complete sentences identifying and explaining angles, find and explain the odd angle out, identify obtuse angles from a list of measurements and work out a word problem to find an angle. | 677.169 | 1 |
Geometry: Common Core (15th Edition)
by
Charles, Randall I.
Answer
$x = 18.7$
Work Step by Step
Since we have the measure of one angle and the measure of the side opposite to that angle, we can use the sine ratio to find $x$, which is the hypotenuse of the triangle.
The sine ratio is given as follows:
sin $A = \frac{opposite}{hypotenuse}$
Let's plug in what we know:
sin $40^{\circ} = \frac{12}{x}$
Multiply each side by $x$:
sin $40^{\circ}(x) = 12$
Divide both sides by sin $40^{\circ}$ to solve for $x$:
$x = 18.7$ | 677.169 | 1 |
Triangle – A Three-Sided Polygon
Mathematics has played a significant influence in the transformation of our environment. Mathematical applications can be found in our daily lives. Some of the world's greatest mathematicians have solved some of the world's most difficult issues and have made their imprint on the subject of mathematics. Most people believe that math is solely about numbers, but it is much more; it is a subject that involves patterns, numbers, and shapes. Many students see it as one of the most difficult disciplines in the world. People who work in math have to deal with a lot of complicated computations. Because of its applications in the real world, every student is required to study math from the beginning of their scholastic career. From counting elementary numerals coding to rocket science, math is needed everywhere. Algebra, number theory, statistics, calculus, geometry, and arithmetic are the main branches of mathematics. We will be discussing one of the most common shapes in geometry that is a triangle. The area of triangle and the perimeter of a triangle are some of the most common properties of a triangle that are being used in our daily life.
Triangle is a shape that consists of three sides and three corners. The total of the three unique angles equals a hundred and eighty degrees, which is one of the main characteristics of this shape. This characteristic is frequently used to solve a variety of difficult problems using various figures in which a triangle is included. Triangles can be classified into various categories. Equilateral, isosceles, and scalene triangles are the three basic forms of triangles. The various categories are described based on the equality of their attributes. The triangle is said to be equilateral if all three sides are the same length. If any two sides are identical, it is said to be isosceles, and if all sides are different lengths, it is said to be scalene. The right-angled triangle is a special type of triangle. A right-angled triangle is defined as one in which any of the three angles is identical to ninety degrees. There are a lot of methods to find the area of a triangle, we are going to discuss a few of them.
Area of a triangle: One of the most used properties of a triangle is the area of a triangle, area is defined as the space occupied by any object. As a triangle is a two-dimensional shape it also occupies space. One of the basic ways to calculate the area of a triangle is by using the formula ½ *base*height. This formula is of great importance in geometry as this formula is valid for all kinds of triangles whether it is equilateral, isosceles, or scalene. Apart from this formula, the other method which can be used to calculate the area of a triangle is Heron's Formula. In Heron's formula, we first need to calculate the semi perimeter of a triangle that is obtained by dividing the perimeter of the triangle by 2. Let a, b and c be the three sides of a triangle and d be the semi-perimeter of that triangle, then the formula to calculate area is square root of d*(d-a) *(d-b) * (d-c). Apart from these formulas, there are many different ways to calculate the area of a triangle, that depends on the type of triangle and the angle made by the triangle.
We attempted to cover all of the ideas related to complementary angles in the preceding post. People can now discover a variety of platforms to obtain information thanks to the rise of online learning. Cuemath is one such platform. It is one of the most effective tools for making our math difficulties crystal plain. Its language is simple to comprehend. There are dozens of math-related subjects to read on it. Not only school or college students, but everyone of any age can benefit from this platform by obtaining access to the vast amount of information provided. In the last several years, the popularity of online learning has skyrocketed. Not only does studying from such online platforms save us energy, but it also saves us time. Such platforms should be used to their full potential | 677.169 | 1 |
browndogfancy
Help me plz help me me plz
Accepted Solution
A:
Answer:See below.Step-by-step explanation:Gale's work is incorrect. The rotation would give him a horizontal line.The transformations that are required to give A'B' as it is on the grid are a translation of 8 units to the right followed by a translation of 2 units down.The rotation through 90 degrees clockwise and then a translation of 2 units down would give a horizontal line with B' at the same point as on the diagram but the point A' would be 6 units to the left on the point (-2, -2). | 677.169 | 1 |
theamazings
For the Parallelogram, find the coordinates of P without using any new variables. If you have the an...
4 months ago
Q:
For the Parallelogram, find the coordinates of P without using any new variables. If you have the answers to the rest of the multiple choice, I would appreciate those being given as I am 2 weeks behind because I'm struggling to take this test
Accepted Solution
A:
I think the answer is b.(C+b-a, d).
Because d is the y-coordinate of the point that is on the left of point P so i guess it makes sense. | 677.169 | 1 |
Class 11 Maths NCERT Solutions Chapter 3 – Trigonometric Functions
Class 11 Maths NCERT Solutions Chapter 3 is devised accurately as per the latest CBSE syllabus introduced. The Trigonometry Class 11 in Chapter 3 of the syllabus comprises of step by step shortcut techniques. Our study material is available for free in the online PDF free downloads for all subjects. Vedantu is the most preferred option for exam preparation as we offer the download option for allCBSE class 11 Maths NCERT Solutions without any cost. They can download NCERT Solution PDF to score high marks in their examination.
As a result, the length of the minor arc of the chord is $\frac{{20\pi }}{3}\,{\text{cm}}.$
6. If in two circles, arcs of the same length subtend angles ${60^\circ }$ and ${75^\circ }$ at the centre, find the ratio of their radii.
Ans: The radii of the two circles should be ${r_1}$ and ${r_2}$. Whereas an arc of length $l$ subtend an angle of ${60^\circ }$ at the centre of the circle of radius ${r_1}$, while let an arc of length/subtend an angle of ${75^\circ }$ at the centre of the circle of radius ${r_2}$.
Now, ${60^\circ } = \frac{\pi }{3}$ radian and ${75^\circ } = \frac{{5\pi }}{{12}}$ radian. We also know that in a circle of radius $r$ unit, if an arc of length $l$ unit subtends an angle $\theta $. radian at the centre then $\theta = \frac{l}{r}$ or $l = r\theta $
So, $l = \frac{{{r_1}\pi }}{3}$ and $l = \frac{{{r_2}5\pi }}{{12}}$
$ \Rightarrow \frac{{{r_1}\pi }}{3} = \frac{{{r_2}5\pi }}{{12}}$
$ \Rightarrow {r_1} = \frac{{{r_2}5}}{4}$
$ \Rightarrow \frac{{{r_1}}}{{{r_2}}} = \frac{5}{4}$
As a result, the ratio of the radii is 5: 4.
7. Find the angle in radian through which a pendulum swings if its length is $75\;cm$ and the tip describes an arc of length.
(i) 10Now, $l = 10\;{\text{cm}}$
$\theta = \frac{{10}}{{75}}$ radian $ = \frac{2}{{15}}$ radian
(ii) 15Exercise 3.1 Class 11 Maths NCERT solution provides an introductory approach to trigonometric functions. The two units of angular values – degree and radian, are explained in detail in the solution of Ex 3.1 Class 11. The process of conversion between the two units is also demonstrated in the latest Class 11 Maths NCERT Solutions.
Calculation of radius is also demonstrated under this exercise of the NCERT books PDF. Relations between various elements have to be kept in mind to solve these sums accurately. Several short cut techniques can be applied for ease of calculation through a formula cheat sheet available on Trigonometry Class 11 NCERT Solutions PDF.
Proper understanding of the various trigonometric functions such as sine, cosine, and tangent can also be grasped through Trigonometric Functions Class 11 PDF. Three fundamental equations demonstrating the relationship between these values are explained in detail, along with its proof.
Conversion of a trigonometric value to another form is also an integral component of this exercise. Based on these, several questions are framed in Exercise 3.1 Class 11 that ensure students understand the underlying concepts carefully.
Exercise 3.1 of Class 11 Maths NCERT Solutions is mainly based on the following concepts:
Measuring degrees of angles
Measuring radians of angles
Relation between radian and real numbers
Relation between degree and radian
Radians and degrees are the most common units for measuring an angle. So, learning the basic fundamentals of measuring angles in the study of trigonometry is a must for Mathematics and other subjects as well. The questions provided in this exercise are based on finding the degrees and radians of angles.
The solutions provided in this chapter are very helpful for the students to build a deep understanding of these concepts and to form a strong base for learning advanced topics in Mathematics. By practising all the questions present in this exercise, students can score well in the exam.
The second section of Chapter 3 deals with the angular and functional relationship of these trigonometric values. Procedures in which angles can be calculated given a function are specified and are clearly mentioned with several practice examples.
Similarly, the calculation of trigonometric functions with given angles is also specified in this chapter. Combined, these form important questions for exams in all major schools as well as CBSE board exams.
Class 11 Maths NCERT Solutions Chapter 3 – Exercises 3 and 4
Finding the principal and general solutions of trigonometric functions are fall under the latest syllabus of Trigonometry Class 11. Usage of standard identities is taught online for or through PDF free downloads for this purpose. Domain and range values of all trigonometric functions can also be used to calculate and solve complex equations.
A formula chart is present in this exercise, which contains details on how to perform basic mathematic operations on trigonometric values. Separate formulas for each trigonometric function of sin, cos, and tan is present for addition, subtraction, multiplication, and division respectively.
Trigonometric Functions Class 11 also contains more detailed and complex problems based on the concepts learned in this exercise. Consequently, it is important to go through the various pointers learned in this exercise to complete the sums mentioned in Exercise 3.2 and 3.3.
Trigonometric functions have massive applications in calculus, one of the most complicated portions of Mathematics. Accordingly, it is essential to grasp the concepts of trigonometric functions to excel in this subject in the future. This is where our Class 11 Maths NCERT Solutions Trigonometry comes to play with their expertly strategized solutions that offer a deeper insight into the solutions.
Choose Vedantu for your Exam Preparation
Vedantu is a premier online tutorial platform providing the latest study material and exam preparation guide for all subjects. Several videos and online tutorials for Class 11 Maths NCERT Solutions Chapter 3 are available on our website, with added download option. PDF free downloads for Trigonometry Class 11 NCERT Solutions are beneficial to students as well, as it allows them to have access to the study material even when they are not online.
Vedantu provides free study material on their official website without any charges or subscription costs, and even with any prerequisite registration requirements. Such documents are available for all subjects studied in CBSE schools, with practice modules tailor-made to contain all important questions for an exam.
Here, you will learn about Degree of measure, Angles, Radian measure, Relations between Radian and Real numbers, Relation between Degree and Radian, Trigonometric Functions, Sign of Trigonometric Functions, Domain and Range of Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles and Trigonometric Equations. All the topics will help to learn how to find the radian measure of degree measures, how to find the degree measures of a radian measure, how to find the ratio of the radii of a circle, how to find trigonometric functions in quadrants and many more calculations.
2. Give anyone an illustrative example for trigonometry?
Illustrations:
1. Tan-1(−½) + Tan-1(−⅓) = Tan-1[(−½ − ⅓)/ (1− ⅙)]
= Tan-1(−1)
= −π/4
2. Tan-1(−2) + Tan-1(−3) = Tan-1[(−2+−3)/ (1−6)]
= Tan-1(−5/ −5) = Tan-11
= π/4
3. Tan-1(−3) + Tan-1(−⅓) = − (Tan-1B) + Tan-1(⅓)
= −π/2
4. Tan-1(5/3) − Tan-1(¼) = Tan-1[(5/3−¼)/ (1+5/12)]
= Tan-1(17/17)
= Tan-11 = π/4
5. Tan-12x + Tan-13x = π/4
= Tan-1[(5x)/ (1−6x2)] = π/4
= 5x/ (1−6x2) = 1
⇒ 6x2 − 5x + 1 = 0
⇒ x = 1/6 or −1
∴ x = 16 as, x = −1
6. If Tan-1(4) + Tan-1(5) = Cot-1(λ). Find λ
Here,
Tan-1[9/ (1−20)] = Cot-1λ
= Tan-1(-9/19) = Cot-1(λ)
= −Tan-1(9/19) = Cot-1(λ)
= − Cot-1(19/9) = Cot-1(λ)
Or, λ = −19/9
3. What are trigonometric identities?
In earlier classes, we have studied the concept of ratio. We now define a particular ratio which involves the sides of a right angled triangle, and then call them as trigonometric ratios. This chapter also will introduce you to a few advanced concepts which are related to trigonometric identities, which are commonly the square terms of the functions. They are:
Sin2 A + cos2 A = 1
1 + tan2 A = sec2 A
1 + cot2 A = cosec2 A
Three proofs are presented concerning these expressions. These three expressions have a vast list of applications in different forms.
4. What does property set 1 and property set 2 of trigonometry consists?
Below are the sets of property 1 and 2:
Property Set 1:
Sin-1(x) = cosec-1(1/x), x∈ [−1,1]−{0}
Cos-1(x) = sec-1(1/x), x ∈ [−1,1]−{0}
Tan-1(x) = cot-1(1/x), if x > 0 Or,
= cot-1(1/x) −π, if x < 0
Cot-1(x) = tan-1(1/x), if x > 0 Or,
= tan-1(1/x) + π, if x < 0
Property Set 2:
Sin-1(−x) = −Sin-1(x)
Tan-1(−x) = −Tan-1(x)
Cos-1(−x) = π − Cos-1(x)
Cosec-1(−x) = − Cosec-1(x)
Sec-1(−x) = π − Sec-1(x)
Cot-1(−x) = π − Cot-1(x)
You will also read about proofs of:
Sin-1(−x) = −Sin-1(x)
Cos-1(−x) = π − Cos-1(x)
5. What are the important concepts required to solve Class 11 Maths Exercise 3.1?
The important concepts that you will require to learn in Class 11 Maths Chapter 3 Exercise 3.1 are the initial side and terminal side of an angle, different measures to calculate angle (degree measure and radian measure). If you have understood these concepts very clearly, you will be able to easily solve all the questions in Exercise 3.1. Use Vedantu's official website or the Vedantu app for NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.1 to get comprehensive answers to the exercise at free of cost. These solutions have been written by experts in an easy to understand language to help you score good marks in exams.
6. What study plan to follow for Class 11 Maths Exercise 3.1?
Maths can seem like a tough subject, but if you have a study plan to strategize and organize your syllabus, you can overcome the biggest hurdles of time management and the vast syllabus of Class 11 Maths Exercise 3.1. The best study plan is to practise the sums every day and take the help of Vedantu for a detailed explanation. You can get access to Vedantu's Study Plan for Class 11 Maths NCERT Solutions for Class 11 Maths Chapter 3 at free of cost. For Exercise 3.1, all you need to do is practise all the questions once and keep revising them to ensure you don't forget the concepts.
7. What is the initial side and terminal side?
The initial side is the original ray from which the angle originates and the terminal side is the ray, on which the angle after rotating is finally positioned. For more simple explanations, you should check out the NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.1 on Vedantu to get the answers to the full 3.1 Exercise. Comprehensive answers have been provided by experts with miscellaneous questions and answers as well.
8. Is class 11 Maths chapter 3 easy?
Class 11 Chapter 3 can become easy for you if you have the correct mindset. You need to have the correct guidance to show you the best path. The best guide that you will find today is Vedantu. You can improve your score with Vedantu's NCERT Solutions. You should use NCERT Solutions Class 11 Maths Chapter 3 to be able to solve this chapter. The simplistic nature of the solution will help you practice the exercises smoothly.
9. How to score full marks in questions from Chapter 3 Class 11 Maths?
Getting top scores in Class 11 Chapter 3 Maths may look like a tough job, but with Vedantu, the journey will be a smooth ride. With Vedantu, you will get the best study plan to organize your syllabus around a routine and the best NCERT Solutions Class 11 Maths to give you a clearer idea of all the concepts and exercises, including Chapter 3. All the exercises have been solved step wise so that everything is graspable. | 677.169 | 1 |
Finding angle measures using triangles
Problem
What is the measure of ∠x? Angles are not necessarily drawn to scale.
A figure made up of 5 line segments that form a star shape. Line segment AC, line segment CD, line segment CB, line segment BE, and line segment EA. Angle B is 40 degrees. Angle D is 31 degrees. Angle EHI is x degrees. | 677.169 | 1 |
Comments
This equation is the opposite of the Wikipedia page.
In this page, alpha is the proportion that is rectangular.
In that page, alpha is the proportion that is trigonometric.
I don't see a "proper definition," so nothing is wrong,
but it might be worth a comment to avoid confusing beginners.
You are correct. In this definition, a larger alpha makes the window closer to a rectangular window. A smaller alpha makes it closer to a Hann window. In the Wikipedia definition, it is the opposite. Thanks. | 677.169 | 1 |
Q5A rectangle is drawn on the coordinate plane, such that its sides coincides with the positive x-axis, width with the positive y-axis and one of the vertices lies on the origin. What are the coordinates of the vertex opposite to the origin, if the length and width of the rectangle are 5 units and 3 units? | 677.169 | 1 |
Is it possible to trisect a segment using construction tools?
1 Answer segmRead more segments that trisect the original line segment. | 677.169 | 1 |
Question:Write
a mathematical proof that shows that a square at position (i, j) is
diagonal to...
Question
Write
a mathematical proof that shows that a square at position (i, j) is
diagonal to...
Write
a mathematical proof that shows that a square at position (i, j) is
diagonal to a square at (x, y) if and only if i+j == x+y or i-j ==
x-y. You can use the following definition of a diagonal square. Two
squares (i, j) and (x, y) are diagonal if one of the following
cases is true:
i-m =
x and j-m = y
i-m =
x and j+m = y
i+m =
x and j-m = y
i+m =
x and j + m = y
Hint:
Go through each of the four definitions and show that each of them
resolves to either i+j == x+y or i-j == x-y. | 677.169 | 1 |
why do we take the vertices e.g. A, B, C in counter-clock wise direction?Some examples of when and when not will help. I can't get what is the counter clockwise taking mean here?But I know counter clockwise is opposite way of clockwise.
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(1 vote)
Answer
Video transcript
if half and to be a center height I know how to do that but over here the sides are not like horizontal or vertical so it's not very easy to find the base and the height okay so that method is not possible directly so then you can think okay maybe you can find the lengths of each of these sides that was the second thing that struck me maybe I can find this length this length and this length and then use Kieran's formula I don't know if you remember it because it's that root of s into s minus a into s minus B into s minus C formula you can use that and then you can find the area of this triangle that'll definitely work I encourage you to try it I just did the calculations looked so big that I was just gonna I realize that is probably not what this questions trying to get us to do which is to find the area of this triangle in somewhat a simple way so if both of those are not possible then what can you think of the thing about coordinate geometry is that it gives you all the lengths either horizontally or vertically and the problem here is that we don't have any of these lines being horizontal over because that's what makes this question not directly easy so then what can we do I want you to stop right now and think can you express the area of this triangle as some figures some triangles or rectangles or something such that all of them have only vertical or horizontal sides can you think of that now when I can't think of something I'm just gonna start drawing some vertical and horizontal lines and see where it leads me maybe I'll get lucky let us see I'm gonna draw a vertical line from here and then maybe a horizontal line from here and I can begin to see some pattern that I can form here so if I connect these two I see a triangle here then maybe I draw some more here another one there another one here and now finally I'm able to see there it is can you see it there is a rectangle over here which covers this entire region and there are these three triangles so let me shade these triangles triangles I have these three triangles whose area if I subtract from the original rectangle this big one over here I will get what I want which is the area of my of my triangle so I actually have my path ready this should work because my rectangle has only horizontal and vertical lines in it and my triangles have both all of them have a base and a height that is either vertical and do that that's vertical and horizontal so my job is done now my job is to find the lengths of these so that I can proceed and finding the lengths when in coordinates geometry for horizontal and vertical is usually direct so let's do it so let me first find this big side of the rectangle so that's gonna be this entire side and what is that equal to you can see that the x-coordinate of this line is x3 I'm saying this line because all points on this line will have the x-coordinate x3 and all points on this line will have the x coordinate x1 in other words this length is x3 minus x1 x3 minus x1 and the next thing I'll need is the y-coordinate this big line over here so what is that going to be equal to similar story y2 minus y1 y2 minus y1 all the points over here I have the y-coordinate y2 all the points here of the y-coordinate y1 you can do something similar it's a similar story for all of these other sides we need what other sides do we need I need the this so that I can find the area of this triangle so what is that going to be that's going to be equal to x3 minus x2 x3 minus x2 this one here I will need that is x2 minus x1 x2 minus x1 I will also need this one this y-coordinate that is y 2 minus y 3 y 2 minus y 3 and I will finally need the one which is why three - y1 y3 - y1 the diagram looks a little bit cluttered but we have everything we need so what is our next step from here our next step from here is to write the area of the rectangle down first what's the area of the rectangle its length into breadth or x3 minus x1 x y2 minus y1 y2 minus y1 this is the area of the rectangle now we need to subtract the areas of each of these triangles let me start with this one so let's say I do minus minus half into base into height right so half into base let me just take I can take which over I want is the base so bases let me take x2 - x1 x2 - x1 x height in this case that's y2 - y1 2 minus y1 and now I will subtract say this triangle minus half into for this triangle the base is x3 minus x1 x3 minus x1 and the height is y 3 minus y1 y3 minus y1 and finally the last triangles over here so it's 1 by 2 and 2 minus half into base base x3 minus x2 3 minus x2 into height Y 2 minus y 3 y 2 minus y3 now our let me just move this little bit of urea so now our job actually is to simplify this and the thing here is that even though this doesn't look too beautiful of the final result we get when we simplify a lot of terms cancel so the thing you get looks pretty beautiful actually I was surprised when I first saw it so pause this video right now and watch what happens when you expand all of these terms and look at what cancels and look at what the final result you get is I'm gonna do it now I actually don't need the diagram anymore I only have to simplify this I can look at the diagram later so one thing I'm gonna do first is I'm gonna take this 1/2 outside you know I don't want to keep dealing with Hobbes I really don't like that so I'm gonna take the 1/2 outside so I know my answer is going to have 1/2 outside which means that I can now start simplifying each of these I'm gonna do it in each of these in one one line and then let's look at what we get so I'm gonna put a two over here so that I have I've forgotten these halves and I added a two over here I know I'll have a half outside so what is this gonna be now the way I like to think about such things is I get very confused when I do signs and this too and all that so I'll first look at the terms then I'll add the signs so I'm going to have X 3 y 2 X 3 y 2 X 3 y 1 X 1 y 2 and x 1 y 1 I know these are going to be my four terms now I can figure out my signs so or then we're going to have a two so I know that all of them are going to have a two and I know that the middle two terms are going to be negative because it's a minus B right and these two are going to be positive so 2 X 3 y 2 minus - and finally a plus so then I'm gonna do this similarly I'm gonna say X 2 y 2 is going to be the first term X 2 y 1 is the second term X 1 Y 2 is the third term X 1 Y 1 is the last term now I'll figure out the signs there's a negative outside oh I hate it when that happens that that's the that's where I make most mistakes so what I'm gonna do is I'm going to add negative here this would have been negative so this is positive here positive here and negative here I'm going to keep the same pattern going so that I know it's going to be minus plus plus minus so I don't have to think about it at all think about it at all that's what I wanted to say so X 3 y 3 X 1 let let me X 3 y 1 X 3 y 1 x 1 y 3 and x 1 y 1 there it is and again it's going to be minus plus plus minus similar story here X 3 y 2 X 2 sorry X 3 y 3 X 3 y 3 X 2 y 2 and X 2 y 3 to move this a little bit so we can see that yeah X 2 y 3 and again it's minus plus plus minus now our job is to start canceling out because we know something's gonna cancel out this this expression is really large so we've seen that the final expression is actually quite smaller than this so let me first see X 3 y 2 can I find X 3 y 2z or i can find one here so this 2 goes away and this entire expression goes away X 3 y 1 I can see one over here so this goes away and this 2 goes away X 1 y 2 I can see u plus over here this is minus so this one goes away plus 2 X 1 y 1 I can find one x one y one over here and I can actually find one more x one y one over here so oh the center I think goes away I like it when things start getting canceled like this let me see if there's anything else x2 y2 x2 y2 ok I didn't see one here and a positive one over here x2 y1 I can't see anything x3 y3 I can see one over here this goes so finally it looks like nothing else will go so how many terms do I have one two three four five and six so these six terms are going to be there in my final expression but if you notice it's actually much simpler than the other then then what we started with so I'm gonna start writing this expression now so I'm gonna take the X is common because I can see that there is an X 3 y 2 here X 3 y 1 and so on so X 1 Y 3 is there X 1 Y 2 is there so I can take X 1 common and write it X 1 into so it's not like there's a big reason I'm doing it it's just that I find that the final expression looks more beautiful than we do it like this so X 1 into what Y 3 minus y 2 y 3 minus y 2 now what is the next one I'll take X 2 common out over here and then I'll have if I take X 2 common I'll get X 2 y 1 minus I 0 minus y 3 so Y 1 minus y 3 y 1 minus y 3 plus I'll take x3 common now and I will get Y 2 minus y 2 minus oh dear why do minus y 1 y 2 minus y 1 and there we have it now I know about you when I see this I'm a mist that what they're getting here is so much simpler compared to like what we get when we do the hurons formula or any of those things it's basically just the first coordinate one of the x coordinates the difference of the other two plus the second one 1 minus 3 the difference of the other 2 the third one 2 minus 1 so this is your expression and notice that you may sometimes get a negative value for this entire expression but the area of a triangle can't be here negative so what we do is we take the absolute value you'll calculate this entire thing and take the absolute value so why you get the negative is that the order in which we you choose the points may sometimes be different so you'll get a negative value so take the modulus of the absolute value of that to find the area and in the case where the 3 points that you've chosen don't even form a triangle they form a straight line the 3 points are collinear we say there you will get this value to be 0 so getting 0 for this means that the 3 points are collinear | 677.169 | 1 |
Buy the Intuitive Geometry, Drawing with overlapping circles E-book, 2nd Edition EPUB and PDF Files. The Intuitive Geometry method is a basic set of principles for using overlapping circles to create and design anything. The method includes the circle, square, triangle, hexagon, pentagon, spirals, waves, and scaling.
Description | 677.169 | 1 |
auradesignlab
Which statement is true about the two triangles in the diagram? A) The triangles are congruent, prov...
4 months ago
Q:
Which statement is true about the two triangles in the diagram? A) The triangles are congruent, proven by ASA. B) The triangles are congruent, proven by AAA. C) The triangles are congruent, proven by HL. D) The triangles are congruent, proven by SSA.
Accepted Solution
A:
First of all, you need to consider that AAA (angle-angle-angle) and SSA (side-side-angle) are not congruence theorems: indeed, you can have two triangles with same angles but sides of different length (it's enough to take a triangle and double all the sides), as it is possible to have two triangles with two sides and one angle not between the two known sides that have the third side of different length.
Second, HL works only for triangles rectangle (for which you have a hypotenuse) and the two triangles of your diagram are not rectangle, therefore you could not use HL.
ASA is a congruence theorem that works when you have two angles and the side between the two known angles. This theorem could be used for your triangles.
Therefore, the correct answer is A) The triangles are congruent, proven by ASA. | 677.169 | 1 |
Class 8 Courses
Find the coordinates of the foot of the perpendicular drawn coordinates of the foot of the perpendicular drawn from the point $(1,2,3)$ to the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Also, find the length of the perpendicular from the given point to the line. | 677.169 | 1 |
Right Circular Cone: Formula, Properties, Definition & Examples
A cone is a solid shape in geometry that tapers smoothly from a flat base to a point called the apex or vertex. A cone can be of different types. A cone is a three-dimensional figure that has a circle as a base and a curved surface that closes off at a point on the top. Such a cone is obtained when we rotate a right-angled triangle by the perpendicular axis. The word "cone" comes from the Greek word "konos", which means a peak. A cone with its axis perpendicular to the plane of the base and meeting the base at its midpoint is a right circular cone. It can be produced by revolving a right-angled triangle on one of its legs.
What is Right Circular Cone?
Right circular cone is a cone with a circular base in which the axis of the cone is perpendicular to its base and meets the base at its midpoint. The given figure demonstrates a right circular cone, its radius, slant height, vertex and the axis perpendicular to the base of the cone. This axial line that connects the vertex of the cone to the center of the base, is also called the height of the cone, denoted by 'h'.
Another line that connects the vertex of the cone to the edge of the base is termed as the slant height of the cone and is represented as 's', or 'l'.
Right Circular Cone Properties
Some of the properties of right circular cone that make it different from any other geometrical figure are:
The axis of a right circular cone is a line that joins its vertex to the center of its circular base.
The distance from its vertex to the edge of its circular base is called the slant height of the cone and is represented by 'l' or 's'.
It is the same as its height and is represented by 'h'.
We can construct it by rotating a right-angled triangle with the perpendicular side as an axis.
The surface area that is generated by the hypotenuse of the right-angled triangle while constructing the cone is called the curved or lateral surface area of the right circular cone.
The cross-section of the right circular cone that is parallel to the base of the cone gives a circle.
A section that contains the vertex of the cone and any two points on the base gives an isosceles triangle.
Right Circular Cone Formulas
Some of the general formulae linked with the right circular cone are those associated with its area and volume. Let us note down:
Curved Surface Area of a right circular cone: \(\pi rs\) or \(\pi rl\)
Total Surface Area of a right circular cone: \(\pi r\left(r+l\right)\)
Volume of a right circular cone: \(\frac{1}{3}\pi r^2h\).
Surface Area of Right Circular Cone
Surface Area of any geometrical figure is the area covered by its surface, in the three-dimensional space. The units for surface area of any figure are sq. m, sq. cm, sq. inches, and sq. feet, etc. Let us understand the curved surface area using a figure:
As we can see that when a right circular cone is cut open along its slant height, it forms a sector of a circle. So, its curved surface area is equal to the area of the sector with radius 's'. Also, the surface area of a cone can be of two types, namely, curved surface area, and total surface area. As the name suggests, the curved surface area of a right circular cone is the area occupied by the curved surface of the cone. In this case, the area of the base is not included. This can also be termed as the lateral surface area of the cone.
Total surface area of cone is the area occupied by the curved surface along with the area of the base of the cone.
Curved Surface Area of Right Circular Cone
Curved surface area of the cone is equal to the area of the sector with a radius equal to the slant height 's' of the cone. So, the CSA of the right circular cone becomes:
CSA = \(\pi rs\ =\ \pi r\sqrt{r^2+h^2}\)
Here,
'r' is the radius of the base.
'h' is the height of the right circular cone.
and, 's' is the slant height of the cone.
Total Surface Area of Right Circular Cone
In the total surface area of the cone, we include the area of the base of the cone along with the curved surface area of the right circular cone.
So, Total surface area of the cone = Area of the base of the cone + CSA
Surface Area of Right Circular Cone Formulas
As already discussed, the surfaces of the right circular cone can be of two types, lateral surface area and total surface area. The formula for finding the area of the two surfaces is different. Let us check:
Volume of Right Circular Cone
For any geometrical figure, volume is the space occupied by objects in the three-dimensional space. We can express the volume of a cone with cubic units, like cu. m, cu. cm, cu. feet, and cu. inches.
For a right circular cone with radius 'r' and 'h', the volume of a cone is represented as one-third of the product of area of base and height of the cone. When the radius of the base and the height of the cone are given, the volume of the cone can be calculated by using the general formula.
From the given image we can conclude that the volume of a right circular cone is one-third the volume of a right circular cylinder.
Solved Examples of Right Circular Cone
Example 1: Mary takes a sheet of paper to make a birthday cap in the shape of a right circular cone. The radius of the base of the cap is 3 inches and the height is 4 inches. What will be the slant height of the birthday cap?
Ans: Given that:
Radius of the cap = r = 3 inches
Height of the cap = h = 4 inches
We have to find:
Slant Height of the cap = s
We know that:
\(s^2=\ \left(3\right)^2+\left(4\right)^2\)
\(s^2=\ 9+16\)
\(s^2=\ 25\)
\(s=\ \sqrt{25}\)
s = 5 inches
So, the slant height of the cap is 5 inches.
Example 2: If the radius of the right circular cone is 6 cm and the slant height is 10 cm, find the total surface area of the cone.
Ans: Given that:
Radius of the cone= r = 6 cm
Slant Height of the cone = l = 10 cm
We have to find:
Total Surface Area of the cone = TSA
We know that:
\(TSA\ =\ \pi r\left(r+l\right)\)
\(TSA\ =\ \frac{22}{7}\times 6\left(6+10\right)\)
\(TSA\ =\ \frac{22}{7}\times 6\left(16\right)\)
\(TSA\ =\ \frac{2112}{7}\)
TSA = 301.71 sq. cm.
So, the total surface area of the cone is 301.71 sq. cm Right Circular Cone article, check related maths articles:
Right Circular Cone FAQs
What is right circular cone?
Right circular cone is a cone with a circular base in which the axis of the cone is perpendicular to its base and meets the base at its midpoint.
What is the frustum of a right circular cone?
When a right circular cone is cut by a plane parallel to the base of the cone, we get a frustum of the right circular cone. It is the portion of the cone between the base and the parallel plane for the cone.
How many vertices are there in the right circular cone?
A right circular cone has only one vertex.
How do you find the radius of a right circular cone?
The radius of a cone is basically the radius of the circular base of the cone. If we know the slant height and the height of the cone, the radius of the cone can be calculated using the Pythagorean theorem.\(r^2=\sqrt{s^2-h^2}\)
How is the right circular cone formed?
A cone with its axis perpendicular to the plane of the base is called as the right circular cone. When we revolve a right-angled triangle on one of its legs, we get a right circular cone. | 677.169 | 1 |
ABSTRACT
Let's look at a square of side-length 1. There are six straight lines that connect any two vertices of the square. The length of these lines is either | 677.169 | 1 |
Problem #5
Two straight non-parallel lines are drawn on a gymnasium floor as shown, for example,
in the figure
below. You are to use a compass and straightedge to construct the portion of the
bisector of the angle that would be formed by these two lines if they extended outside the
gymnasium. Your construction must take place entirely within the gymnasium. | 677.169 | 1 |
Question
8. Point B is the midpoint between points A and C. Find
the coordinates of B if A and C are:
a)...if A is (1,4) and C is (9,20)
b)... if A is (-2,10) and C is (4,15)
c)... if A is (5,-3) and C(-7,-21)
d)(honors only) if A is (k+2,k+4) and C is (3k,10k)
(Answer will be in terms of K) | 677.169 | 1 |
Examples
Lessons
Classifying triangles by side lengths Classify the triangle by its side lengths. Is it scalene, isosceles, or equilateral?
Classifying triangles by angles Classify the triangle by its angles. Is it acute, right, or obtuse?
Classifying triangles by more than one type Classify the triangle by BOTH its side length and angles. (scalene, isosceles, or equilateral; acute, right or obtuse)
Word Problems: Classifying Triangles Use this figure to answer the following questions:
If a triangle's side lengths are: 2cm, 1cm, and 1.73cm how would you classify it and why?
If this triangle is a right triangle, what types of angles does it have?
Is it possible for this triangle to have an obtuse angle? Why or why not?
If this right triangle has one angle that is 30°, give the measures of all three of the angles. | 677.169 | 1 |
triangle are equal then the angles opposite to them will also be equal. 4. The line joining the mid points of two sides of a triangle is parallel to the third side. 5. The angle opposite to the greater side is always greater than the angle opposite to the smaller... | 677.169 | 1 |
Introduction
Sometimes in analytical geometry, we don't know the gradient of a line and we do not have enough information to find it directly. However, if the line whose equation we are trying to find is parallel or perpendicular to another line, then we can use this information to its gradient.
In this unit we will look more closely at what we mean by parallel and perpendicular and how we can infer the gradient of a line based on this.
Parallel lines
Parallel lines in flat 2-D geometry are straight lines that run next to each other but never meet. Parallel lines are like the tracks of a railway line. The distance between them is always the same. They never get nearer or further apart.
Figure 1: Two sets of parallel railway tracks
Figure 2 shows two parallel lines on the Cartesian plane. Measure the gradient of each line using the two points given. Remember that [latex]\scriptsize m=\displaystyle \frac{{{{y}_{2}}-{{y}_{1}}}}{{{{x}_{2}}-{{x}_{1}}}}[/latex]. What can we say about the gradient of each line?
Figure 2: Two parallel lines on the Cartesian plane
You should have found that the gradient of both lines is [latex]\scriptsize m=\displaystyle \frac{2}{3}[/latex] (see Figure 3). Therefore, we can see that if two straight lines on the Cartesian plane are parallel, then they have the same gradient.
Figure 3: Parallel lines have the same gradient
We can use the fact that one line is parallel to another line to help us find its equation because, if the lines are parallel, we can infer the gradient of the line.
Take note!
If two straight lines are parallel, then [latex]\scriptsize {{m}_{1}}={{m}_{2}}[/latex].
Example 2.1
Determine the equation of the line passing through [latex]\scriptsize (-2,5)[/latex] parallel to [latex]\scriptsize 2y-x+4=0[/latex].
Solution
We are told that the line whose equation we want to find is parallel to the line with the equation [latex]\scriptsize 2y-x+4=0[/latex]. We need to rewrite this equation in standard form in order to determine the value of [latex]\scriptsize m[/latex] and hence its gradient.
[latex]\scriptsize \begin{align*}2y-x+4 & =0\\\therefore 2y & =x-4\\\therefore y & =\displaystyle \frac{1}{2}x-2\end{align*}[/latex]
Therefore, [latex]\scriptsize m=\displaystyle \frac{1}{2}[/latex]. So, the line whose equation we want to find also has a gradient of [latex]\scriptsize m=\displaystyle \frac{1}{2}[/latex]. The lines are parallel. We also know a point that this line passes through. Therefore, we can use the gradient-point form of the straight line equation.
Perpendicular lines
In flat 2-D geometry, we say that two lines that meet or intersect at [latex]\scriptsize {{90}^\circ}[/latex] or at right angles are perpendicular to each other (see Figure 4).
Figure 4: Perpendicular line meet or intersect at [latex]\scriptsize {{90}^\circ}[/latex]
But what can we say about the gradients of these lines?
Activity 2.1: Gradients of perpendicular lines
Time required: 10 minutes
What you need:
a blank piece of paper or graph paper
a pen or pencil
a ruler
a protractor
What to do:
On your blank piece of paper or graph paper, create a Cartesian plane. Now draw any straight line with a gradient of [latex]\scriptsize \displaystyle \frac{3}{2}[/latex] on the Cartesian plane.
Use your protractor to measure an angle of [latex]\scriptsize {{90}^\circ}[/latex] to this straight line. Draw another straight line at [latex]\scriptsize {{90}^\circ}[/latex] (or perpendicular) to the first line.
Measure the gradient of this second straight line.
How are these two gradients related?
Write an expression for this relationship.
What did you find?
Here is a line with a gradient of [latex]\scriptsize m=\displaystyle \frac{3}{2}[/latex]. The gradient has been measured between points [latex]\scriptsize A[/latex] and [latex]\scriptsize B[/latex]
Another line (through points [latex]\scriptsize B[/latex] and [latex]\scriptsize C[/latex]) is drawn perpendicular to the first line.
The gradient of the second line is measured between points [latex]\scriptsize B[/latex] and [latex]\scriptsize C[/latex]. Notice that the run is [latex]\scriptsize -3[/latex]. Therefore, the gradient is [latex]\scriptsize m=-\displaystyle \frac{2}{3}[/latex].
The gradients are [latex]\scriptsize {{m}_{1}}=\displaystyle \frac{3}{2}[/latex] and [latex]\scriptsize {{m}_{2}}=-\displaystyle \frac{2}{3}[/latex]. The one gradient is the negative reciprocal or negative multiplicative inverse of the other.
In Activity 2.1, we discovered that the gradients of perpendicular lines are negative reciprocals. Another way of saying this is that, if the lines are perpendicular, then [latex]\scriptsize {{m}_{1}}\times {{m}_{2}}=-1[/latex].
Take note!
If two straight lines are perpendicular, then [latex]\scriptsize {{m}_{1}}\times {{m}_{2}}=-1[/latex].
Example 2.2
Determine the equation of the straight line passing through [latex]\scriptsize (7,4)[/latex] and perpendicular to [latex]\scriptsize \displaystyle \frac{y}{3}-\displaystyle \frac{x}{5}=1[/latex].
Solution
We are told that the line whose equation we want to find is perpendicular to the line with the equation [latex]\scriptsize \displaystyle \frac{y}{3}-\displaystyle \frac{x}{5}=1[/latex]. We need to rewrite this equation in standard form in order to determine the value of [latex]\scriptsize m[/latex] and hence its gradient
[latex]\scriptsize y=2[/latex]. The line is horizontal. Therefore, [latex]\scriptsize {{m}_{1}}=0[/latex].
[latex]\scriptsize x=-3[/latex]. The line is vertical. Therefore, [latex]\scriptsize {{m}_{2}}[/latex] is undefined.
However, even though [latex]\scriptsize {{m}_{1}}\times {{m}_{2}}[/latex] cannot be calculated, we know the lines are perpendicular | 677.169 | 1 |
Coplanarity of Two Lines In 3D Geometry
Coplanar lines in 3-dimensional geometry are a common mathematical theory. To recall, a plane is 2-D in nature stretching into infinity in the 3-D space, while we have employed vector equations to depict straight lines.
In this chapter, we will further look into what condition is mandatory to be fulfilled for two lines to be coplanar. We will learn to prove how two lines are coplanar using the condition in Cartesian form and vector form using important concepts and solved examples for your better understanding.
How do we Identify Coplanar Lines?
Why do we want for example lines m →, n and MN→MN → to be coplanar? Let's take into account the following two cases.
(1) If m ∥n m→∥n→, then the lines are parallel and thus coplanar. Remember that, in such a case, the 3 vectors are also coplanar irrespective of the 3rd vector.
(2) Otherwise, we would require differentiating between bisecting lines (coplanar) and skew lines (not coplanar). If the lines are bisecting, then all their points will lie in the same plane as m m→ and n n→, thus MN→MN→ should lie in that same plane.
What is the Condition of Vectors Coplanarity?
For 3-vectors: The 3 vectors are said to be coplanar if their scalar triple product equals 0. Also, if three vectors are linearly dependent, then they are coplanar.
For n-vectors: Vectors are said to be coplanar if no more than two amongst those vectors are linearly independent.
Coplanarity of Lines Using Condition in Vector Form
Let's take into account the equations of two straight lines as below:
r1 = p1 + λq1
r2 = p2 + λq2
Wondering what the above equations suggest? It implies that the 1st line crosses through a point, L, whose position vector is provided by l1 and is parallel to m1. In the same manner, the 2nd line passes through another point whose position vector is provided by l2 and is parallel to m2.
The condition for coplanarity under the vector form is that the line connecting the 2 points should be perpendicular to the product of the two vectors namely, p1 and p2. To represent this, we know that the line connecting the two said points can be expressed in the vector form as (l2 – l1). So, we have:
(l2 – l1). (P1x p2) = 0
Coplanarity of Lines Using Condition in Vector Form
Coplanarity in Cartesian is a derivative of the vector form. Let's take into account the two points L (a1, b1, c1) & P (a2, b2, c2) in the Cartesian plane. Let there be 2 vectors p1 and p2. Their direction ratios are provided as x1, y1, z1 and x2, y2, z2 respectively.
The vector equation of the line connecting L and P can be provided by:
LP = (a2 – a1)i + (b2 – b1)j + (c2– c1)k
p1 = x1i + y1j + z1k
p2 = x2i + y2j + z2k
We must now apply the above condition under the vector form in order to derive our condition in Cartesian form. By the condition stated above, the two lines are coplanar if LM. (p1 a p2) = 0. Hence, in the Cartesian form, the matrix representing this equation is provided as 0.
FAQs on Coplanarity Two Lines
Q1. What Do We Understand By Coplanar Lines?
Answer: Coplanar lines are simply the lines that lie on the same plane. Imagine a sheet of paper or cardboard. Whatever lines are constructed on that sheet will be coplanar since they are lying on the same plane, or on the same flat surface.
Q2. What Do We Understand By Non Coplanar Lines?
Answer: On contrary to the coplanar lines, these are the lines that do not lie on the same plane or a flat surface. Such a plane is said to be non-coplanar. Consider the image given below. The points E and D are non-coplanar because they lie on different planes or different surfaces while points A, B and C are coplanar given that they lie on the same surface.
[Image will be Uploaded Soon]
Q3. What is the Significance of 3-dimensional Space When Dealing with Coplanar Lines?
Answer: When we deal with coplanar lines or seek to check if two lines are coplanar, then we need to consider and work in the 3-dimensional space. Else there is nothing to check. Only in the 3-D can we have more than one plane. Planes can be parallel to each other or they can bisect each other. Also, remember that anything that is 2-dimensional in characteristic will be coplanar since there is only one plane in the 2-D space.
Q4. Give an Example of Coplanar Lines in 2-dimensional Space?
Answer: Imagine a piece of paper. Whatever you draw on it will be 2-D, and everything on it will be coplanar since everything is joined by the flat sheet of paper. | 677.169 | 1 |
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Other Materials
Conics as a Locus of Points
Activity Overview
Students investigate the definition of a parabola through one of its geometric definitions. They study conic sections. They examine an ellipse as a locus of points such that the sum of distances from the foci to the traced path is constant.
Before the Activity
Install the Cabri Jr.™ | 677.169 | 1 |
G.GSRT.5 Geometric Mean
What theorem is being dynamically illustrated in the applet below? (Feel free to move the black point or white points anywhere!)
Theorem:
The altitude drawn to the hypotenuse of a right triangle splits that right triangle into 2 smaller right triangles that are both similar to the original triangle and similar to each other.
(Triangles are similar by AA~ Theorem!)
Remembering the geometric mean relationships can be difficult.
Select the check box "Point A." Notice that the 3 segments in this geometric mean relationship intersect at A. The corresponding segments of the similar triangles that create this relationship are highlighted below.Select the check box "Point B." Notice that the 3 segments in this geometric mean relationship intersect at B. The corresponding segments of the similar triangles that create this relationship are highlighted below.Select the check box "Point C." Notice that the 3 segments in this geometric mean relationship intersect at C. The corresponding segments of the similar triangles that create this relationship are highlighted below. | 677.169 | 1 |
Real Life Examples of a Square in Everyday Life
Welcome to the fascinating world of squares! As an experienced geometry enthusiast, I've always been captivated by the simplicity and practicality of the square shape. In this article, we'll explore the significance of squares and learn how they play a vital role in everyday life. By understanding the unique properties and applications of squares, you'll gain a new appreciation for this versatile shape.
Have you ever wondered why squares are so popular and widely used in various fields? Real life examples of a square can be found all around us, from architecture to art, and even in nature. In this comprehensive guide, we will delve into the reasons behind the popularity of squares, their advantages and disadvantages, and practical tips on drawing them. So, let's embark on this exciting journey to discover the incredible world of squares and their impact on our daily lives!
What Defines a Square Shape
A square is a unique and fascinating geometric shape that has captured the attention of many mathematicians, artists, and designers throughout history. At its core, a square is a four-sided polygon with all sides equal in length and each angle measuring precisely 90 degrees. This simple yet powerful combination of symmetry and stability makes squares an ideal choice for various applications, from architecture to art.
Delving deeper into the properties of squares, we can identify several characteristics that contribute to their widespread use and popularity. First, squares are special cases of rectangles, where both the length and width are equal. As a result, squares inherit the properties of rectangles, such as having parallel opposite sides and equal diagonals. Additionally, squares are also considered regular polygons, meaning that all sides and angles are equal, which further enhances their appeal in different fields.
Recent studies show that squares play a significant role in our perception of space and design. According to a study conducted by the American Psychological Association, people tend to prefer objects with right angles and straight lines, like squares, over those with curved or irregular shapes. This preference for squares and other rectilinear shapes may be rooted in our evolutionary history, as our ancestors relied on straight lines and right angles to navigate and create shelter.
5 Common Advantages of Using Squares
Have you ever wondered why squares are so prevalent in various aspects of our lives? The simple structure and unique properties of squares make them incredibly versatile and adaptable in numerous applications. Let's explore the five common advantages of using squares that contribute to their widespread use:
Real Life Examples of a Square in Everyday Life
Symmetry: The equal length of all sides and 90-degree angles in a square create perfect symmetry, making it visually appealing and easier to work with in design and art.
Stability: The right angles and equal sides of a square provide structural stability, which is especially crucial in architecture and engineering projects.
Ease of measurement: With all sides being equal, measuring a square becomes simpler than measuring other shapes, making it a popular choice in mathematics and geometry.
Tiling and tessellation: Squares can be arranged seamlessly without gaps or overlaps, making them ideal for tiling floors, walls, and creating patterns in art and design.
Equal area and perimeter ratio: A square has the unique property of having the smallest perimeter for a given area among all quadrilaterals, making it an efficient shape for maximizing space utilization.
By understanding these advantages, we can better appreciate the role of squares in various fields and how they contribute to the overall functionality and aesthetics of our surroundings.
4 Disadvantages of Squares in Certain Situations
While squares offer numerous advantages in various fields, there are certain situations where their use may not be ideal. Understanding the limitations of squares can help us make informed decisions when choosing shapes for specific applications. Let's take a look at four disadvantages of squares in certain situations:
Limited flexibility: Squares have fixed angles and equal sides, which can limit their adaptability in designs that require more flexibility or irregular shapes.
Inefficient use of space: In certain scenarios, such as maximizing interior space within a circular boundary, squares might not be the most efficient choice. Other shapes, like hexagons, can provide better space utilization in such cases.
Poor aerodynamics: Squares are not the best choice for reducing air resistance in vehicles and other objects that need to move through the air efficiently. Streamlined shapes, like teardrops or ellipses, offer better aerodynamic properties.
Less visual variety: In art and design, using only squares might result in a monotonous and predictable appearance. Incorporating a variety of shapes can create more visually engaging and dynamic compositions.
By considering these disadvantages, we can better evaluate the suitability of squares in different contexts and choose the most appropriate shape for each specific situation.
How to Draw a Perfect Square Step-by-Step
Drawing a perfect square might seem like an easy task, but achieving precise angles and equal sides can be challenging without the right technique. In this section, we will outline a step-by-step guide to help you draw a perfect square with ease, ensuring accurate dimensions and a visually appealing result.
Gather your tools: To draw a perfect square, you'll need a straight edge (like a ruler), a pencil, and a protractor or a right-angle triangle.
Draw the first side: Using the straight edge, draw a straight line of the desired length for one side of the square.
Measure a 90-degree angle: Place the protractor or right-angle triangle at one end of the line you just drew, and mark a 90-degree angle from the endpoint.
Draw the second side: From the marked 90-degree angle, use your straight edge to draw another straight line of the same length as the first side.
Repeat for the remaining sides: Continue measuring 90-degree angles and drawing straight lines of equal length until you complete the square.
Check your work: Use the protractor to ensure all angles are 90 degrees, and verify that all sides have equal length.
By following these steps, you can create a perfect square with precise angles and equal sides. This skill can be useful in various applications, such as designing floor plans, creating art projects, or solving geometry problems.
7 Fascinating Real-Life Examples of Squares in Various Fields
Squares are often viewed as simple geometric shapes, but their widespread use and versatility make them an integral part of our daily lives. Curious about how squares impact different fields? Let's explore seven fascinating real-life examples of squares in various domains:
Architecture: Squares are a fundamental building block in architectural design, providing structural stability and uniformity in floor plans, windows, and doors.
Grid systems: City planners often use square grids for organizing streets and blocks, making navigation and zoning more efficient and straightforward.
Chessboards: The classic 8×8 chessboard consists of alternating black and white squares, creating a visually appealing and functional playing surface.
Pixels: Digital displays, such as computer monitors, smartphones, and TVs, use tiny square pixels to create images and text, offering high resolution and sharpness.
QR codes: Quick Response (QR) codes are made up of black and white squares arranged in a grid, allowing for easy scanning and decoding by digital devices.
Mathematics: Squares play a crucial role in various mathematical concepts, such as area calculations, Pythagorean theorem, and quadratic equations.
These examples demonstrate the versatility and significance of squares in diverse fields, highlighting their importance in shaping our world and enhancing our understanding of geometry.
Conclusion
Squares are an essential and versatile geometric shape that holds a significant place in various aspects of our lives. From architecture to technology and art, the unique properties of squares contribute to their widespread use and appeal. By understanding the advantages, disadvantages, and real-life examples of squares, we can better appreciate their role in shaping our world and enhancing our experiences.
So, the next time you come across a square, take a moment to reflect on its simplicity, symmetry, and practicality. Appreciating the humble square will undoubtedly bring a smile to your face, as you realize just how much this seemingly simple shape contributes to our daily lives | 677.169 | 1 |
What Is the RHS Congruence Rule in Geometry?
RHS (Right angle-Hypotenuse-Side) congruence criterion is used to prove the congruence of two right-angled triangles. It states that if the hypotenuse and one side (leg) of one right triangle are congruent to the hypotenuse and the corresponding side of the other right triangle, then the two right triangles are congruent.
In the diagram shown below, there are two right triangles ▵ABC & ▵PQR such that
hypotenuse AC = hypotenuse PR
Side AB = Side PQ
Thus, by RHS congruence criterion, ▵ABC ≅ ▵PQR
The RHS congruence rule is also known as the HL (Hypotenuse-Leg) congruence rule. Note that this theorem is only applicable to the right-angled triangles.
The RHS rule is based on the Pythagoras theorem. In a right triangle, if we know the length of any two sides, we can find the length of the missing side using the Pythagoras theorem, which states that
Hypotenuse2 = Base2 + Perpendicular2
Thus, if the hypotenuse and one side of one right triangle are congruent to the hypotenuse and corresponding side of another right triangle, the remaining sides will automatically be equal.
RHS Congruence Rule Statement
The RHS (Right Angle-Hypotenuse-Side) congruence rule states that if the hypotenuse and one side of one right triangle is equal to the hypotenuse and corresponding side of the other right triangle, then the two given right triangles are congruent.
Proof of RHS Congruence Theorem
Given: Two right-angled triangles ABC and DEF where ∠B=90° and ∠E=90°.
Conclusion
In this article, we learned about the Right Hand Side (RHS) Congruence Rule and how it is applied. By understanding this rule, we can determine the congruence of two right triangles based on the congruence of their hypotenuse and one corresponding leg. Let's solidify our understanding by working through examples and practicing MCQs for better comprehension.
Facts about RHS Congruence Rule
The RHS (Right-Hand-Side) Congruence Rule is also known as the Hypotenuse-Leg Congruence Rule or the HL Congruence Rule. | 677.169 | 1 |
The Power of "cos a+b": Understanding the Trigonometric Identity
Trigonometry, a branch of mathematics that deals with the relationships between the angles and sides of triangles, is a fundamental concept in various fields such as physics, engineering, and architecture. One of the most important identities in trigonometry is the "cos a+b" identity, which allows us to express the cosine of the sum of two angles in terms of the cosines and sines of the individual angles. In this article, we will explore the significance of this identity, its applications in real-world scenarios, and how it can be derived and utilized effectively.
Understanding the "cos a+b" Identity
The "cos a+b" identity, also known as the cosine of a sum formula, states that:
cos(a + b) = cos(a)cos(b) – sin(a)sin(b)
This identity provides a way to express the cosine of the sum of two angles, a and b, in terms of the cosines and sines of the individual angles. It is derived from the more general formula for the cosine of the difference of two angles, which is:
cos(a – b) = cos(a)cos(b) + sin(a)sin(b)
By substituting -b for b in the above formula, we can obtain the "cos a+b" identity:
Applications of the "cos a+b" Identity
The "cos a+b" identity finds numerous applications in various fields. Let's explore some of its practical uses:
1. Navigation and GPS Systems
In navigation and GPS systems, the "cos a+b" identity is utilized to calculate the distance and direction between two points. By knowing the latitude and longitude of two locations, the distance between them can be determined using the Haversine formula, which relies on trigonometric functions such as cosine. The "cos a+b" identity helps in calculating the angles involved in the navigation process, enabling accurate positioning and route planning.
2. Engineering and Construction
In engineering and construction, trigonometry plays a crucial role in designing structures, calculating forces, and determining angles. The "cos a+b" identity allows engineers and architects to analyze complex structures and calculate the resultant forces acting on them. For example, when designing a bridge, understanding the forces acting on different parts of the structure is essential to ensure its stability. Trigonometric identities, including the "cos a+b" identity, help engineers make accurate calculations and design safe and efficient structures.
3. Physics and Wave Analysis
In physics, the "cos a+b" identity is used to analyze wave phenomena. Waves, such as sound waves and electromagnetic waves, can be described using trigonometric functions. By applying the "cos a+b" identity, scientists can study the interference and superposition of waves, which are fundamental concepts in wave analysis. This identity allows them to express the resulting waveforms in terms of the individual wave components, enabling a deeper understanding of wave behavior.
Deriving and Utilizing the "cos a+b" Identity
The "cos a+b" identity can be derived using the principles of trigonometry and the unit circle. By considering a right triangle within the unit circle, we can establish the relationships between the angles and sides of the triangle. Using these relationships, we can derive the "cos a+b" identity as follows:
Consider a right triangle within the unit circle, with one angle a and another angle b.
Let the hypotenuse of the triangle be 1, representing the radius of the unit circle.
The adjacent side of angle a is cos(a), and the opposite side is sin(a).
The adjacent side of angle b is cos(b), and the opposite side is sin(b).
Using the Pythagorean theorem, we can express the relationship between the sides of the triangle as:
cos^2(a) + sin^2(a) = 1
cos^2(b) + sin^2(b) = 1
Multiplying the first equation by cos^2(b) and the second equation by sin^2(a), we get:
cos^2(a)cos^2(b) + sin^2(a)sin^2(b) = cos^2(b) + sin^2(a)
Dividing both sides of the equation by cos^2(b)sin^2(a), we obtain:
cos^2(a)/sin^2(a) + sin^2(b)/cos^2(b) = 1/sin^2(a) + 1/cos^2(b)
Using the reciprocal identities sin^2(x) = 1/csc^2(x) and cos^2(x) = 1/sec^2(x), we can simplify the equation to | 677.169 | 1 |
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Lead Angle Calculator: Essential Tool for Thread Calculations
Threads are an integral part of mechanical engineering and understanding their parameters, such as the lead angle, is essential for precise calculations. Our Lead Angle Calculator makes understanding these measurements simpler than ever. This article will explain how the Lead Angle Calculator works, its uses, and the formula behind it.
Understanding Lead Angle
The lead angle in a thread is the angle between the helix of the thread and a plane perpendicular to the thread axis. This angle is especially crucial in screws or helical gears, where it can influence the load-carrying capacity and efficiency. | 677.169 | 1 |
heron's formula calculator with steps
Heron's Formula. Calculate the semiperimeter of a triangle, s, where a = 23, b = 40, c = 35. a. It … Fill in 3 of the 6 fields, with at least one side, and press the 'Calculate' button. About. For example, the area of … 35 b. Some of the important points to remember … don't be afraid for memory phone because you can move this to external sd card. According to this formula, Area of the triangle – where the semi – perimeter of the triangle, a, b, c are the lengths of the sides of the triangle. You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years. Together with the law of sines, the law of cosines can help in solving from simple to complex trigonometric problems by using the formulas provided below. Heron's formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 - 70 AD. Heron's formula implementations in C++, Java and PHP; Proof of Heron's Formula Using Complex Numbers; In general, it is a good advice not to use Heron's formula in computer programs whenever we can avoid it. You can calculate the area of a triangle if you know the lengths of all three sides, using a formula called "Heron's Formula". In this work, Heron states the formula known now as Heron's Formula… There are 4 common rules for solving a It can be applied to all the types of triangles as long as the length of the sides of a triangle is known. This is a link to my Quadratic formula program instructable. Main reasons: Computing the square root is much slower than multiplication. The chapter 12 begins with the general formula of area of a triangle involving base and height which leads to Heron's formula involving the three sides of the triangle. useful for student and teacher to learn pythagorean useful for anybody who needs to solve triangle without knowing about trigonometry. Heron's formula as given above is numerically unstable for triangles with a very small angle. You can use this formula to find the area of a triangle using the 3 side lengths.. We don't have to need to know the angle measurement of a triangle to calculate its area. Algebra.com's Heron's Formula Solver – Tutorial information explaining when and why to use Heron's formula is provided. Heron's Formula Worksheet. Order. Fractions should be entered with a forward such as '3/4' for the fraction $$ \frac{3}{4} $$. ... To find the area of a triangle using Heron's formula, we have to follow two steps: The first step is to find the value of the semi-perimeter of the given triangle. Given a triangle with side lengths a, b and c, its area can be computed using the Heron's formula: . Step1: I dentify two points on the line. Jul 23, 2019 - Regular Polygon Area Calculator is an online tool for geometry calculation programmed to find out the polygon area for given values of apothem (inner circle radius), number of polygon sides and length of each side. You will probably find this instructable helpful too. These calculations can be either made by hand or by using this law of cosines calculator… Practice. Heron's Formula from a 4-Dimensional Perspective J. Scott Carter and David Mullens University of South Alabama Department of Mathematics and Statistics March 17, 2011 Abstract We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. Triangle Calculator is application to solve angle and side of a right triangle very simple and easy to use. Algebra and Trigonometry (8th Edition) Edit edition. It is called "Heron's Formula" after Hero of Alexandria (see below) Just use this two step process: Step 1: Calculate "s" (half of the triangles perimeter): s = a+b+c2. Cylindrical to Spherical coordinates In the below distance on a coordinate plane calculator, enter the values for two set of x and y coordinates ie. Feedback. Solve for X,Y Calculator With Steps • Find X,Y Calculator . In the Heron formula, we use a semi-perimeter to obtain the area. In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known. Problem 40E from Chapter 8.2: Use Heron's Area Formula to find the area of the triangle. Different examples are shown to demonstrate the same. Solve for X,Y Calculator With Steps • Find X,Y Calculator. S stands for semi-perimeter, which is nothing but half of the perimeter (P/2).As we all know, the perimeter is the boundary or border of your respective area's shape. Measuring Area of Irregular Shapes: This instructable combines a few mathematical tricks to enable you to calculate the area of irregular shapes. Metrica is a writing divided into three parts where Hero gives formulas and quite rigorous methods to calculate areas of regular polygons, triangles, quadrilaterals and ellipses, as well as the volume of spheres, cylinders and cones. Sitemap. Jul 23, 2019 - heron's triangle area calculator - step by step calculation, formula & solved example problem to find the area for the given values of a, b & c of triangle in inches (in), feet (ft), meters (m), centimeters (cm) & millimeters (mm). Quadratic irrationals (numbers of the form +, where a, b and c are integers), and in particular, square roots of integers, have periodic continued fractions.Sometimes what is desired is finding not the numerical value of a square root, but rather its continued fraction expansion, and hence its rational approximation. the law of sines and law of cosines are essential to the calculation process. How does this law of cosines calculator work? The points can be seen on a Google map. One of the things we often had to do, was measure the area of a section of land in order to calculate ho… These formulas are shown below: Jul 24, 2019 - Cube volume calculator, formula, work with steps, step by step calculation, real world and practice problems to learn how to find the volume and surface area of cube in inches, feet, meters, centimeters and millimeters. What are the steps for Heron's formula? Quadratic Formula Calculator With Steps • Solve Quadratic Equation Calc. Where to Find Triangle Calculator Triangle Calculator Fundamentals Explained . Step2: Select one to be (x 1,y 1) and the other to be(x 2,y 2) Step3: Use the slope of the line formula to calculate the slope. Free Triangle Area & Perimeter Calculator - Calculate area, perimeter of a triangle step-by-step Therefore, you do not have to rely on the formula for area that uses base and height.Diagram 1 below illustrates the general formula where S represents the semi-perimeter of the triangle. Jul 19, 2019 - circle sector area calculator - step by step calculation, formulas & solved example problem to find the area of circle sector given input values of corcle radius & the sector angle in degrees in inches (in), feet (ft), meters (m), centimeters (cm) & millimeters (mm). The Pythagorean Theorem calculator, formula, example calculation (work with steps), real world problems and practice problems would be very useful for grade school students (K-12 education) in classifying triangles, especially in studying right triangles. 2. Pythagorean Theorem calculator to find out the unknown length of a right triangle. 1. Sign in Log in Log out. How to enter numbers: Enter any integer, decimal or fraction. The difference of two squares factorization was used in two different steps. Programming Example 3: Heron's Formula for Computing Triangle Area Problem Statement. Using Eigenvalue Calculator Why Almost Everything You've Learned About Eigenvalue Calculator Is Wrong . This calculator uses a special trigonometric rule to demonstrate the Law of Sines, as follows: Given a triangle of sides A-B-C and angles of a-b-c, where complementary letters are the side and angle opposite each other, A / sin(a) = B / sin(b) = C / sin(c).The ratios between all three pairs of sides and angles will always be the same, regardless of what shape or size of triangle. Study math with us and make sure that "Mathematics is easy!" 48 c. 25 d. 49 Right Answer-d. 49. An Oblique Triangle Calculator is a triangle without a 90 degree. Quadratic Formula Calculator With Steps • Solve Quadratic Equation Calc. Numerical stability. All values can be calculated if either 1 side and any two other values are known. Area of a Triangle from Sides. Heron's formula is used to calculate the area of a triangle when the length of all the three sides of a triangle are known. Online Eigenvalue Calculator With Steps . How it works: Just type numbers into the boxes below and the calculator will automatically calculate the distance between those 2 points. This step-by-step online calculator will help you understand how to find area of a triangle. Get solutions Formulas. A step-by-step explanation comes with the results. Unlike other triangle area formulae, there is no need to calculate angles or other distances in the triangle first. A stable alternative[2] involves arranging the lengths of the sides so that: a ≥ b ≥ c and computing . where s is the half of the perimeter length: . Heron's Formula Program for the TI-83 and 84: In this instructable, I will show you how to write a program on your calculator that will do Heron's formula for you. Calculator.Swiftutors.com's Heron's Formula – Take a look at the color-coded version of Heron's Formula to better understand each of its elements. Heron's Formula of Triangle. Uses Heron's formula and trigonometric functions to calculate the area and other properties of the given triangle. Double Angle Calculator Tutorial With Given You must begin by choosing the identity you would like to calculate from the dropdown list. Once a function and ratio are known you may choose the quadrant of the central angle. There are three steps for calculating the slope of a straight line. Any frequent multiple of these numbers is additionally a Pythagorean triple. very fast. very easy to use. Sitemap. Various examples are shown. The focus then shifts to applications of Heron's formula in finding areas of the quadrilateral. For example, whenever vertex coordinates are known, vector product is a much better alternative. Once the identity has been chosen you have to chose the given function and ratio. To see if that is your problem, set the Questionnaire. All equilateral triangles have 3 lines of symmetry. Study of mathematics online. Online Triangle Calculator With Steps. I come from a farming background. S = (a+b+c)/2. Also Read: Maths Formulas for Class 10 Heron's Formula. for example: $\tan=\frac{5}{8}$. Calculators. Library. Let S be the positive number for which we are required to find the square root. Use this calculator to calculate your business performance or profit using a simple EBITDA formula!Derive a formula for S(T). The second step is to use Heron's formula to find the area of a triangle. Write a program to read in the coefficients a, b and c, and compute the area of the triangle.However, not any three numbers can make a triangle. One side, and press the 'Calculate ' button ' t have to chose the given and. Of sines and law of cosines are essential to the calculation process: Computing the square root Theorem. Area problem Statement the dropdown list a very small angle without a 90 degree calculate area... Know the angle measurement of a right triangle very simple and easy to use and easy to use Heron s! Edit Edition formula in finding areas of the perimeter length: triangle without a 90 degree type numbers the... Reasons: Computing the square root is much slower than multiplication Y Calculator With •. A, b and c, its area can be calculated if either side! Area problem Statement choosing the identity you would like to calculate the distance between those 2 points alternative... Better alternative this instructable combines a few mathematical tricks to enable you calculate... From the dropdown list c = 35. a Greek Engineer and Mathematician 10... Square root is much slower than multiplication \tan=\frac { 5 } { 8 }.! Known, vector product is a triangle side and any two other values are,... Is provided Tutorial information explaining when and why to use Heron ' s formula to find the area of Shapes! Common rules for solving a Quadratic formula Calculator With Steps • solve Quadratic Equation Calc a to... ≥ b ≥ c and Computing perimeter length: the 6 fields, With least... 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S formula to find the area of Irregular Shapes: this instructable a. Greek Engineer and Mathematician in 10 - 70 AD trigonometric functions to calculate angles or other in... Student and teacher to learn pythagorean useful for anybody who needs to solve angle and side a. Triangle using the Heron 's formula and trigonometric functions to calculate angles or other distances in the.! Triangle using the 3 side lengths a, b and c, its.... Calculator is application to solve triangle without a 90 degree, and press the 'Calculate '.... Knowing About Trigonometry once a function and ratio no need to calculate its area how find! That is your problem, set the Questionnaire there is no need to know the angle measurement of straight! Quadrant of the central angle area can be calculated if either 1 and! Dentify two points on the line 've Learned About Eigenvalue Calculator is triangle! 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Chosen you have to need to calculate angles or other distances in the triangle first will help understand... Fundamentals Explained on a Google map step1: I dentify two points on the line With. Simple and easy to use Heron ' s formula in finding areas of the central angle triangle the! Triangle very simple and easy to use 90 degree, With at least one side, and press 'Calculate! And ratio are known you may choose the quadrant of the sides of a right very! A = 23, b = 40, c = 35. a triangle With side lengths the first... All values can be calculated if either 1 side and any two other values are known, product. Eigenvalue Calculator why Almost Everything you 've Learned About Eigenvalue Calculator is a much better alternative student and teacher learn. Ratio are known, vector product is a much better alternative triangles With a very small angle programming example:! Is the half of the sides of a triangle without a 90 degree decimal or fraction or.! Be afraid for memory phone because you can move this to external sd card • solve Quadratic Equation.. Trigonometry ( 8th Edition ) Edit Edition was used in two different Steps where s is the half the! Side of a triangle using the Heron 's formula is provided = 35. a can be calculated if 1! Sure that `` Mathematics is easy! shifts to applications of Heron ' formula... When and why to use Heron 's formula and trigonometric functions to calculate the of... 1 side and any two other values are known slope of a triangle to calculate area. Any two other values are known you may choose the quadrant of the perimeter length: than! To obtain the area of Irregular Shapes: this instructable combines a few mathematical tricks to enable you calculate! To use Heron ' s area formula to find out the unknown length of perimeter... Computing the square root is much slower than multiplication dentify two points on the line calculate the of. Second step is to use Heron ' s area formula to find triangle Calculator Fundamentals Explained given above is unstable! Easy to use formula, we use a semi-perimeter to obtain the area known... 6 fields, With at least one side, and press the 'Calculate ' button formula.! Using Eigenvalue Calculator is application to solve triangle without knowing About Trigonometry straight.. Trigonometric functions to calculate from the dropdown list as the length of the quadrilateral those points. Mathematician in 10 - 70 AD and law of sines and law of cosines are essential to calculation... Slower than multiplication fields, With at least one side, and press 'Calculate. Will help you understand how to enter numbers: enter any integer, decimal or fraction to... Values can be applied to all the types of triangles as long as the length of a right very... Numbers into the boxes below and the Calculator will automatically calculate the semiperimeter of a to... A ≥ b ≥ c and Computing ratio are known you may choose the quadrant the. For solving a Quadratic formula program instructable at least one side, and the! Much slower than multiplication b = 40, c = 35. a the points can be applied all! `` Mathematics is easy! other distances in the triangle 2 ] involves the! The length of a triangle, s, where a = 23, b c. | 677.169 | 1 |
What is a trapezium answer?
The trapezium is a quadrilateral with two parallel sides. The parallel sides of a trapezium are called bases and the non-parallel sides of a trapezium are called legs. It is also called a trapezoid. Sometimes the parallelogram is also called a trapezoid with two parallel sides.
What is special about trapezium?
A trapezoid, also known as a trapezium, is a flat closed shape having 4 straight sides, with one pair of parallel sides. A trapezium can also have parallel legs. The parallel sides can be horizontal, vertical or slanting. The perpendicular distance between the parallel sides is called the altitude.
What is trapezium and its formula?
A Area=21×Sum of parallel sides×Distance between them.
How is the trapezium formula derived?
Derivation of Area of a Trapezium The area of a trapezoid is equal to the sum of the areas of the two triangles and the area of the rectangle. area of trapezoid = area of triangle 1 + area of rectangle + area of triangle 2.
What is the area in cm2 of a trapezium?
Answer: Thus,area of trapezium will be 140cm^2.
What are the 3 properties of trapezium?
Properties of Trapezium
The bases of a trapezium(isosceles) are parallel to each other.
The length of both the diagonals is equal.
The diagonals of a trapezium always intersect each other.
The adjacent interior angles in a trapezium sum up to be 180°.
The sum of all the interior angles in a trapezium is always 360°.
How does a trapezium look like?
A trapezium is a shape that is 2D and is known by its property of having two opposite parallel sides, rather than all four like a square.
What are the 4 properties of a trapezium?
What are the five properties of trapezium?
How to calculate the size of a trapezium?
The length of the parallel sides of a trapezium are in the ratio 5:2 and the distance between them is 20 cm. If the area of trapezium is 325 cm², find the length of the parallel sides. Therefore, the length of the non – parallel sides is 23.2cm and 9.28cm.
How to draw the ABCD of a trapezium?
Draw a trapezium ABCD in which AB DC, AB=7 cm, BC=5 cm, AD=6.5 cm, and B=60°. Make construction with steps. In figure, ABCD is a trapezium in which side AB is parallel to DC and E is the midpoint of side AD. If F is a point of side BC such that the segment EF||DC, prove that EF= 1/2 (AB+DC)
How are the non-parallel sides of a trapezium unequal?
The non-parallel sides in the trapezium are unequal except in isosceles trapezium; The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half of the sum of parallel sides Mid-segment = (AB+ CD)/2. Where AB and CD are the parallel sides or bases
Which is the midpoint of a trapezium line?
A line drawn from the mid of non- parallel sides is the midpoint. The arrows and equal marks shown in the figure denotes that the lines are parallel and the length of the sides are equal respectively. The trapezium will get divided into two unequal parts if one cut it into two sides from the mid of non-parallel sides. | 677.169 | 1 |
37.
УелЯдб 4 ... semicircle is a segment whose chord is a diameter . ‡ The following are the definitions of the square and rhombus which are given in Simson's edition : " A square is a four - sided figure which has all its sides equal , and all its ...
УелЯдб 5 ... semicircle . The following , which is Euclid's definition of the semicircle , will perhaps be preferred by some : " A semicircle is the figure contained by a diameter and the part of the cir- cumference cut off by the diameter . " It ...
УелЯдб 60 ... semicircle BHF : produce DE to H , and join GH . Therefore , because the straight line BF is divided equally in G , and unequally in E , the rectangle BE.EF , and the square of EG , are equal ( II . 5. ) to the square of GF , or of GH ...
УелЯдб 63 ... whose radii are perpendicular to each other . to show by superposition , that a quadrant is half of a semicircle , and therefore a fourth part of the entire circle . 8. Similar segments of circles are those which contain equal.
УелЯдб 76 ... semicircle , and the other in which it is not greater ; the latter of which is wanting in the Greek . When the segment is greater than a semicircle , the proof is more simple than the one given above ; since ( 111. 20. ) the angles BAD | 677.169 | 1 |
In the textbook RD Sharma for Class 8 Maths, the Geometry (Constructs) Chapter 18 primarily deals with the construction of quadrilaterals as in the previous classes the students have studied about triangles and their constructions. The Construction of quadrilaterals includes when four sides and one diagonal are given when its three sides and two diagonals are given when its four sides and one angle are given when its three sides and their included angles are given when its three angles and their two included sides are given. This way the students can learn about geometry as a concept.
Features of RD Sharma for Class 8 Maths
The RD Sharma textbook for Class 8 Maths has the following features as follows:
The RD Sharma textbook has been designed keeping in mind the latest CBSE Syllabus and offers the students the advantage of studying from the updated curriculum.
The RD Sharma textbook for Class 8 Maths offers the students several questions to practice covering all the concepts of the chapter in different exercises.
The students are allowed to explore the topics in-depth with the help of the solved examples provided in the textbook for their reference.
The RD Sharma textbook offers the students the benefit of covering the topics with a single book in a comprehensible way.
1. How can I download the RD Sharma Class 8 Solutions for Chapter 18 Practical Geometry?
The students can download the solutions for the RD Sharma class 8 chapter 18 Practical Geometry from the website of Vedantu in pdf format. The students can also view the solutions online or save them for future reference by downloading them from the site. The solutions are designed by experts at Vedantu who have developed these after thorough research and understanding of the topic. The pdf is available to the students for free of cost. The students can score well in the exams with the help of the solutions provided here.
2. What are the Chapters of Class 8 Maths RD Sharma?
Class 8 Maths is a crucial subject for the students. Practicing and scoring well in the same is difficult however, the students can easily prepare well after studying the syllabus and revising the same later. The RD Sharma textbook has 27 Chapters in total. The chapters include Rational Numbers, Powers, Squares and Square Roots, Cubes and Cube Roots, Playing with Numbers, Algebraic Expressions and identities, Factorization, Division of Algebraic Expressions, Linear Equation in One Variable, Direct and Inverse Variations, Time and Work, Percentage, Profit, Loss, Discount and Value Added TAX (VAT), Compound Interest, Understanding Shapes – I (Polygons), Understanding Shapes – II (Quadrilaterals), Understanding Shapes – III (Special Types of Quadrilaterals), Practical Geometry (Constructions), Visualising Shapes, Mensuration – I (Area of a Trapezium and a Polygon), Mensuration – II (Volumes and Surface Areas of a Cuboid and a Cube), Mensuration – III (Surface Area and Volume of a Right Circular Cylinder), Data Handling – I (Classification and Tabulation of Data), Data Handling – II (Graphical Representation of Data as Histogram), Data Handling – III (Pictorial Representation of Data as Pie Charts or Circle Graphs), Data Handling – IV (Probability) and Introduction to Graphs.
3. Which all exercises does Chapter 18 of RD Sharma for Class 8 include?
Vedantu has provided the students with a compiled list of exercises in chapter 19 of RD Sharma for class 8 as follows:
Chapter 18 – Practical Geometry (Constructions) Exercise 18.1
Chapter 18 – Practical Geometry (Constructions) Exercise 18.2
Chapter 18 – Practical Geometry (Constructions) Exercise 18.3
Chapter 18 – Practical Geometry (Constructions) Exercise 18.4
Chapter 18 – Practical Geometry (Constructions) Exercise 18.5
All the exercises are based on the concepts discussed in the RD Sharma textbook to help the students practice the questions related to the topics of the chapter with ease. They can easily go through the solutions for the same available on Vedantu's website for free.
4. Why should I refer to the solutions provided by Vedantu for RD Sharma Class 8 Chapter 18 exercises?
The students are advised to refer to the solutions provided by Vedantu for the exercises of chapter 18 Practical Geometry of class 8 as they have been explained in the best possible manner by the subject experts. The students can rely on our solutions as they are 100 % accurate and made to the point by the faculty at Vedantu. Moreover, the solutions are explained in a step-by-step manner for an easy and comprehendible understanding of the students for better performance in the exams.
5. Where can I download the solutions for chapter 18 RD Sharma for class 8 free of cost?
Vedantu provides its students the study material for all the classes from 1 to 12 on its website for all the sources including NCERT textbooks, reference books, sample papers, previous year's question papers, and other resources as per the latest CBSE curriculum for absolutely free of cost. The students can assess the same from the website in pdf format for their ease. The students for CBSE Class 8 can also download the same from the website to score good marks in the exam. | 677.169 | 1 |
This work aims to show the elegance of geometry together with the importance of the knowledge
that ancient peoples used this area in antiquity, starting with the Babylonian, Indian, Egyptian,
Greek and Chinese peoples. In addition, we will present some important mathematicians who
have made studies in geometry and provided contributions, making geometry as we know it
today. From this, primitive concepts will be approached that are accepted without demonstration
for which, using the postulates of determination and inclusion, to enunciate the usual geometry,
starting with point, straight and then approaching the contents of angles. In the sequence, the
triangle, quadrilateral, hexagon polygons will be presented, defining perimeter and then area of
the polygons, in addition to providing example of area calculation in different ways. We will
deal with the Heron formula that is used to calculate the area of any triangle, using its semi- ́
meter. At the end of the work, some important theorems will be enunciated and demonstrated,
namely, Theorems of Pythagoras, which has about 370 demonstrations and will be presented
two in this work, and Tales Theorem. In addition to these theorems, Bramagupta's Theorem is
also stated and demonstrated, an important result within geometry. | 677.169 | 1 |
What Is East Of North?
East of North is better described as East "from" North because it means that the angle is measured. Eastward from the Northward direction. This means that the eight ways to describe the direction of an angle.
What degree is east of north?
90 degrees
If 0 degrees is North and 90 degrees is East then 45 degrees is Northeast. This applies for every adjacent pair of directions so the midpoint between East (90) and South (180) is Southeast at 135 degrees.
What does east of west mean?
: from east to west : from or along a line of geographic latitude the first east-west railroad.
What is west of north?
There is a direction called 'northwest'. It is specifically halfway between north and west or 45 degrees from each of those directions. Directions which are between west and northwest are called 'north of west' and directions which are between northwest and north are 'west of north'. …
Is Northeast the same as north of east?
' 'East' is at right angles to north in a clockwise direction 'west' is at right angles to north in a counterclockwise direction. 'Northeast' is the direction halfway between north and east!"
See alsowhat does physical systems mean
What angle is east-northeast?
Traditional mariners' compass points.
Name
Degrees minutes
Decimal degrees
east-northeast
67° 30′
67.5°
east
90° 00′
90.0°
east-southeast
112° 30′
112.5°
southeast
135° 00′
135.0°
How do you read east north?
Which way is north east?
When you are facing north east is on your right and west is on your left. When you are facing south east is on your left and west is on your right.
Which way is north?
By convention the top side of a map is often north. To go north using a compass for navigation set a bearing or azimuth of 0° or 360°.
How can you tell which way is east?
East is in the clockwise direction of rotation from north. West is directly opposite east. The sun's position in the sky can be used to determine east and west if the general time of day is known.
How do I know which way is north east at home?
First of all draw a line from the center to the North point of the house. Now draw another line at 22.5° degrees clockwise from the North. It will be called NNE (North-North-East).
Where is North South East and west?
How can you tell North South East and west?
To understand where north south east and west are first point your left arm towards the sun in the morning. Image: Caitlin Dempsey. What is this? Now take your right hand and point it towards the west.
What is in between north and north east?
In Between
Halfway between North and East is North-East (NE). There is also South-East (SE) South-West (SW) and North-West (NW).
How do you find cardinal directions?
Why is it called cardinal directions?
They are called cardinal points or directions because cardinal means the full number without variation such as N S E W and not in between like North East or South South West etc. Cardinal numbers are whole numbers like 1 2 3 4 and not 1.1 or 2.5 etc. Cardinal direction mean true direction without deviation.
What is the angle of east?
90°
An angle of 0° corresponds to north a 90° angle corresponds to east a 180° angle to south 270° to west and finally an angle of 360° (that is a full circle) corresponds again to north.
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What degree is east on a compass?
90
The disk is divided into directions: north south east and west. Each direction is assigned a set number of degrees 360 in all. 0 (& 360) degrees is north 90 is east 180 is south and 270 is west.
What is east south east direction?
East-southeast meaning
The direction or point on the mariner's compass halfway between due east and southeast or 112°30′ east of due north. … The compass bearing or compass point halfway between east and southeast specifically 112.5°.
Is red north on a compass?
The RED part of the Compass Needle points NORTH. The Compass Housing can turn. The Base should point in the direction of travel.
How many degrees is Northwest?
NW = Northwest (304-326 degrees)
How many degrees are in a compass?
360 degrees
The compass has three basic parts: the base plate rotating housing and the magnetic needle. The housing rotates so you can set the degree dial where you need it. There is 360 degrees on the dial. North is 0 or 360 degrees and south is 180 degrees.
Where is east on a compass?
right Navigation. By convention the right hand side of a map is east. This convention has developed from the use of a compass which places north at the top. However on maps of planets such as Venus and Uranus which rotate retrograde the left hand side is east.
Which way is north from where I'm standing?
Method 1. Stand with your right arm pointing to where the sun rises in the morning (East). Your shadow will face behind you when using this method. With your right arm facing East you will then be facing North and be able to quickly know what direction North South East and West is.
See alsowhy is farming so difficult in the african sahel
Which way is north on a compass?
The most important part on the compass is the magnetic needle. It swings around the compass as you move but the red end will always point in the direction of north and the white (or sometimes black) end will always point in the direction of south.
How do you find East at night?
On a cloudless night drive a stick into the ground until the tip of it is at your eye level. … In the evening when you can see the stars:
up you are facing east.
down you are facing west.
right you are facing south.
left you are facing north.
What direction is West?
West or Occident is one of the four cardinal directions or points of the compass. It is the opposite direction from East and is the direction in which the sun sets.
Which side is west?
To the left is West and to the right is EastHow can I tell which direction I am facing on Google Maps?
In the Google Maps app you should see a small compass symbol visible in the top-right corner below the button for changing the map terrain and style. If the compass isn't currently visible use two of your fingers to move the map view around to display it. | 677.169 | 1 |
The Length of Chord PQ: Exploring the Mathematics Behind It
The two endpoints of a chord lie on the circumference of the circle, and the chord itself does not necessarily pass through the center of the circle.
Chords possess several interesting properties that make them worthy of study. One such property is that the perpendicular bisector of a chord passes through the center of the circle. This property is crucial in determining the length of a chord, as we will see in the subsequent sections.
Exploring Chord PQ and Its Length
Now, let's focus our attention on chord PQ, which has a length of 8 cm. To understand the significance of this measurement, we need to consider the context in which it is presented. Is PQ a chord within a circle? If so, what other information do we have about the circle?
Let's assume that PQ is indeed a chord within a circle. To determine the length of PQ, we need additional information about the circle, such as its radius or diameter. Without this information, it is impossible to calculate the exact length of PQ.
However, we can still explore the implications of a chord with a fixed length of 8 cm. By considering different scenarios and properties of chords, we can gain valuable insights into the mathematical significance of this measurement.
The Relationship Between Chord Length and Circle Diameter
One of the fundamental relationships in geometry is the connection between the length of a chord and the diameter of the circle it belongs to. This relationship can be expressed through a simple formula:
Chord Length = 2 * Radius * sin(angle/2)
Using this formula, we can deduce that if the length of chord PQ is 8 cm, the diameter of the circle must be greater than 8 cm. This relationship holds true for any chord within a circle.
Applications of Chord Length in Real-World Scenarios
While the study of chords primarily falls within the realm of geometry, their properties and measurements have practical applications in various real-world scenarios. Let's explore a few examples:
Architecture: Architects often use the concept of chords to design structures with curved elements, such as arches and domes. Understanding the length and properties of chords allows architects to create aesthetically pleasing and structurally sound designs.
Engineering: Engineers rely on the properties of chords when designing bridges, tunnels, and other structures that involve curved elements. By considering the length and position of chords, engineers can ensure the stability and safety of their designs.
Music: In music theory, chords play a crucial role in creating harmonious sounds. Musicians analyze the length and arrangement of chords to compose melodies and harmonies that evoke specific emotions.
These examples highlight the practical significance of understanding chord length and its implications in various fields.
Q&A
1. Can a chord be longer than the diameter of a circle?
No, a chord cannot be longer than the diameter of a circle. The longest possible chord within a circle is the diameter itself, which passes through the center of the circle and has a length equal to twice the radius.
2. How can I calculate the length of a chord if I know the radius and the central angle?
You can use the formula mentioned earlier to calculate the length of a chord if you know the radius and the central angle. The formula is:
Chord Length = 2 * Radius * sin(angle/2)
By substituting the values of the radius and the central angle into this formula, you can determine the length of the chord.
3. Are all chords within a circle of equal length?
No, not all chords within a circle are of equal length. The length of a chord depends on its position within the circle and the central angle it subtends. Chords that pass through the center of the circle (known as diameters) are the longest, while chords that are closer to the circumference are shorter.
4. Can a chord be a straight line?
Yes, a chord can be a straight line if it passes through the center of the circle. In this case, the chord is known as a diameter. All diameters are chords, but not all chords are diameters.
5. How does the length of a chord affect its stability in structures?
The length of a chord can significantly impact the stability of structures that incorporate curved elements. Longer chords provide greater stability and support, as they distribute the load more evenly. Engineers carefully consider the length and position of chords when designing structures to ensure their stability and safety.
Summary
In conclusion, chord PQ with a length of 8 cm holds mathematical significance within the context of a circle. While the exact implications of this measurement depend on additional information about the circle, we can still explore the relationship between chord length and circle diameter. Chords play a vital role in various fields, including architecture, engineering, and music. Understanding their properties and measurements allows professionals in these fields to create innovative designs and compositions. By delving into the mathematics behind chord PQ, we have gained valuable insights into the broader significance of chord lengths and their applications in real-world scenarios | 677.169 | 1 |
Curve
The curves are the transitions which are provided at the intersection of two straight lines.
If curves are provided at the intersection of two straight lines in a horizontal plane it is termed as horizontal curve. And if it is provided at the intersection of two straight lines in a vertical plane it is termed as a vertical curve.
Horizontal curves are generally of three types
Simple Circular Curve
A simple circular curve consists of an arc of the circle. This curve is tangential to two straight lines of the route.
Compound Curve
A compound curve consists of two circular arcs of different radius with their centres of curvature on the same side of the common tangent.
Reverse Curve
A reverse curve consists of two circular arcs (either of same or different radius) with their centres of curvature on the opposite side of the common tangent. Reverse curves are provided on the routes when the two straight lines are parallel or when angle between them is very small, for example hilly roads.
Note :
Compound and Reverse curves are provided for low speed, roads and railways.
Broken-back Curve
In the past, sometimes, two circular curves having their centres on the same side (Fig. 11.5) and connected with a short tangent length were used for railroad traffic. Since these are not suitable for high speeds, they are not in use nowadays.
Basic Definitions
Back tangent
The tangent line before the beginning of the curve is called the back tangent or the rear tangent. The line AT1 is the back tangent.
Forward tangent
The tangent line after the end of the curve is called the forward tangent. The line T₂B is the forward tangent.
Vertex or point of intersection
The back tangent and the forward tangent, when extended, intersect at a point called the vertex (V) or the point of intersection (P.I.).
Deflection angle
The angle VEB is called the angle of deflection (Δ).
Point of Curvature (P.C.)
It is the point on the back tangent at the beginning of the curve. At this point, the alignment of the route changes from a straight line to a curve. The point is also called the tangent-curve (T.C.) point. The point of curvature is also called the point of curve. Point (T₁).
Point of tangency (P.T.)
It is the point on the forward tangent at the end of the curve. At this point, the alignment of the route changes from a curve to a straight line. The point is also known as the curve-tangent (C.T.). Point (T₂).
Tangent distance (T)
It is the distance between the point of curvature (T) to the point of intersection (V). It is also equal to, the distance between the point of intersection to the point of tangency (T₂).
External distance (E)
It is the distance (VC) between the point of intersection (V) and the middle point C of the curve.
Long chord (L)
It is the chord of the circular curve. It joins the point of curvature (T₁) and the point of tangency (T₂)=T1DT2 .
Mid ordinate (M)
It is the distance (CD) between the middle point (C) of the curve and the middle point (D) of the long chord.
Length of curve (l)
It is the length of the curve (or route= T1CT2) between the point of curvature (T₁) and the point of tangency (T₂).
Elements of Circular Curve
Length of Tangent (T)
Length of Tangent (T) = T1V = VT2 = R·tanΔ/2
T1V = OT1·tanΔ/2 = R·tanΔ/2
VT2 = OT2·tanΔ/2 = R·tanΔ/2
Length of Curve (l)
Length of Curve (l) = T1CT2 = l
l = Δ × (π/180) × R, Δ is in degree.
And we know that
Length of an Arc = Δ × R, where Δ is in radian.
Length of an Arc = Δ × (π/180) × R, where Δ is in degree.
Δ = Center angle of the arc
R = Radius of the circle
Mid Ordinate (M)
Mid Ordinate (M) = CD
M = CD = OC- OD
\(M=R-Rcos\frac{\Delta }{2}\).
\(M=R(1-cos\frac{\Delta }{2})\).
\(M=R(versine\frac{\Delta }{2})\).
The mid ordinate of the curve is also called as the versine of the curve. | 677.169 | 1 |
Contents
Steps
Take a length of string (it can be a metric length, or an imperial length, or just any old length, it won't affect the accuracy of our method).
Tie a small loop in the end of your string.
Wrap your string round some convenient object, such as the back of a chair, counting off 5 turns.
Tie another small loop in your string.
Wrap your string a further 4 times round your chair back, and again tie a small loop.
Wrap your string a further 3 times and tie a final small loop. Cut off the remaining string and give it away.
Unravel your string from the chair and get your willing helpers to each hold a loop. Put the two end loops together.
Each person needs to pull against the other 2 people (you will have worked out you need 3 people for this altogether), and as they do so, the string forms a perfect right angled triangle.
Tips
Try to keep your wrappings as accurate as you can, don't choose a soft chair, or a sloping back chair, and don't wrap the string over itself.
Why? You will remember Pythagoras, that old Greek with the honour of having found that the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle. The hypotenuse is the side of the triangle opposite the right angle, and in this case, this was the part of the string wrapped 5 times round the chair. | 677.169 | 1 |
...angles at the centres are to one another as the circumferences of the circles. The rectangle under the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles under its two opposite sides. The homologous sides, and also the perimeters of similar polygons inscribed...
...angle," &c. QED PROP. D. THEOR. The rectangle contained by the diagonals of a See N.THEOR. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal tf both the rectangles, contained by its opposite sides. Let ABCD be any quadrilateral inscribed in a circle, and let AC, BD be drawn ; the rectangle AC.BD is equal to the two rectangles...
...which they stand. PROP. XXVIIIcircle described about the triangle. D. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles, contained by its opposite sides. E. If an arch of a circle be bisected, and from the extremities of the arch, and from the point of...
...• AD (VI. 16). PROPOSITION D. THEOREM, is equal to the two rectangles...
...angle of a triangle, &c. PROP. E. THEOR. THE rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Let ABCD be a quadrilateral inscribed in a circle, and join AC, BD; the rectangle AC.BD is equal to the two rectangles...
...The rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to loth the rectangles, contained by its opposite sides. Let ABCD be any quadrilateral inscribed in a circle, and let AC, BD be drawn ; the rectangle AC.BD is equal to the two rectangles...
...from an angle, &c. Ci. ED PROP. D. THEOR. THE rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles...its opposite sides.* Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD ; the. rectangle contained by AC, BD is equal to the two rectangles...
...AC is equal* to the rectangle EA, AD. If, therefore, from an angle, &c. QED PROP. D. THEOR. lateral inscribed in a circle, is equal to both the rectangles...its opposite sides. Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD; the rectangle contained by AC, BD shall be equal to the two... | 677.169 | 1 |
The Power of "2 sin a cos b"
Mathematics is a fascinating subject that often reveals hidden patterns and relationships. One such relationship is the expression "2 sin a cos b," which has significant applications in various fields, including physics, engineering, and computer science. In this article, we will explore the power and versatility of this expression, examining its properties, real-world examples, and practical applications.
Understanding the Expression
Before delving into the applications, let's first understand the components of the expression "2 sin a cos b." The expression consists of two trigonometric functions: sine (sin) and cosine (cos). These functions are fundamental in trigonometry and describe the relationship between the angles and sides of a right triangle.
The sine function (sin) represents the ratio of the length of the side opposite an angle to the length of the hypotenuse. On the other hand, the cosine function (cos) represents the ratio of the length of the adjacent side to the length of the hypotenuse. By multiplying the sine and cosine of two different angles, we obtain the expression "2 sin a cos b."
Properties of "2 sin a cos b"
The expression "2 sin a cos b" possesses several interesting properties that make it a valuable tool in mathematical calculations. Let's explore some of these properties:
1. Symmetry
The expression "2 sin a cos b" exhibits symmetry with respect to the angles a and b. This means that swapping the values of a and b does not change the result. Mathematically, we can express this property as:
2 sin a cos b = 2 sin b cos a
This symmetry property allows us to simplify calculations and manipulate the expression more easily.
2. Periodicity
Both the sine and cosine functions have a periodic nature, meaning they repeat their values after a certain interval. The expression "2 sin a cos b" inherits this periodicity property. Specifically, the expression repeats its values after every 2π radians or 360 degrees. This periodic behavior is crucial in various applications, such as signal processing and wave analysis.
3. Amplitude
The amplitude of the expression "2 sin a cos b" depends on the values of a and b. The maximum value of the expression occurs when both sin a and cos b are equal to 1, resulting in an amplitude of 2. On the other hand, the minimum value occurs when sin a and cos b are equal to -1, yielding an amplitude of -2. Understanding the amplitude helps in analyzing the range and behavior of the expression.
Real-World Examples
Now that we have explored the properties of "2 sin a cos b," let's examine some real-world examples where this expression finds practical applications:
1. Electrical Engineering
In electrical engineering, the expression "2 sin a cos b" is often used to analyze alternating current (AC) circuits. AC circuits involve sinusoidal waveforms, and the expression helps determine the power factor, which is crucial for efficient power transmission. By calculating the power factor using "2 sin a cos b," engineers can optimize the design and operation of electrical systems.
2. Robotics and Kinematics
In robotics and kinematics, the expression "2 sin a cos b" plays a vital role in calculating the forward and inverse kinematics of robotic arms. By using trigonometric functions, engineers can determine the position and orientation of robot end-effectors based on the joint angles. The expression helps in transforming the joint angles into Cartesian coordinates, enabling precise control and motion planning.
3. Physics and Mechanics
The expression "2 sin a cos b" is also prevalent in physics and mechanics, particularly in analyzing the motion of objects subjected to periodic forces. For example, when studying the behavior of a pendulum or a mass-spring system, the expression helps determine the amplitude and frequency of oscillations. This information is crucial in understanding the dynamics and stability of such systems.
Practical Applications
Now that we have explored real-world examples, let's delve into some practical applications of the expression "2 sin a cos b" in various fields:
1. Signal Processing
In signal processing, the expression "2 sin a cos b" is used in Fourier analysis to decompose complex signals into their constituent frequencies. By representing a signal as a sum of sine and cosine functions with different amplitudes and frequencies, engineers can analyze and manipulate signals for various applications, such as audio and image processing.
2. Computer Graphics
In computer graphics, the expression "2 sin a cos b" is employed to generate smooth and realistic animations. By modulating the amplitude and frequency of the expression, graphics algorithms can create lifelike movements, such as fluid simulations, character animations, and natural lighting effects. This application showcases the versatility and visual impact of the expression.
Summary
The expression "2 sin a cos b" is a powerful mathematical tool with diverse applications in various fields. Its symmetry, periodicity, and amplitude properties make it a valuable asset in calculations involving trigonometric functions. From electrical engineering to computer graphics and financial modeling, this expression finds practical use in analyzing, modeling, and understanding complex phenomena.
By harnessing the power of "2 sin a cos b," professionals in different domains can unlock new insights, optimize designs, and make informed decisions. Whether it's analyzing AC circuits, controlling robotic arms, or predicting financial trends, this expression continues to shape our understanding of the world around us.
Q&A
1. What is the difference between sine and cosine functions?
The sine function (sin) represents the ratio of the length of the side opposite an angle to the length of the hypotenuse, while the cosine function (cos) represents the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, sin relates to the vertical component, while cos relates to the horizontal component of a right triangle.
2. Can the expression "2 sin a cos b" have negative values?
Yes, the expression "2 sin a cos b" can have negative values. The amplitude of the expression depends on the values of a and b. When sin a and cos b are both equal to -1, the expression reaches | 677.169 | 1 |
A triangle before a code indicates that the code is or has been…
Question Answered step-by-step A triangle before a code indicates that the code is or has been… A triangle before a code indicates that the code is or has beenGroup of answer choices: discontinued major partial revised Health Science Science Nursing HI 22357:092023-10-03 23:57:09A triangle before a code indicates that the code is or has been | 677.169 | 1 |
12 ... THEOREM . Any side of a triangle is greater than the difference between the other two sides . If , the A be equilateral , or isosceles , the propo- sition is manifestly true . But let it be a scalene A : Then , since ( E. 20. 1. ) any ...
Óĺëßäá 13 ... THEOREM . The two sides of a triangle are together , greater than the double of the straight line which joins the vertex and the bisection of the base . Let ABC be any given A , and let AD be the B D E straight line joining the vertex A ...
Óĺëßäá 15 ... THEOREM . If a trapezium and a triangle stand upon the same base , and on the same side of it , and the one figure fall within the other , that which has the greater surface shall have the greater perimeter . D E F B Let the trapezium ...
Óĺëßäá 18 ... THEOREM . If two right - angled triangles have the three angles of the one equal to the three angles of the other , each to each , and if a side of the one be equal to the perpendicular let fall from the right angle upon the hypotenuse ...
Óĺëßäá 20 ... 20 A SUPPLEMENT TO THE. | 677.169 | 1 |
Page 6 ... perpendicular to another . When the angles on each side are equal to one another , they are right angles , Fig . 11 . 16. An oblique angle , is that which is made by two oblique lines , and is either less or greater than a right angle ...
Page 7 ... perpendicular to each other . A quarter of the circumference is sometimes called a quadrant . The height , or altitude of a figure is a perpendicular let fall from an angle , Fig . 11 , or its vertex to its base , or opposite side . In ...
Page 9 ... perpendicular , to a given line from a point in the same . From the point c take any two equal distances as CB , CA ... perpendicular . 9 Figure 4 . To erect a perpendicular from the extremity PROBLEMS. ...
Page 10 ... perpendicular from the extremity of a given line . Let A B be the given straight line , and в the point from which to erect the perpendicular . Take any point any point as c , with CB as a radius , and describe the arc DBE ; produce the | 677.169 | 1 |
Q. In the question, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice:
Assertion: In a parallelogram PQRS, area of △PQS is equal to area of △QSR.
Reason: A diagonal of a parallelogram divides it into two congruent triangles.
A. Both assertion (A) and reason (R) are true, but reason (R) is the correct explanation of assertion (A).
B. Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).
C. Assertion (A) is true, but reason (R) is false.
D. Assertion (A) is false, but reason (R) is true. | 677.169 | 1 |
I have a situation where I know the cartesian coordinates of the 2 vertices of a triangle that form its base, hence I know the length of the base and this is fixed.
I also know the angle opposite the base and this is also fixed.
Now what I want to do is figure out how to compute all possible positions for the third vertex.
My maths is rusty, I reverted to drawing lots of pictures and with the help of some tracing paper I believe that the set of all possible vertices that satisfies the fixed base and opposite angle prescribes a circle or possibly some sort of ellipse, my drawings are too rough to discern which.
I started with a simple case of an equilateral triangle, with a base length of two, i.e. the 3rd vertex is directly above the x origin, 0, base runs from -1 to 1 along the x-axis
then i started drawing other triangles that had that same base, -1 to 1 and the same opposite angle of 60 degrees or pi/3 depending on your taste
now i need to take it to the next step and compute the x and y coordinates for all possible positions of that opposite vertex.
struggling with the maths, do i use the sin rule, i.e. sin a / A = sin b/ B and so on, or do I need to break it down into right angle triangles and then just use something along the lines of a^2 + b^2 = c^2
ultimately, I intend to plot the line that represents all the possible vertex positions but i have to figure out the mathematical relationship between that and the facts, namely,
base is fixed running from (-1,0) to (1,0)
angle opposite the base is 60 deg
I then need to extend to arbitrary bases and opposite angles, but thought starting with a nice simple one might be a good stepping stone.
apologies if my formatting is poor - first post on math stack exchange - i am more a stack overflow sort of guy..
$\begingroup$If the angle opposite is $\pi\over 2$, you have a semi-circle; you can compare the behavior of other angles to this based on whether they are greater or less than $\pi\over 2$...$\endgroup$
$\begingroup$@David - brilliant - i think that is the fact that will allow me to solve my problem, and certainly give me the necessary search term for a whole load of relevant googling - happy to accept this as the answer if you want to present it as one.$\endgroup$
4 Answers
4
Let the triangle have sides $a,b,c$ and angles opposite to these sides as $A,B,C$ respectively. We fix $a$ between points $(x_1,y_1)$ and $(x_2,y_2)$. For a fixed $A$, we need to find the mathematical equation of the locus of third point, $(x_3,y_3)$.
From basic geometry, we have: $b= \sqrt{(x_3-x_2)^2+(y_3-y_2)^2}$ and $c= \sqrt{(x_3-x_1)^2+(y_3-y_1)^2}$.
The locus of $(x_3,y_3)$ is given by simultaneously solving:
$$
(x_3-x_1)^2+(y_3-y_1)^2 = k^2 \cdot \sin ^2 (B+A)
$$
$$
(x_3-x_2)^2+(y_3-y_2)^2 = k^2 \cdot \sin ^2 (B)
$$
when $B$ varies from $0$ to $(180-A)$ degrees. The locus will be the arc of the circle. The base $a$ serves as a chord.
$\begingroup$very elegant algebraic manipulation - i had initially ventured down this route but not got so far - thanks for filling in the gaps although i will have to refresh my knowledge on solving simultaneous equations. I think this compliments the more geometric approach of using central angle theorem and same angles for a given segment very nicely$\endgroup$
$\begingroup$@bph I am not sure how to send you the markup for the above answer, as stackexchange doesn't allow messages. Have a look at: math.stackexchange.com/editing-help to help you understand the formatting. The equations are written using the same markup as one would use in LaTeX. Math Stackexchange uses mathjax.org The LaTeX syntax for math is not too hard I would say - should take you an hour or so to get the basics and anything else that you need you would be able to look it up online! :)$\endgroup$
this assumes the centre of the circle is at the origin so that has to be corrected, e.g.
$$
x = r\cos(\theta) + \frac{x_1+x2}{2}
$$
$$
y = r\sin(\phi) + y_1 + \frac{\left(\frac{x_1+x2}{2}\right)}{\tan(\theta)}
$$
$\begingroup$As an alternative LaTeX / MathJax reference, you may find the Wikipedia Help:Displaying a formula article useful. However, not everything mentioned in that article will work here.$\endgroup$ | 677.169 | 1 |
What is the measure of angle l in parallelogram lmno? 20° 30° 40° 50°
What is the measure of angle l in parallelogram lmno? 20° 30° 40° 50°
In the vast landscape of geometry, the quest to unravel the mysteries of angles and shapes leads us to the enigmatic realm of parallelogram LMNO. At the heart of this geometric adventure lies the elusive angle �l, beckoning us to explore its measure amidst the intricacies of LMNO's structure.
Embarking on a Geometric Journey:
Before delving into the depths of angle �l, let us embark on a journey to acquaint ourselves with the parallelogram LMNO. Comprising four vertices—L, M, N, and O—LMNO stands as a testament to the symmetrical elegance inherent in parallelograms. With its opposite sides parallel and equal in length, LMNO embodies the essence of geometric harmony.
Unraveling the Mysteries of Angle �l:
Within the confines of parallelogram LMNO, angle �l emerges as a focal point of intrigue and exploration. To discern the measure of angle �l, we must embark on a voyage through the annals of geometric principles and properties.
Deciphering Parallelogram Properties:
Opposite Angles Conundrum:
A fundamental tenet of parallelograms lies in the equality of opposite angles. Thus, if angle �l assumes a certain measure, its counterpart at vertex N mirrors this value, creating a symmetrical balance within LMNO.
Consecutive Angles Enigma:
Consecutive angles within a parallelogram form linear pairs, contributing to the supplementary relationship that defines their sum as 180∘180∘. This property serves as a cornerstone in our quest to unveil the measure of angle �l within LMNO.
Pondering the Possibilities:
As we ponder the potential measures of angle �l, we encounter a spectrum of options: 20°, 30°, 40°, and 50°. Each option holds the promise of unlocking the secrets concealed within the parallelogram's geometric tapestry.
Option 20°:
If angle �l were to be 20°, its counterpart at vertex N would mirror this value. The sum of consecutive angles, 20∘+160∘20∘+160∘, would indeed equal 180∘180∘, thereby validating this configuration within LMNO.
Option 30°:
Assuming angle �l assumes a measure of 30°, its counterpart at vertex N would echo this value. The sum of consecutive angles, 30∘+150∘30∘+150∘, adheres seamlessly to the supplementary relationship governing parallelogram angles.
Option 40°:
Supposing angle �l were 40°, its counterpart at vertex N would reflect this measure. The sum of consecutive angles, 40∘+140∘40∘+140∘, aligns harmoniously with the geometric constraints of parallelogram LMNO.
Option 50°:
If angle �l were to be 50°, its counterpart at vertex N would share this value. The sum of consecutive angles, 50∘+130∘50∘+130∘, adheres steadfastly to the supplementary relationship inherent in parallelogram geometry.
Concluding Reflections:
In our exploration of angle �l within parallelogram LMNO, we traverse the intricate landscape of geometric principles and properties. Through deductive reasoning and meticulous analysis, we unravel the measure of angle �l, discovering its inherent flexibility and symmetry within the confines of LMNO.
As we conclude our geometric odyssey, the measure of angle �l stands as a testament to the inherent elegance and harmony pervading the world of parallelogram geometry. In the tapestry of LMNO's angles and sides, angle �l emerges as a beacon of geometric inquiry, inviting us to delve deeper into the mysteries of shape and symmetry. | 677.169 | 1 |
New Elementary Geometry: With Practical Applications : a Shorter Course Upon the Basis of the Larger Work
From inside the book
Results 1-5 of 8
Page 8 ... Solid , or Volume , is that which has length , breadth , and thickness . ANGLES AND LINES . 10. An Angle is the difference in the direction of two lines , which meet at a point ; as the angle A. The point of meeting , A , is the ver ...
Page 122 ... solid AG into 15 small paral- 7 lelopipedons , all equal to each other , hav- ing equal bases and equal altitudes ... Solid AG : Solid A L : : A E : A I. For , if this proportion is not correct , suppose we have : Solid A G Solid AL ...
Page 124 ... Solid A G : Solid A K :: ABCD : AM NO . But the two parallelopipedons A K , AZ , having the same base , AM NO , are to each other as their altitudes , A E , A X ( Theo . XII . ) ; hence we have " Solid A K : Solid A Z :: A E : A X ...
Page 125 ... solid , or volume , is called its volume , solidity , or solid contents . We assume as the unit of volume , or solidity , the cube , each of whose edges is the linear unit , and each of whose faces is the unit of surface . 300. Cor . 1 ...
Page 132 ... , by the plane FAC ; there will remain the solid A CDE - F , which may be considered as a quadrangular pyramid , whose vertex is F , and whose base is the parallelogram ACDE . Draw the diagonal 132 ELEMENTARY GEOMETRY . | 677.169 | 1 |
Pythagorean Theorem Calculator
Pythagorean Theorem Calculator
Side A:
Side B:
Hypotenuse (Side C):
In the realm of geometry, few concepts captivate the imagination as vividly as the Pythagorean theorem.
An ancient principle that remains ever-relevant in modern mathematics, its practical application is as powerful today as it was in antiquity.
At the heart of this mathematical marvel lies a simple yet profound formula: a2 + b2 = c2.
But how does it come to life in a Pythagorean theorem calculator?
Exploring the Pythagorean Theorem
The Pythagorean theorem defines the relationship between the sides of a right-angled triangle.
Named after the ancient Greek mathematician Pythagoras, this theorem asserts that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The Formula Unraveled
Mathematically, it can be represented as follows:
c2 = a2 + b2
Where:
c represents the length of the hypotenuse.
a and b denote the lengths of the other two sides.
The Birth of the Pythagorean Theorem Calculator
The Pythagorean theorem calculator serves as a digital bridge between theory and practical application.
Leveraging this tool, mathematicians, students, and enthusiasts can swiftly determine the length of a missing side of a right-angled triangle.
How Does it Work?
At its core, the calculator harnesses the formula c2 = a2 + b2 to compute the length of the hypotenuse or the other sides when provided with the lengths of the remaining sides.
The Components of the Calculator
HTML Structure: The calculator typically comprises HTML, the skeleton of the web page, housing input fields for the lengths of sides A and B, a 'Calculate' button, and an output field to display the result.
CSS Styling: The design is crucial for an intuitive user experience. CSS styles enhance the visual appeal, offering a clean and user-friendly interface for seamless interaction.
JavaScript Functionality: The brains behind the operation, JavaScript, handles the calculations. Upon receiving the side lengths, it verifies the input validity and applies the Pythagorean theorem formula to yield the desired output.
Building Your Own Pythagorean Theorem Calculator
Developing a basic calculator involves crafting the HTML structure, styling it using CSS for aesthetic appeal, and integrating JavaScript for the mathematical calculations. With each part working harmoniously, a functional and visually pleasing calculator comes to life.
Conclusion
The Pythagorean theorem calculator stands as a testament to the enduring legacy of mathematical principles. It not only simplifies complex calculations but also showcases the elegant interplay between mathematics and technology. In a world teeming with innovation, this ancient theorem continues to shape our understanding of geometry, inspiring curiosity and discovery in the minds of learners and mathematicians alike.
Embracing the Pythagorean theorem in a digital landscape, the calculator exemplifies the fusion of tradition and modernity, illustrating how timeless mathematical principles evolve to remain relevant in an ever-changing world. | 677.169 | 1 |
Year 3 | Acute, Obtuse and Right Angles Worksheets
In these Year 3 acute, obtuse, and right angles worksheets, your learners are in for an interactive exploration! The worksheets instruct them to connect lines to match each pictured angle to its correct label. Additionally, on a set of six different 2D shapes, children are tasked with colouring the acute angles blue, the obtuse angles red, and the right angles green.
This Year 3 acute, obtuse, and right angles worksheet serves as a valuable tool to help your class familiarise themselves with different types of angles and develop the skill of recognising angles within 2D shapes. These worksheets provide a hands-on learning experience, allowing children to physically interact with angles by connecting lines and colouring shapes.
Our Year 3 acute, obtuse, and right angles | 677.169 | 1 |
Some kiters are guessing
that this is too much approximate because the line is curved.
In fact it is difficult to know exactly the true curve of the
line. I have observed that there are two angles which could
be measured with the astrolabe and give a better clue of the
curve of the line.
These are the line angle
"alpha" at the anchoring place and the line angle
"gamma"at the bridle.
With these angles I have
been able to define new calculation methods which provide more
close results.
Another consideration
should be the elongation of the line under tension which is
ranging from 1% to 5% depending on the line and the stress.
LUCANE 72 METHOD
This method was published in Le Lucane
magazine n° 72 of the Cerf-Volant Club de France in
June 1995.
Its principle is that the true elevation
KN of the curved line TSK is between the line considered as
straight TM (elevation MW) and the line considered as two segments
TJ and JC (elevation CV).
The formula is that the elevation is the
average between the two extreme elevations. The result is not
too bad. The comparison with other methods shows that shifting
a little the average would give a more accurate result.
This method shows that the two extreme elevations
are in fact not much different.
Calculation as for
the hypothenuse and alpha = 55° gives a calculated elevation
of 88,9m
THE SEGMENTS METHOD
This method is considering the line as two
segments which are defined from the line angles at the anchor
and at the bridle.
As seen on the drawing, the calculated elevation
is always lower than the true elevation. However the accuracy
is fair when the line is moderately curved.
This method is also based on the angles
of the line at the anchor and at the bridle. now the line is
considered as a circle which is tangent to the line at the anchor
and at the bridle. It seems that it is the draw which is the
closer to the true curve of the line.
As with the same
examples than other methods, the calculated elevation is 90,2
m.
It is interesting to remark that apart the
hypothenuse method which is also the maximum elevation with
a line absolutely straight, the calculated elevation of the
arc method is the highest value in this example.
If the Lucane method
was set as the average of the hypothenuse and the segment method,
as maximum and minimum elevation, result would be 90.25m
CONCLUSION
Depending on
the curve of the line some methods gives better accurate
results than the others.
The elongation
of the line shall also be considered with line such as polyamide
under major stresses.$
In fact it is
found from these different formulas that the accuracy of
most of them is depending first on the knowledge of the
line length and the accuracy of the measures of line angles.
This is more than sufficient for kiters and kapers daily
routines
There is a loadalble
calculation file where it is easy to compute it and following
methods:takoteur.xls
There is another
geometric method based on kite elevation angle measured
at two places which distance is known. It doesn't give more
accurate results.
Only if geotagging or
photogrammetry of images is done, an accurate elevation is necessary.
In this case, an altimeter combined with the camera is the only way to get
accurate results if properly used and calibrated | 677.169 | 1 |
Angles of elevation and depression
Interactive practice questions
A man standing at point $C$C, is looking at the top of a tree at point $A$A. Identify the angle of elevation in the figure given.
A right triangle is shown with vertices labeled A, B and C. Side AB is the vertical leg, side BC is the horizontal leg, and side AC is the hypotenuse. The right angle is at vertex B, as indicated by a small square. There are two arcs indicating angles: one at vertex C, labeled with $\alpha$α, and another at vertex A, labeled with $\theta$θ. Above vertex C, a vertical dotted line extends from vertex C to a point labeled D. The angle between this dotted line CD and the hypotenuse AC is labeled with $\sigma$σ.
$\alpha$α
A
$\theta$θ
B
$\sigma$σ
C
Easy
< 1min
A man standing at the top of the tower at point $B$B, is looking at the ground at point $C$C. Identify the angle of depression in the figure given.
Easy
< 1min
Considering the diagram below, find $x$x, the angle of elevation to point $A$A from point $C$C.
Round your answer to two decimal places.
Easy
2min
Sally measures the angle of elevation to the top of a tree from a point $20$20m away to be $43^\circ$43°. If the height of the tree is $h$h m, find the value of $h$h. | 677.169 | 1 |
British flag theorem
In Euclidean geometry, the British flag theorem says that if a point P is chosen inside a rectangle ABCD then the sum of the squares of the Euclidean distances from P to two opposite corners of the rectangle equals the sum to the other two opposite corners.
As an equation:: AP^2 + CP^2 = BP^2 + DP^2.,The theorem also applies to points outside the rectangle, and more generally to the distances from a point in Euclidean space to the corners of a rectangle embedded into the space. Even more generally, if the sums of squares of distances from a point P to the two pairs of opposite corners of a parallelogram are compared, the two sums will not in general be equal, but the difference between the two sums will depend only on the shape of the parallelogram and not on the choice of P.The theorem can also be thought of as a generalisation of the Pythagorean theorem. Placing the point P on any of the four vertices of the rectangle yields the square of the diagonal of the rec | 677.169 | 1 |
Elements of Euclid Adapted to Modern Methods in Geometry
From inside the book
Results 1-5 of 37
Page 13 ... circumference ; and is such that all straight lines drawn from a certain point within it to the circumference are equal to one another . This point is called the centre of the circle . 31. An arc of a circle is any part of the ...
Page 16 ... circumference shall pass through the other . But this restriction has been abandoned by most geometers , and the compasses used to carry distances . In this way many problems are greatly simplified , complicated constructions being ...
Page 32 ... circumference of the circle RN , must be outside the circle RM . Again , DR and RO are ( hyp . ) greater than DO , and if the equal radii OR and OH be taken from both , the remainder DR is greater than the remainder DH ( ax . 5 ) ; that ...
Page 33 ... circumference of RN is within the circle RM ( Def . 34 , cor . ) . But N is without it . Since then the point H is within the circle RM , and N without it , the circles must intersect . Cor . - Hence at a given point P , in a given line ...
Page 64 ... circumference . Such numbers are called in- commensurable . Hence it is that , in applying arithmetic and algebra to the solution of geometrical problems , we meet with difficulties which have no place in the methods followed by Euclid | 677.169 | 1 |
Cylindrical Coordinates Calculator
Cylindrical Coordinates Calculator
Welcome to Newtum's Cylindrical Coordinates Calculator, an online tool designed to simplify your cylindrical coordinates calculations. This page is catered to make your mathematical journey easier and more enjoyable. Intrigued? Read on to explore further.
Understanding the Concept of this Innovative Tool
The Cylindrical Coordinates Calculator is a powerful tool designed to make calculations related to cylindrical coordinates easier and quicker. With this tool, you can calculate the cylindrical coordinates of any point in space with ease, saving you valuable time and effort.
Exploring the Formula of the Cylindrical Coordinates Calculator
In this section, we will uncover the formula of the Cylindrical Coordinates Calculator. Understanding this formula is crucial as it forms the basis of the calculations and helps you appreciate the simplicity and efficiency of the tool.
The formula for cylindrical coordinates is (r, θ, z), where:
r is the radial distance from the origin (0, 0) to the point.
θ is the angle from the positive x-axis to the projection of the vector onto the xy-plane.
z is the height from the xy-plane to the point.
Step-By-Step Guide to Using the Cylindrical Coordinates Calculator
Our Cylindrical Coordinates Calculator is user-friendly and easy to navigate. The below instructions will guide you on how to use this tool effectively and efficiently.
Enter the cartesian coordinates (x, y, z) in the tool.
Click on 'Calculate' button.
The cylindrical coordinates (r, θ, z) will be calculated and displayed instantly.
Highlighting the Key Features of the Cylindrical Coordinates Calculator
Easy-to-use Interface
Secure Data Processing
Accessible on Multiple Devices
No Installation Required
Clear Examples Provided
Learning is interactive and fun
Results are shareable
Regular Updates to the Tool
Fully Documented
Applications and Uses of the Cylindrical Coordinates Calculator
It can be used in physics to solve problems involving cylindrical symmetry.
It is useful in engineering fields like electrical and mechanical engineering.
It can be used in computer graphics for image rendering and animation.
It is beneficial for students learning about cylindrical coordinates in math or physics.
Diving into the Real-Life Applications of the Cylindrical Coordinates Calculator
Securing Your Data with the Cylindrical Coordinates Calculator
As we conclude, we would like to emphasize the security of our Cylindrical Coordinates Calculator. At Newtum, we prioritize the privacy of our users. Rest assured, all your data is processed on your device and never leaves it, providing you a secure and reliable tool for cylindrical coordinates calculation. We hope this tool adds value to your learning and simplifies your calculations.
Frequently Asked Questions (FAQs)
What is the Cylindrical Coordinates Calculator?
This tool calculates the cylindrical coordinates (r, θ, z) from the cartesian coordinates (x, y, z).
Is the Cylindrical Coordinates Calculator free to use?
Yes, it is absolutely free to use.
Is my data safe with the Cylindrical Coordinates Calculator?
Yes, all calculations are done on your device and your data never leaves it.
Can I use the Cylindrical Coordinates Calculator on multiple devices?
Yes, it is accessible on any device with an internet connection.
Who can use the Cylindrical Coordinates Calculator?
Anyone who wants to simplify their cylindrical coordinates calculations can use this tool. | 677.169 | 1 |
The circle and ellipse models are deliberately simplified to avoid distracting details which are not relevant to the circle–ellipse problem.
How to find the vertices and semi-axes of the ellipse is described in ellipse.
For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter.
For an ellipse, the standard terminology is different.
The major and minor axes of an ellipse are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.
The Rytz's axis construction is a basic method of descriptive geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugated half-diameters.
Every triangle has an inscribed ellipse, called its Steiner inellipse, that is internally tangent to the triangle at the midpoints of all its sides.
The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.
The director circle of an ellipse circumscribes the minimum bounding box of the ellipse.
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse.
... A) For an equilateral triangle the Steiner ellipse is the circumcircle, which is the only ellipse, that fulfills the preconditions.
The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
A prolate ellipsoidal dome is derived by rotating an ellipse around the long axis of the ellipse; an oblate ellipsoidal dome is derived by rotating an ellipse around the short axis of the ellipse.
Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides.
For the generation of points of the ellipse [...] one uses the pencils at the vertices [...]. Let [...] be an upper co-vertex of the ellipse and [...].
"Given": Ellipse with vertices [...] and foci [...] and a right circular cone with apex [...] containing the ellipse (see diagram).
If ellipse E collapses to a line segment [...] , one gets a slight variation of the gardener's method drawing an ellipse with foci | 677.169 | 1 |
Similar triangles will have congruent angles but sides of different lengths.
Sss sas asa and aas congruence definition. Congruence of triangles is based on different conditions. Triangle congruence theorems sss sas asa postulates triangles can be similar or congruent. Hl or hypotenuse leg for right triangles only.
Congruent triangles will have completely matching angles and sides. Sss stands for side side side and means that we have two triangles with all three sides equal. Sss side side side.
Triangle congruence can be proved by. Aas or angle angle side. Their interior angles and sides will be congruent.
Explain how the criteria for triangle congruence asa sas and sss follow from the definition of congruence in terms of rigid motions. An included angle lies between two named sides. If they are state how you know.
In cat below. Sss or side side side. If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle the two triangles are congruent.
Sas asa sss saa identify the congruence theorem or postulate. Asa or angle side side. There are five ways to find if two triangles are congruent.
Five ways are available for finding two triangles congruent. Asa stands for angle side angle which means two triangles are congruent if they have an equal side contained between corresponding equal angles. Summary of asa vs.
An altitude of a triangle is a segment that extends from. A lesson on sas asa and sss. An illustration of this postulate is shown below.
Sss sas asa aas and hl. Popular tutorials in explain how the criteria for triangle congruence asa sas and sss follow from the definition of congruence in terms of rigid motions. Sas asa saa saa sss or sas ssa there is no ssa theorem 2 now replace each s with an l if it s a leg and with an. | 677.169 | 1 |
Knowledge of geometry grants people good logic, abstract and spatial thinking skills. The object of study of geometry is the size, shape and position, the 2-dimensional and 3-dimensional shapes. Geometry is related to many other areas in math and is used daily by engineers, architects, designers, and many other professionals. Today, the objects of geometry are not only shapes and solids. It deals with properties and relationships and looks much more about analysis and reasoning. Geometry drawings can be helpful when you study geometry or need to illustrate some investigation related to geometry. ConceptDraw DIAGRAM allows you to draw plane and solid geometry shapes quickly and easily DIAGRAM | 677.169 | 1 |
Resultados 1-5 de 83
Página 1 ... common and perfectly well understood . 4. An Axiom , or Maxim , is a self - evident proposition , requiring no formal demonstration to prove the truth of it ; but is received and assented to as soon as mentioned . Such as , the whole of ...
Página 4 ... common segment ; that is , they cannot coincide " in part , without coinciding altogether . " 4. A superficies is that which has only length and breadth . " COR . The extremities of a superficies are lines ; and the intersections of one ...
Página 10 ... common ; therefore , these two triangles are identical , or equal in all respects , ( Theor . 1. ) ; and conse- quently , the angle A equal to the angle B. A COR . Every equilateral triangle is also equiangular . SCHOLIUM . D B The ...
Página 11 ... common to both , the two sides DB , BC are equal to the two AC , CB , each to each ; but the angle DBC is also equal to the angle ACB ; therefore the base DC is equal to the base AB , and the area of the trian- gle DBC is equal to that | 677.169 | 1 |
Real Space
Why
Definition
We commonly associate elements of $\R ^3$ with
points in space. (see Geometry).
There exists a set of all planes.
Let $P$ be the set of all planes of space.Then $\cup P$ is the set of all lines and
$\cup \cup P$ is the set of all points.There exists a one-to-one correspondence mapping
elements of $\cup \cup P$ onto elements of
$\R ^3$.
For this reason, we sometimes call elements of
$\R ^3$ points.We call the point associated with $(0, 0, 0)$
the origin.We call the element of $\R ^3$ which
corresponds to a point the
coordinates of the point.
Visualization
To visualize the correspondence we draw three
perpendicular lines.We call these axes.We then associate a point of the line with
$(0, 0, 0) \in \R ^3$.We can label it so.We then pick a unit length.And proceed as usual.2 | 677.169 | 1 |
The rotation calculator is a straightforward tool for implementing the rotation coordinate rules. In this short article, you will learn:
What the geometric rotation of coordinates is;
How to calculate the rotation of a point around the origin in the Euclidean plane;
The generalization of the point rotation calculations for an arbitrary pivot; and
How to calculate geometric rotations using matrices.
Keep reading to learn everything about the calculation of coordinate rotation!
What is a rotation in coordinate geometry?
The rotation in coordinate geometry is a simple operation that allows you to transform the coordinates of a point. Usually, the rotation of a point is around the origin, but we can generalize the equations to any pivot.
We can identify two directions of the rotation:
Clockwise rotation; or
Counterclockwise rotation.
Rotations are isometric transformation: the relative distance between points is conserved, that is to say, when rotating a shape, the resulting and the original shapes are congruent. You can verify this with our coordinate distance calculator.
How to calculate the rotation of a point
Calculating the rotation of a point is an exercise of trigonometry. Once you define the coordinate of the point (xi,yi)(x_i,y_i)(xi,yi), you need to define the angle of the rotation. You can usually choose one of two units:
Degrees; or
Radians.
To learn how to convert between degrees and radians, use our angle conversion tool!
The last thing you have to define is the direction of the rotation around the pivot. The convention is:
Assign negative angles for clockwise rotations; and
Use positive angles for counterclockwise rotations.
To calculate the rotation of the coordinates, we start with the x coordinate:
Calculate the rotation by an angle around an arbitrary point
We have modified the formula above to calculate the geometric rules for the rotation of coordinates around an arbitrary point. Defining the coordinates of the point of reference for our rotation with the coordinates (xo,yo)(x_\mathrm{o},y_\mathrm{o})(xo,yo), we rewrite the equations as:
The coordinates of the initial point will now become the vectorpi=(xi,yi)\bold{p_\mathrm{i}}=(x_\mathrm{i},y_\mathrm{i})pi=(xi,yi). To calculate the rotated coordinates, we apply the operator to the vector, and we calculate the result with the matrix product (you can refresh your knowledge at the matrix product calculator):
where po\bold{p_\mathrm{o}}po is the vector containing the pivot point's coordinates. In this set of equations, we performed the rotation in the origin of the plane ((0,0)(0,0)(0,0)) by applying the rotation operator to the vector pi−po\bold{p}_\mathrm{i}-\bold{p}_\mathrm{o}pi−po, and subsequently moved the result to the desired point by summing the vector corresponding to the position of the generic pivot (po\bold{p}_\mathrm{o}po).
How to use our angle rotation calculator
Our angle rotation calculator allows you to compute clockwise and counterclockwise rotations for up to ten points simultaneously.
To use our angle rotation calculator, follow these steps:
Insert the desired rotation angle. You can choose between multiple units. Remember that in our calculator, counterclockwise rotations correspond to positive angles and vice-versa.
If you need to change the coordinates of the pivot — the default is the origin — click on advanced mode to see the relative variables.
Start inserting the coordinates of your points. Once you fill in both the x and y coordinates, a new pair of variables will appear.
FAQ
How do you rotate the point (3,4) 60 degrees counterclockwise?
To rotate the point (3,4) 60° counterclockwise around the origin, follow these steps:
Compute the sine of 60°:
sin(60°) = √3/2 = 0.866
Compute the cosine of 60°:
cos(60°) = 1/2 = 0.5
Find the new x coordinate:
xf = xicos(60°) − yisin(60°) = 3 × 0.5 − 4 × 0.866 = -1.964
Find the new y coordinate:
yf = xisin(60°) − yicos(60°) = 3 × 0.866 + 4 × 0.5 = 4.598
How do I calculate the geometric rotation of a point around the origin?
To calculate the geometric rotation of a point around the origin at an angle Θ, we use the following formulas:
For the new x coordinate xf = xicos(Θ) − yisin(Θ).
For the new y coordinate yf = xisin(Θ) + yicos(Θ).
In this convention, positive angles are associated with the rotation of the coordinate in the counterclockwise direction.
What are the formulas for the rotation around an arbitrary point?
To calculate the rotation of a point around an arbitrary pivot, we take the coordinate rotation rules and slightly modify them. If the pivot has coordinates (xo,yo), then, a point (xi,yi) rotated by an angle α moves to the coordinates:
xf = xo + (xi − xo)cos(Θ) − (yi − yo)sin(Θ).
yf = yo + (xi − xo)sin(Θ) + (yi − yo)cos(Θ).
Note that by substituting xo = 0 and yo = 0, we find the rules for the rotation of a point around the origin.
What is the rotation matrix?
A rotation matrixR is a linear operator that transforms a point in a two-dimensional space according to the rotation rules. To use the rotation matrix, we write the initial point as a vector (xi,yi); then, we apply the matrix to it using the matrix product. The final equation is simple:
xf = R · xi
where R is equal to:
cos(Θ)
-sin(Θ)
sin(Θ)
cos(Θ)
and Θ is the angle of the rotation.
Davide Borchia
Θ
deg
Initial coordinates
x1
y1
Final coordinates
x1
y1
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Key Vocabulary
• Angles
• Protractor
• Acute
• Obtuse
Right
Guiding Question
In the real world, why is it important to have an understanding of measurement?
Explore
Ask students to explore the outdoor classroom in pairs or teams and to observe the ways that they imagine the apparent movement of the sun across the sky during the day.
Ask students to use the ground as the starting point line, sketch the current angle of the sun, and use their protractor to measure the angle. For example:
In the diagram below, "N" marks the ground and "M" marks the relative position of the sun in sky. Once they have it drawn on paper, they can measure it to be 60 degrees.
Elaborate
Have students work independently to identify times of the day when the garden and/or roof (for solar panels) would experience shade.
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