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Exterior Angle Inequality
The exterior angle inequality theorem states that a triangle's
exterior angle is greater than either of the non-adjacent interior
angles. Before delving deeper into this topic, let's take a closer
look at the exterior angle theorem.
The exterior angle theorem
The
exterior angle theorem
states that the measure of an exterior angle of a
triangle equals
the sum of the measures of the two non-adjacent interior angles of
the triangle. Remember, a triangle has 3 internal angles that always
add up to 180 degrees. The triangle also has six exterior angles.
The exterior angle theorem applies to each of the 6 exterior angles.
Let's take a look at the following example:
In this figure, the exterior angle is
∠4
, the adjacent angle is
∠3
, and the non-adjacent interior angles are
∠1
and
∠2
. According to the exterior angle theorem,
m∠4=m∠1+m∠2
.
Let's look at how to prove this:
Proof:
Given:
∆PQR
To Prove:
m∠4=m∠1+m∠2
Statement
Reason
1
∆PQR
is a triangle
Given
2
m∠1+m∠2+m∠3=180°
Triangle Sum Theorem
3
∠3
and
∠4
forms a linear pair.
Definition of a linear pair.
4
∠3
and
∠4
are supplementary.
If two angles form a linear pair, they are
supplementary.
5
m∠3+m∠4=180°
Definition of supplementary angles.
6
m∠3+m∠4=m∠1+m∠2+m∠3
Statements 2, 5, and Substitution Property.
7
m∠4=m∠1+m∠2
Subtraction Property
The exterior angle inequality theorem
Remember, the exterior angle inequality theorem states that any
exterior angle of a triangle is greater than either of the opposite
(non-adjacent) interior angles. Let's take a look at the following
example:
In the above figure, the exterior angle is
∠QRS
. Its adjacent angle is
∠R
and the non-adjacent angles are
∠Q
and
∠P
. You can see that
m∠QRS
is larger than
m∠Q
. Since the exterior angle theorem states that
m∠QRS=m∠Q+m∠P
,
m∠QRS
must also be greater than
m∠P
.
Practice tests covering the Exterior Angle Inequality
Get help learning about exterior angle inequality
Getting familiar with the exterior angle inequality theorem can be
exciting, but also challenging. Whether your student wants to more
easily identify non-adjacent angles or would like to brush up on
their knowledge of the exterior angle theorem, it helps to have the
assistance of a professional educator. Working alongside a tutor can
make a positive difference in your student's academic progress. Find
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HTML canvas arc() Method – W3Schools
The arc() method creates an arc/curve (used to create circles, or parts of circles). Tip: To create a circle with arc(): Set start angle to 0 and end angle to 2*Math.PI. Tip: Use the stroke() or the fill() method to actually draw the arc on the canvas.
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HTML | canvas arc() Method – GeeksforGeeks
The arc() method is used to create an arc/curve i.e. circles, or parts of circles.. Syntax: context.arc(x, y, r, sAngle, eAngle, counterclockwise); Parameters: x: This parameter specifies the x-coordinate of the center of the circle. y: This parameter specifies the y-coordinate of the center of the circle. r: This parameter specifies the radius of the circle.
CanvasRenderingContext2D.arc() – Web APIs | MDN
This example draws a complete circle with the arc() method. HTML < canvas > </ canvas > JavaScript. The arc is given an x-coordinate of 100, a y-coordinate of 75, and a radius of 50. To make a full circle, the arc begins at an angle of 0 radians (0°), and ends at an angle of 2π radians (360°).
The Canvasarc method for drawing arcs and circles with …
The canvasarc method can be used in conjunction with other line methods such as cts.mobveTo, and ctx.lineTo, more on that later. There is also the ctx.stroke, and ctx.fill methods and setting the style of lines and fills as well that you should be familiar with as well.
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Python – Tkinter Canvas – Tutorialspoint
The Canvas widget can support the following standard items − arc − Creates an arc item, which can be a chord, a pieslice or a simple arc. coord = 10, 50, 240, 210 arc = canvas.create_arc(coord, start=0, extent=150, fill="blue") image − Creates an image item, which can be an instance of either the BitmapImage or the PhotoImage classes.
Conclusion:
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I'm veryconfused with an a level maths q
In summary, a triangle with sides that can be written in the form n^2+1, n^2-1, and 2n (where n>1) is right angled. This is proven by showing that the side n^2+1 is greater than or equal to the side 2n. However, the converse is false as shown by a counter example. The rest of the proof involves using Pythagoras' theorem, but the details are not fully understood.
Apr 11, 2005
#1
liz
23
0
the question:
(im presuming n^2 means n squared)
prove the following result: "a traingle with sides that can be written in the form n^2+1, n^2-1, and 2n (where n>1) is right angled.
show, by means of a counter example, that the converse is false.
this q was taken from the back of the book "the curious incedent of the dog in the night-time" and there was a full proof but i don't understand some of it.
it start by exmplaining we need to prove which side is the longest by doing:
n^2+1 - 2n = (n-1)^2
if n>1 then (n-1)^2 >0
therefore n^2+1 > 2n
similarily (n^2+1) - (n^2-1) = 2
therefore n^2+1 > n^2-1
so n^2+1 > n^2-1.
the rest is worked out using pythagoras but then the converse bit has completely lost me.
if anyone would like to explain, then i would be very grateful. thanks!
1. What is an A level maths question?
An A level maths question is a type of mathematical problem that is typically included in the curriculum for advanced level mathematics courses. It can cover a wide range of topics, including algebra, calculus, statistics, and more.
2. Why am I confused with an A level maths question?
Complexity and ambiguity are common characteristics of A level maths questions, which can make them challenging and confusing for students. Additionally, the pressure to perform well and the fear of making mistakes can also contribute to confusion.
3. How can I improve my understanding of A level maths questions?
One effective way to improve your understanding of A level maths questions is to practice regularly. This will help you become more familiar with different types of problems and develop problem-solving skills. You can also seek help from your teacher or a tutor if you need further clarification on certain topics.
4. What strategies can I use to tackle A level maths questions?
There are several strategies that can help you tackle A level maths questions more effectively. These include breaking down the problem into smaller, more manageable parts, using visual aids or diagrams to better understand the problem, and checking your work for mistakes.
5. How important is it to understand A level maths questions?
Understanding A level maths questions is crucial for success in this subject and for pursuing further studies or careers in fields that require strong mathematical skills. Additionally, improving your understanding of maths can also help develop critical thinking and problem-solving abilities that can be applied in other areas of life. | 677.169 | 1 |
$P$ is any point within triangle $ABC$. $Q$ is a point outside triangle $ABC$ such that $\angle CBQ = \angle ABP$ and $\angle BCQ = \angle BAP$ . Show that the triangles $PBQ$ and $ABC$ are similar.
Sir here I think that the diagram I formed is not according to question!
From $\angle CBQ=\angle ABP$ and $\angle BCQ=\angle BAP$, you have $\triangle CBQ\sim\triangle ABP$, so $BQ/BP=BC/AB$.
So $\triangle PBQ$ and $\triangle ABC$ are similar (same angle and proportional sides).
Edit: The thought process that goes with this solution is as follows. If the conclusion is true, then either $CB$ or $AB$ must divide $\angle PBQ$. Since the given conditions on angles involve only $\triangle CBQ$ and $\triangle ABP$, it is immediate that the two are similar. Then the correct mental picture emerges --- you rotate the triangle $ABC$ about point $B$ so ray $BC$ becomes ray $BQ$. Hence the opening statement "$BC$ divides $\angle PBQ$ internally". By then there is no need to draw any any pictures. | 677.169 | 1 |
What are some geometry words that begin with the letter X?
The horizontal x-axis on the Cartesian plane
What are the two root words of geometry?
The root word is "metric", which is related to "measurement"."geo" is a prefix here, or a second root word, just as "geo" is one of these in "geography", "geology", "geophysics", and "geostationary".Read more: What_is_the_root_of_the_word_geometry | 677.169 | 1 |
The scatter plot below shows the relationship between the number of hours a student listens to music and the student's test scores. Which of the following types of correlation is shown on the scatter plot?
A.
Positive
B.
Positive and negative
C.
No correlation
D.
Negative
Correct Answer : C
Let the unknown length be x.
Applying the Pythagoras theorem, the value of x is found as:
The unknown length of the triangle is approximately 8.9 feet | 677.169 | 1 |
A spherical gas balloon of radius 16 meter subtends an angle 60$$^\circ$$ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75$$^\circ$$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :
A
$$8(2 + 2\sqrt 3 + \sqrt 2 )$$
B
$$8(\sqrt 6 + \sqrt 2 + 2)$$
C
$$8(\sqrt 2 + 2 + \sqrt 3 )$$
D
$$8(\sqrt 6 - \sqrt 2 + 2)$$
2
JEE Main 2021 (Online) 20th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Let in a right angled triangle, the smallest angle be $$\theta$$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then sin$$\theta$$ is equal to :
A
$${{\sqrt 5 + 1} \over 4}$$
B
$${{\sqrt 5 - 1} \over 2}$$
C
$${{\sqrt 2 - 1} \over 2}$$
D
$${{\sqrt 5 - 1} \over 4}$$
3
JEE Main 2021 (Online) 18th March Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be $${\pi \over 3}$$. If the radius of the circumcircle of $$\Delta$$ABC is 2, then the height of the pole is equal to :
A
$${{1 \over {\sqrt 3 }}}$$
B
2$${\sqrt 3 }$$
C
$${\sqrt 3 }$$
D
$${{{2\sqrt 3 } \over 3}}$$
4
JEE Main 2021 (Online) 25th February Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
A man is observing, from the top of a tower, a boat speeding towards the lower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30$$^\circ$$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point B, where the angle of depression is 45$$^\circ$$. Then the time taken (in seconds) by the boat from B to reach the base of the tower is : | 677.169 | 1 |
As a quasitruncated square
Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t{4}={8}. A quasitruncated square, inverted as {4/3}, is an octagram, t{4/3}={8/3}.[2]
Another three-dimensional version of the octagram is the nonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{4/3,3}.
Star polygon compounds
There are two regular octagrammic star figures (compounds) of the form {8/k}, the first constructed as two squares {8/2}=2{4}, and second as four degenerate digons, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.
Used as a parol or star for the 2010 ABS-CBN Christmas Station ID Ngayong Pasko Magniningning Ang Pilipino (lit.'This Christmas, the Filipinos will Shine') due to the usage of a sun from the Philippine flag, making it also a nationalism and patriotism-themed song aside from being a Christmas song. | 677.169 | 1 |
How Is the Magnitude of the Cross Product Related to Parallelogram Area?
In summary, the cross product of two vectors is a vector that is perpendicular to both original vectors and has a magnitude equal to the product of their magnitudes multiplied by the sine of the angle between them. It is calculated using the formula a x b = |a||b|sin(θ). The cross product can be interpreted as the area of a parallelogram formed by the two vectors and is often used in physics, engineering, computer graphics, and mathematics. The dot product and cross product are different ways of multiplying vectors, with the dot product resulting in a scalar and the cross product resulting in a vector. However, the dot product can be used to calculate the magnitude of the cross product.
Mar 1, 2010
#1
ƒ(x)
328
0
How do you prove that ||VxU|| is the area of the parallelegram they form?
1. What is the cross product of two vectors?
The cross product of two vectors is a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of the magnitudes of the two vectors multiplied by the sine of the angle between them.
2. How is the cross product calculated?
The cross product of two vectors, a and b, is calculated using the following formula: a x b = |a||b|sin(θ), where |a| and |b| represent the magnitudes of the vectors and θ is the angle between them.
3. What is the geometric interpretation of the cross product?
The cross product of two vectors can be interpreted as the area of the parallelogram formed by the two vectors. It also gives the direction in which the resulting vector points.
4. What is the relationship between the cross product and the dot product?
The dot product and the cross product are two different ways of multiplying vectors. The dot product results in a scalar, while the cross product results in a vector. However, the dot product can be used to calculate the magnitude of the cross product.
5. In what applications is the cross product used?
The cross product is commonly used in physics and engineering, particularly in mechanics and electromagnetism. It is also used in computer graphics to calculate lighting and shading effects. Additionally, it has applications in geometry and linear algebra. | 677.169 | 1 |
Math Grade 4 Quiz Measuring and Sketching Angles – 4.MD.C.6
This standard requires students to understand how to measure angles in whole-number degrees using a protractor. They should also be able to sketch angles when given a specific measurement in degrees. This skill is foundational in geometry and necessary for advanced studies in mathematics and various fields of science and engineering.
Time limit: 0
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Which of these lengths can be used to form a triangle give reasons for your answer
Answers
Answered by mauryanilay6
0
Answer:
Can you write the question properly
Answered by jjaajjajaja36
0
Answer:
To determine if 3 side lengths are a triangle, use the triangle inequality theorem, which states that the sum of 2 sides of a triangle must be greater than the third side. Therefore, all you have to do is add together each combination of 2 sides to see if it's greater than the third side. | 677.169 | 1 |
...any triangle, the square on the side opposite an acute angle is equal to the sum of the squares on the other two sides minus twice the product of one of these sides and the projection of the oiher sule upon it. AA Given : A ABC, an acute ZC, and the projection DC of AC on BC. To Prove : AB2...
...the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, plus twice the product of one of these sides and the projection of the other side upon it. Draw A ABC having an obtuse angle at C; draw AD _L BC, meeting BC extended at D. We then have : Givenside opposite the obtuse angle is equal to the sum of the squares of the other two sides increased by twice the product of one of these sides and the projection of the other upon that side. 11. DE is a line parallel to AB, the hypotenuse of the right triangle ABC, meeting...
...sin A sin B sin С 75. LAW OF COSINES. In any triangle the square of one side |g equal to the sum of the squares of the other two sides minus twice the product of these two sides times the cosine of their included angle. Given : A ABC whose sides are a, o, e. /...
...I. 437. THEOREM. Tlie square of a side opposite an acute angle of a triangle is equal to the sum of the squares of the other two sides minus twice the...one of these sides and the projection of the other upon it. om Outline of Proof : In either figure let ZB be the given acute angle, and in each case BD...
...I. 437. THEOREM. The square of a side opposite an acute angle of a triangle is equal to the sum of the squares of the other two sides minus twice the...one of these sides and the projection of the other upon it. CC cm Outline of Proof : In either figure let ZB be the given acute angle, and in each case...
...XLVIII 195. In any triangle, the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides, minus twice the...sides and the projection of the other side upon it. Draw A ABC, either acute-angled or obtuse-angled at B. Draw CE.LAB. Then AE (p) is the projection of...
...side opposite the obtuse angle is equal to the sum of the squares of the other two sides increased by twice the product of one of these sides and the projection of the other side upon it. A Given A BAX with ZX obtuse, and p, the projection of b upon a. To prove a? = of + b2 + 2 op. 1. 2....
...of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides plus twice the product of one of these sides and the projection of the other side upon it 190 Polygons THEOREM XXXIII 149. The sum of the angles of any polygon is equal to twice as many right... | 677.169 | 1 |
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Study Flashcards
12 Questions
Which geometric shape has perpendicular tangents intersecting at right angles?
What is the primary focus of coordinate geometry?
In three-dimensional geometry, what do cubes, spheres, and pyramids have in common?
What is the fundamental principle of Euclidean geometry introduced in Chapter 1?
Which property of lines is NOT covered in Chapter 2 of the textbook?
What theorem related to triangles is discussed in Chapter 2?
Which type of angle is NOT mentioned in the essential principles of Euclidean geometry in Chapter 1?
In Chapter 3, what property is used to solve problems related to triangles?
Which concept is NOT part of Chapter 2 on Lines and Angles?
Summary
Mathematics Class 9: Exploring Geometry
As you embark on your mathematical journey in Class 9, you'll find that geometry plays a significant role in building your foundational knowledge and problem-solving skills. In this article, we'll delve into the essential chapters of geometry from Class 9 that will help you understand the world around you and prepare you for more advanced topics.
Ch. 1: Introduction to Euclidean Geometry
In this chapter, you'll be introduced to the fundamental principles of Euclidean geometry, such as:
Definitions of points, lines, and planes.
Types of angles, including acute, right, and obtuse angles.
Properties of parallel and perpendicular lines, which will be essential when dealing with triangles and other geometric shapes.
Ch. 2: Lines and Angles
This chapter builds upon the foundation laid in Chapter 1. In it, you'll learn:
Properties of lines, such as collinearity, intersection, and concurrency.
Theorems related to interior and exterior angles of triangles, including the sum of the angles in a triangle is 180°.
By mastering these topics in geometry, you'll build a strong foundation in mathematics that will prepare you for more advanced topics and problem-solving techniques in your future mathematical journey.
Description
Explore the fundamental principles and properties of geometry in Class 9, including points, lines, angles, triangles, quadrilaterals, circles, and coordinate geometry. Learn about two-dimensional and three-dimensional shapes, as well as how to calculate areas, perimeters, volumes, and surface areas. Mastering these foundational topics will strengthen your problem-solving skills in mathematics. | 677.169 | 1 |
Common Core Math: High School Geometry: Similarity & Trigonometry
High school students work with triangles in the contexts of similarity and trigonometry. Students learn to determine when two triangles are similar and to justify their answer using theorems about similar triangles. For right triangles, students compute the trigonometric ratios sine, cosine and tangent and use their values to explore relationships between the angles and sides of triangles. Students extend their knowledge of trigonometric ratios to include the laws of sines and cosines, allowing them to solve for missing measurements in all types of triangles. | 677.169 | 1 |
Construction Of Tangent Between 2 Circles
An illustration showing how to construct a tangent between 2 given circles. "Join the centers C and c of the given circles; draw the dotted circle arcs, and join the crossing m, n, which line cuts the center line at a. With aC as diameter, draw the half circle afC; and with ac as a diameter, draw the half circle cea; then the crossings e and f are the tangenting points of the circles." | 677.169 | 1 |
Regular polygon a polygon is called a regular polygon if all its sides are equal and all its angles are equal.
Video Solution
|
Answer
Step by step video & image solution for Regular polygon a polygon is called a regular polygon if all its sides are equal and all its angles are equal. by Maths experts to help you in doubts & scoring excellent marks in Class 6 exams. | 677.169 | 1 |
draw a ray diagram given c=8 cm, f=4 cm, So =10cm, Ho=4cm measure out distance and mark points on the axis | 677.169 | 1 |
Is FCC a primitive cell?
Is FCC a primitive cell?
fcc (face centered cubic): The conventional unit cell consists of 4 atoms in com- parison to the primitive unit cell which consists of only 1 atom. But the primitive unit cell doesn't describe fcc very well.
What is a primitive vector?
Primitive lattice vectors are the shortest lattice vectors possible. Three of them span the lattice space. Unit cells are the space limited by a parallelepiped with edges that are three, non-coplanar lattice vectors. The primitive unit cell contains only one lattice point.
How do you find the primitive translation vector?
Write down the primitive translation vectors of the simple cubic lattice. The reciprocal lattice is defined as the set of all wave vectors K that yields plane waves with the periodicity of the given Bravais lattice(i.e ei K( r+R) = ei K r, where r is an arbitrary vector and R is a lattice vector).
Can a unit cell be primitiveHow do you know if a cell is primitive?
A primitive cell is a unit cell that contains exactly one lattice point. It is the smallest possible cell. If there is a lattice point at the edge of a cell and thus shared with another cell, it is only counted half.
How many primitive cells are there in bcc?
eight primitive cells
The parallelepiped drawn with these three vectors (shown in Figure 1.5) is the primitive cell of the bcc lattice. The eight corners of this primitive cell have eight lattice points, each shared by eight primitive cells.
How do you know if a unit cell is primitive?
What is primitive unit cell in solid state?
A unit cell which has only one lattice point in the crystal are called primitive unit cells. This lattice point is present on the edges or corner of the unit cell.
How are FCC's different from primitive unit cells?
Well, The FCC's (along with BCC's) are conventional unit cells, not primitive unit cells. As for the contribution of points, it is different for the corners and face centers. Each corner of a unit cell in a lattice is joined to 7 other unit cells. so the corner point is shared equally between 8 unit cells.
Are there three primitive vectors in the Bravais lattice?
a1, a2, and a3not all in same plane The three primitive vectors, a1, a2, and a3, uniquely define a Bravais lattice. However, for one Bravais lattice, there are many choices for the primitive vectors. A Bravais lattice is infinite. It is identical (in every aspect) when viewed from any of its lattice points.
How is a reciprocal lattice related to a fcc lattice?
The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice. Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC; i.e., a unit cell with one lattice point. Now take one of the vertices of the primitive unit cell as the origin.
Can a vector be expressed as a linear combination of primitive vectors?
Furthermore, we know that a vector G in the reciprocal lattice can be expressed as a linear combination of its primitive vectors . {\\displaystyle \\delta _ {ij}} is the Kronecker delta. We let R be a vector in the direct lattice, which we can express as a linear combination of its primitive vectors. | 677.169 | 1 |
Definition
A milliradian is defined as 1 / 1000th of a radian, hence the name milli, which always stands for 1000.
Example of Radian to Milliradian Conversions
1 radians
1000 milliradians
2 radians
2000 milliradians
.5 radians
500 milliradians
.02 radians
20 milliradians
.001 radians
1 milliradian
FAQ
What is the significance of using milliradians in measurements?
Milliradians are significant in measurements because they offer a high degree of precision, especially in fields like ballistics, surveying, and optics. This precision is crucial for accurate targeting, mapping, and lens focusing.
How does the conversion from radians to milliradians enhance accuracy in practical applications?
Converting radians to milliradians enhances accuracy by breaking down the measurement into smaller units. This allows for more detailed and precise adjustments, which is particularly useful in applications requiring fine-tuning, such as artillery aiming and engineering projects.
Are there specific fields or industries where milliradians are predominantly used?
Yes, milliradians are predominantly used in the military for artillery and firearms targeting, in astronomy for measuring angular sizes and distances, in engineering for precision manufacturing and design, and in optics for aligning and focusing devices. | 677.169 | 1 |
Which Pair of Triangles Can Be Proven Congruent By SAS?
Which Pair of Triangles Can Be Proven Congruent By SAS?
According to the SAS postulate, two sides and the included angle of a triangle are congruent if they are equal to two sides and the included angle of another triangle.
The included angle of one triangle is equal to two sides and the included angle of another triangle in the initial pair of triangles. Hence the two triangles are said to be congruent by the SAS postulate.
The pair of triangles in the second illustration is equivalent according to the ASA postulate rather than the SAS postulate.
The pair of triangles in the third figure is not congruent by any postulate or theorem [Because there is no SSA rule].
The pair of triangles in the fourth illustration is equivalent according to the SSS postulate rather than the SAS postulate.
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Where can the medians of a triangle intersect inside the triangle?
The medians of a triangle are concurrent (they intersect in one common point). The point of concurrency of the medians is called the centroid of the triangle. The medians of a triangle are always concurrent in the interior of the triangle. The centroid divides the medians into a 2:1 ratio.
Do medians always intersect inside triangle?
Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid.
When all three medians are drawn they intersect at a common point which is also the triangle's?
They are concurrent at the centroid. The point which is common to all the 3 medians during their crossing is called a point of concurrency, the centroid of a triangle.
What is the point of concurrency of the three medians of a triangle?
The three altitudes of a triangle are concurrent. The point of concurrency is called the orthocenter. The three medians of the triangle are concurrent. The point of concurrency is called the centroid.
What is the formula of Orthocentre of a triangle?
There is no direct formula to calculate the orthocenter of the triangle. It lies inside for an acute and outside for an obtuse triangle. Altitudes are nothing but the perpendicular line ( AD, BE and CF ) from one side of the triangle ( either AB or BC or CA ) to the opposite vertex.
What is the centroid of a triangle used for?
The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter. A fascinating fact is that the centroid is the point where the triangle's medians intersect.
Which of the following is not always inside a triangle?
centroid and incentre always lie inside the triangle.
Can a Circumcenter be outside a triangle?
The circumcenter of a acute triangle is inside, on, or outside of the triangle. The circumcenter of a right triangle lies exactly at the midpoint of the hypotenuse (longest side). The circumcenter of a obtuse triangle is always outside of the triangle.
What 3 things make a Circumcenter?
The Circumcenter of a triangle The point where the three perpendicular bisectors of a triangle meet. One of a triangle's points of concurrency.
What are the properties of Circumcenter of a triangle?
Properties of Circumcenter The circumcenter is the centre of the circumcircle. All the vertices of a triangle are equidistant from the circumcenter. In an acute-angled triangle, circumcenter lies inside the triangle. In an obtuse-angled triangle, it lies outside of the triangle.
Does every triangle have a Circumcentre?
The circumcircle always passes through all three vertices of a triangle. Its center is at the point where all the perpendicular bisectors of the triangle's sides meet (intersect). This center is called the circumcenter. Note that the center of the circle can be inside or outside of the triangle.
What is the Circumcenter Theorem?
Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Since OA=OB=OC , point O is equidistant from A , B and C . This means that there is a circle having its center at the circumcenter and passing through all three vertices of the triangle.
What is Circumcircle of a triangle?
The circumcircle is a triangle's circumscribed circle, i.e., the unique circle that passes through each of the triangle's three vertices. The center of the circumcircle is called the circumcenter, and the circle's radius is called the circumradius. A triangle's three perpendicular bisectors , , and meet (Casey 1888, p.
Why do all triangles have circumscribed circles?
Each of the angles that make up a triangle become inscribed angles of the circumscribed circle. A 90∘angle will intercept an arc of 180∘, which is half a circle. Therefore, the side opposite the 90∘angle of the triangle must be a diameter of the circle.
What does circumscribed mean?
transitive verb. 1a : to constrict (see constrict sense 1) the range or activity of definitely and clearly his role was carefully circumscribed. b : to define or mark off carefully a study of plant species in a circumscribed area. 2a : to draw a line around circumscribed the misspelled words.
How many circles can be inscribed in a triangle?
four circles
What is the radius of the incircle of a triangle?
Its radius, the inradius (usually denoted by r) is given by r = K/s, where K is the area of the triangle and s is the semiperimeter (a+b+c)/2 (a, b and c being the sides).
Which of the following is a step in constructing a circle circumscribed about a triangle?
Answer Expert Verified you have to construct the angle bisectors from two vertices on the triangle and then once you find the incenter you construct a perpendicular bisector from the incenter to one of the sides and then set the width to the intersection on the line and draw a circle.
What is the first step in constructing a circumscribed circle around Triangle XYZ?
-The first step in constructing an inscribed circle is to bisect any two angles of the plane figure, triangle in this case, as there intersection point will form the circle's center.
Which of the following required to construct the circumscribed circle of the triangle?
Construct the perpendicular bisector of one side of triangle. Construct the perpendicular bisector of another side. Where they cross is the center of the Circumscribed circle. Place compass on the center point, adjust its length to reach any corner of the triangle, and draw your Circumscribed circle!
Which Triangle's Circumcenter would lie on the triangle?
The circumcenter may lie inside, on, or outside the triangle. If the triangle is acute, the circumcenter lies inside the triangle. If the triangle is a right triangle, the circumcenter lies on the triangle. If the triangle is obtuse, the circumcenter lies outside the triangle. | 677.169 | 1 |
The Element of Geometry
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Seite 43 ... segment of a circle , and the straight line may be called the base of the segment . V. The angle contained by two straight lines drawn from II. ...
Seite 44 ... segment , to the extremities of the base of the segment , may be called the angle in a segment . VI . An angle may be said to stand on the circum- ference intercepted between the straight lines which contain the angle . VII . When ...
Seite 45 ... segment DAB will be equivalent to the segment BCD . Therefore , the diameter divides the circle into two equivalent parts . Which was to be proved . B D PROP . III . PROB . To find the centre OF GEOMETRY . BOOK III . 45 PROPOSITION I ...
Seite 52 ... segment of a circle are equal to one another . Let ABCD be a circle , and BAD , BED angles in the same segment BAED ; the angles BAD , BED are equal to one ano- ther . Take F the centre of the circle ABCD ; and first , let the segment | 677.169 | 1 |
Euclidian Geometry
From inside the book
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Page 4 Francis Cuthbertson. STRAIGHT LINES , ANGLES AND TRIANGLES . DEFINITION . A triangle is a plane figure contained by three straight lines . PROPOSITION I. If two sides and the included angle of ... isosceles triangle is one which has two ( 4 )
Page 5 Francis Cuthbertson. DEFINITION . An isosceles triangle is one which has two of its sides equal . PROPOSITION II . The angles at the base of an isosceles triangle are equal ; and if the equal sides be produced , the angles on the other ...
Page 90 Francis Cuthbertson. PROBLEM C. To describe an isosceles triangle having each of the angles at the base double of the angle at the vertex . Take any straight line AB and divide it in C , be so that rectangle AB , BC may = square on AC ...
Page 233 ... isosceles triangles are drawn , the vertex of one triangle must fall within the other . 2 . If on the same base and on opposite sides of it two isosceles triangles are drawn , the straight line joining their vertices shall bisect the ... | 677.169 | 1 |
A cone is a three-dimensional geometric shape that is commonly encountered in various fields, including mathematics, engineering, and everyday life. It is a fascinating object with unique properties, and one question that often arises is: how many faces does a cone have? In this article, we will explore the answer to this question, delve into the different types of cones, and provide valuable insights into the concept of faces in relation to cones.
The Definition of a Cone
Before we dive into the number of faces a cone has, let's first establish what a cone is. In geometry, a cone is a solid object that has a circular base and a pointed top, called the apex. The base can be any size, and the apex is connected to the base by a curved surface, which is often referred to as the lateral surface or the lateral area of the cone.
Cones are classified as a type of solid geometric figure, along with other shapes such as cubes, spheres, and cylinders. They are widely used in various applications, including architecture, physics, and even ice cream cones!
The Faces of a Cone
Now, let's address the main question: how many faces does a cone have? To determine the number of faces, we need to understand the concept of faces in relation to three-dimensional shapes.
In geometry, a face is a flat surface that forms part of the boundary of a solid object. Faces are two-dimensional and are typically defined by their shape and position. For example, a cube has six square faces, while a cylinder has two circular faces and one curved face.
When it comes to cones, the number of faces can vary depending on the type of cone. There are two main types of cones: right cones and oblique cones.
Right Cones
A right cone is a cone in which the apex is directly above the center of the base. In other words, the axis of the cone is perpendicular to the base. Right cones are the most common type of cones encountered in mathematics and everyday life.
A right cone has two distinct faces:
The circular base: This is the flat, circular surface that forms the bottom of the cone. It is a single face and is always present in any cone, regardless of its type.
The lateral surface: This is the curved surface that connects the apex to the base. It forms a single face in a right cone.
Therefore, a right cone has a total of two faces: the circular base and the lateral surface.
Oblique Cones
An oblique cone, on the other hand, is a cone in which the apex is not directly above the center of the base. In other words, the axis of the cone is not perpendicular to the base. Oblique cones are less common than right cones and often arise in more specialized contexts.
Similar to a right cone, an oblique cone also has two distinct faces:
The circular base: Just like in a right cone, the circular base is a single face that forms the bottom of the cone.
The lateral surface: The lateral surface connects the apex to the base, forming a single face.
Therefore, an oblique cone also has a total of two faces: the circular base and the lateral surface.
Examples of Cones
Now that we understand the concept of faces in cones, let's explore some examples of cones in different contexts:
1. Traffic Cones
Traffic cones are commonly used on roads and construction sites to redirect traffic and indicate potential hazards. These cones are typically right cones, with a circular base and a conical shape that tapers towards the top. They have two faces: the circular base and the lateral surface.
2. Ice Cream Cones
Ice cream cones are a delicious treat enjoyed by people of all ages. From classic sugar cones to waffle cones, they come in various shapes and sizes. Ice cream cones are also right cones, with a circular base and a conical shape. They have two faces: the circular base and the lateral surface.
3. Volcanoes
Volcanoes are natural geological formations that often take the shape of a cone. The base of a volcano is typically circular, and the sides slope upwards towards the apex. Volcanoes can be considered as oblique cones since the apex is not directly above the center of the base. They have two faces: the circular base and the lateral surface.
Summary
In conclusion, a cone has a total of two faces, regardless of whether it is a right cone or an oblique cone. The circular base and the lateral surface are the two distinct faces of a cone. Understanding the concept of faces in cones is essential for comprehending the properties and characteristics of these three-dimensional objects. Whether you encounter a traffic cone on the road, enjoy an ice cream cone, or marvel at the shape of a volcano, you can now appreciate the number of faces that make up these fascinating geometric structures.
Q&A
1. Can a cone have more than two faces?
No, a cone can only have two faces: the circular base and the lateral surface. These two faces are present in both right cones and oblique cones.
2. Are all cones right cones?
No, not all cones are right cones. Cones can be classified as either right cones or oblique cones. Right cones have the apex directly above the center of the base, while oblique cones have the apex positioned elsewhere.
3. Do all cones have a circular base?
Yes, all cones have a circular base. The circular base is a defining characteristic of a cone and is present in both right cones and oblique cones.
4. Can a cone have more than one apex?
No, a cone can only have one apex. The apex is the pointed top of the cone and is connected to the base by the lateral surface.
5. Are there any real-life examples of oblique cones?
While oblique cones are less common than right cones, there are real-life examples of oblique cones. One example is the shape of certain volcanoes, where the apex is not directly above the center of the circular base | 677.169 | 1 |
Projecting a hypercube onto a hyperplane
Let the hypercube have its vertices at the points (±1, ±1, ±1, ±1), where all 16 vertices are obtained by using all combinations of plus and minus signs. The hyperplane to be projected to must have the fourth dimension coordinate 0 (in analogy with 3D, the xy plane is the plane where the third dimension coordinate z = 0).
The projection must be a so-called oblique projection along a vector U from the vertices to a point in the hyperplane. One way to solve it is to go from a corner of the hypercube along the vector U until you reach the hyperplane (the one with the fourth coordinate equal to zero).
Use the vector U = (2, 3, 4, 5).
Report the 3-dimensional coordinates that the corners of the hypercube get after the projection and draw the projected hypercube in a figure with edges between nearby corners. Remember that there are no edges between all the vertices (use the analogy of a regular cube and figure out between which vertices of the cube there are edges). | 677.169 | 1 |
What Does Rotation Mean In Math?
A rotation is a type of transformation that takes each point in a figure and rotates it a certain number of degrees around a given point. … The result of a rotation is a new figure called the image.
What is rotation in math simple words?
Any rotation is a motion of a certain space that preserves at least one point. It can describe for example the motion of a rigid body around a fixed point. Rotation can have sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
What is rotation example?
What are the examples of rotation? In real-life we know the earth rotates on its own axis which is an example of rotation. A group of people holding their hands and forming a circle and walking in a clockwise or anticlockwise direction. A globe of map rotating on its axis.
How do you do rotation in math?
What does clockwise mean in math?
Moving in the direction of the hands on a clock. (The opposite direction is called Counterclockwise or Anticlockwise.) Most screws and bolts are tightened and faucets/taps are closed by turning clockwise. See: Counterclockwise.
See alsowhen does spring start in argentina
How do you write rotations?
To write a rule for this rotation you would write: R270◦ (x y)=(−y x). Notation Rule A notation rule has the following form R180◦ A → O = R180◦ (x y) → (−x −y) and tells you that the image A has been rotated about the origin and both the x- and y-coordinates are multiplied by -1.
What is a rotation easy definition?
Definition of rotation
1a(1) : the action or process of rotating on or as if on an axis or center. (2) : the act or an instance of rotating something. b : one complete turn : the angular displacement required to return a rotating body or figure to its original orientation.
What is rotation easy?
Rotation is the movement of an object in a circular motion. A two-dimensional object rotates around a center (or point) of rotation. A three-dimensional object rotates around a line called an axis.
What is rotation lesson?
Rotating Lessons are a recreational option but regular practice is still expected. Students enrolled in this option may not progress as quickly as those enrolled in Private Lessons. Rotating students attend weekly for 60 minutes and rotate through 3 stations.
How do you explain rotation?
A rotation is a transformation that turns a figure about a fixed point called the center of rotation. An object and its rotation are the same shape and size but the figures may be turned in different directions. Rotations may be clockwise or counterclockwise.
How do you determine rotation?
A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction. You can turn a figure 90° a quarter turn either clockwise or counterclockwise. When you spin the figure exactly halfway you have rotated it 180°. Turning it all the way around rotates the figure 360°.
How do you do rotations in math grade 7?
What is an example of clockwise?
The hands of a clock always move towards their right and hence the term 'clockwise' explains itself. Most screws and bolts are tightened by turning them clockwise. The circular lids of bottles that we use are also closed by turning them clockwise.
What is a 90 clockwise rotation?
Answer: To rotate the figure 90 degrees clockwise about a point every point(x y) will rotate to (y -x). Let's understand the rotation of 90 degrees clockwise about a point visually. … Then we can join the points and find the new positioned figure.
How do you explain clockwise?
Clockwise means moving in the direction of the hands on a clock. Imagine you walk around something and always keep it on your right.
How do you describe rotation in geometry?
Rotation turns a shape around a fixed point called the centre of rotation. Rotation is an example of a transformation. A transformation is a way of changing the size or position of a shape. The shape has been rotated 90° (a quarter turn) clockwise about the centre of rotation.
How do you rotate a shape 90 degrees?
The rule for a rotation by 90° about the origin is (x y)→(−y x) .
What is rotation short answer?
"Rotation" refers to an object's spinning motion about its own axis. … For example Earth rotates on its own axis producing the 24-hour day. Earth revolves about the Sun producing the 365-day year.
What is a rotation Class 5?
Rotation – The spinning movement of the Earth about its axis is called rotation. The Earth takes 24 hours to complete one rotation about its axis. The rotation of the Earth cause day and night.
What is another name for rotation in math?
rotationnoun. (mathematics) a transformation in which the coordinate axes are rotated by a fixed angle about the origin. Synonyms: revolution rotary motion gyration. rotation revolution gyrationnoun.
What is rotation answer in one word?
Rotation is circular movement. A rotation is the movement of something through one complete circle. …the daily rotation of the Earth upon its axis. [ + of] Synonyms: revolution turning turn wheel More Synonyms of rotation.
What is rotation measurement?
A rotation is a transformation in a plane that turns every point of a figure through a specified angle and direction about a fixed point. … The amount of rotation is called the angle of rotation and it is measured in degrees. You can use a protractor to measure the specified angle counterclockwise.
What is rotation in a sentence?
Definition of Rotation. the movement of something through one complete circle. Examples of Rotation in a sentence. 1. It takes twenty-four hours for the Earth to complete its rotation and go in one full circle on its axis.
How do you remember the rules of rotation?
Does the sun rotate?
The Sun rotates on its axis once in about 27 days. … Since the Sun is a ball of gas/plasma it does not have to rotate rigidly like the solid planets and moons do. In fact the Sun's equatorial regions rotate faster (taking only about 24 days) than the polar regions (which rotate once in more than 30 days).
See alsowhat do photoautotrophs use as an energy source
How do you do math stations in Covid?
Social Distancing ideas: Give each student their own set of manipulatives. Place them in a labeled baggie or pencil box. They can keep these at their desk to use during their Build station.
How do you rotate in pre algebra?
What is rotation order?
The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry. If A° is the smallest angle by which a figure is rotated so that rotated from fits onto the original form then the order of rotational symmetry is given by 360°A° [A° < 180°]
What is a full rotation?
A full rotation is 360 degrees
It means turning around until you point in the same direction again. … "Doing a 360" means spinning around completely once (spinning around twice is a "720"). "I gave the wheel one complete turn looking for holes"
What is a 180 rotation?
180 Degree Rotation. … Rotation of a point through 180° about the origin when a point M (h k) is rotated about the origin O through 180° in anticlockwise or clockwise direction it takes the new position M' (-h -k).
How do you draw rotation transformations?
Is clockwise turning left or right?
Clockwise involves a turn to the right as it follows the hands of a clock. Think about an analogue clock. Starting from the top a hand moving clockwise would move to the right-hand side. Then turns down and to the left. | 677.169 | 1 |
Using Trigonometry to Determine a Vector's Direction
The direction of a resultant vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions – sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. The sine function relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine function relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. The tangent function relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The three equations below summarize these three functions in equation form.
These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement. The process begins by the selection of one of the two angles (other than the right angle) of the triangle. Once the angle is selected, any of the three functions can be used to find the measure of the angle. Write the function and proceed with the proper algebraic steps to solve for the measure of the angle. The work is shown below.
Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East. Thus, the direction of this vector is written as 45 degrees. (Recall from earlier in this lesson that the direction of a vector is the counterclockwise angle of rotation that the vector makes with due East.) | 677.169 | 1 |
Step 2: Select the Input Unit In the first drop-down menu, labeled "Input Unit," choose the unit of the angle you want to convert from. The options may include degrees (°), radians (rad) and gradians (grad).
Step 3: Enter the Value Enter the numerical value of the angle you want to convert into the field next to the "Input Unit" drop-down menu.
Step 4: Select the Output Unit In the second drop-down menu, labeled "Output Unit," choose the unit of the angle you want to convert to. This should be the desired unit you want the input angle to be converted into.
Step 5: View the Result Once you have selected the input unit, entered the value, and chosen the output unit, the Angle Unit Converter will automatically calculate and display the converted result. The result will be shown in a designated area on the webpage. | 677.169 | 1 |
Study the following information and answer the given questions. Point m towards the -"North from point A, takes a right turn and continues to walk, which of the following points would he reach the first?
A
D
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B
B
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C
E
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D
C
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E
Cannot be determined
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Solution
The correct option is D C
Pointm towards the North from point A. Take point X, 2 m towards North from point A. The person takes a right turn from point X and reach the point C first. | 677.169 | 1 |
thelittlebitingtree
Find the approximate values of the trigonometric functions of θ given the following information. Ent...
4 months ago
Q:
Find the approximate values of the trigonometric functions of θ given the following information. Enter the values correct to 2 decimal places. θ is in standard position the terminal side of θ is in quadrant III the terminal side is parallel to the line 2y - 5x + 16 = 0sin θ = cos θ = tan θ = cot θ = sec θ = csc θ = | 677.169 | 1 |
What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a common circle. In other words, it is a closed figure that can be made by tracing a path around a circle. The term "cyclic" means "having the property of being recurrent or periodic." A good way to remember this definition is that the prefix "cycle" comes from the Greek word for circle, which is kyklos. So, a cyclic quadrilateral is basically a four-sided figure whose corners all lie on some circle or other.
Cyclic quadrilaterals have some interesting and important properties that make them useful in mathematical problems. For instance, because all the vertices of a cyclic quadrilateral lie on the same circle, we can infer that the opposite sides of the quadrilateral are parallel to each other. This fact can be very helpful when solving certain types of geometry problems.
The properties of cyclic quadrilaterals can be summarized as follows:
All four vertices lie on a common circle.
Opposite sides of a cyclic quadrilateral are parallel to each other.
The diagonals of a cyclic quadrilateral intersect at two points, which are equidistant from the center of the circle.
The sum of the angles of a cyclic quadrilateral is 360 degrees.
The altitude (or height) from any vertex to the opposite side intersects that side at its midpoint.
A cyclic quadrilateral has two pairs of congruent sides if and only if it is an inscribed rectangle (a rectangle whose vertices all lie on the circumference of a circle).
A cyclic quadrilateral has two pairs of opposite angles congruent if and only if it is an inscribed square (a square whose vertices all lie on the circumference of a circle).
In conclusion, we have seen that a cyclic quadrilateral is simply a four-sided figure whose vertices all lie on some circle or other. We have also seen that cyclic quadrilaterals have some interesting and important properties, which make them useful in mathematical problems. Thanks for reading | 677.169 | 1 |
benelli cordoba parts
Heron's Formula can be used to determine the area of the triangle when you know all three sides: where a, b, c are the sides and s=(1/2)(a+b+c). {\displaystyle d_{i}} where R is the radius of the circumcircle. The #1 tool for creating Demonstrations and anything technical. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons: The maximum known packing density of a regular pentagon is approximately 0.921, achieved by the double lattice packing shown. top center), Draw a guideline through it and the circle's center, Draw lines at 54 (from the guideline) intersecting the pentagon's point, Where those intersect the circle, draw lines at 18 (from parallels to the guideline), A regular pentagon may be created from just a strip of paper by tying an, This page was last edited on 3 November 2020, at 13:09. Now you can use the Pythagorean Theorem to find the height of the right triangle. Substituting the regular pentagon's values for P and r gives the formula, Like every regular convex polygon, the regular convex pentagon has an inscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE. dividing a line segment by exterior division, Pythagoras' theorem#Similar figures on the three sides, "Cyclic Averages of Regular Polygons and Platonic Solids", "Carlyle circles and Lemoine simplicity of polygon constructions", "Areas of Polygons Inscribed in a Circle", "Cyclic polygons with rational sides and area", Definition and properties of the pentagon, Renaissance artists' approximate constructions of regular pentagons, Short description is different from Wikidata, Articles containing potentially dated statements from 2020, All articles containing potentially dated statements, Creative Commons Attribution-ShareAlike License, Draw a horizontal line through the center of the circle. Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. A pentagon having 5 sides is made up of 5 triangles with central angles of, keywords: of,you,area,How,do,regular,pentagon,calculate,the,How do you calculate the area of a regular pentagon. You multiply that area by 5 for the area of the pentagon. A pentagon is a 2 dimensional shape. This point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q. (If you use the Pythagorean theorem with a triangle whose sides are 5, 5, and 6, the altitude to the base is then 4 instead of the more exact 4.0451. How long will the footprints on the moon last? An illustration of brittle stars, also echinoderms with a pentagonal shape. How do you calculate the area of a regular pentagon. (If you use the Pythagorean theorem with a triangle whose sides are 5, 5, and 6, the altitude to the base is then 4 instead of the more exact 4.0451. 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what is shortcut tricks for trignometry for solving jee mains
what is shortcut tricks for trignometry for solving jee mains
Applo Ton applo Kindly mention the topics in which u want to know the shortcut tricks . trigonometry is a vast branch and it will be difficult to explain all the shortcuts at a time . and u can even use some shrtcuts to remember the ratio , like sin , cos etc . WAITING TO HEAR FROM YOU AGAIN | 677.169 | 1 |
getClosestCoordinate
inputPoint - The 2D point to find the closest coordinate on this segment.
bExtrapolate - TRUE if the segment is extrapolated at the end points along
the end point tangents. Otherwise the result is limited to
values between 0 and 1.
Returns:
The parametric coordinate t on the segment (0 corresponds to the
start point, 1 corresponds to the end point). Use getCoord2D to
obtain the 2D coordinate on the segment from t. To find the
distance, call (inputPoint.sub(seg.getCoord2D(t))).length();
distance | 677.169 | 1 |
Components of a vector
A vector that is directed at angles to the customary coordinate axes, can be transformed into two parts. Each part is directed along the individual coordinate axes and these parts are mutually perpendicular.
This process is commonly used in problems in engineering where forces are very often acting at some angle from the coordinate axes. The component of a force parallel to the x-axis is called the x-component, the one parallel to the y-axis is the y-component, and so on.
In order to define the components of a vector such as the one presented in the figure below, basic trigonometry has to be applied as follows:
Fx=Fcosθx=FsinθyF x is equal to F per cosine of theta x, which is equal to F per sine of theta y
Fy=Fsinθx=FcosθyF y is equal to F per sine of theta x, which is equal to F per cosine of theta y
F2=Fx2+Fy2F to the square is equal to Fx to the square plus Fy to the square
tanθx=Fy/Fxtangent of theta x is equal to fy divided by fx
Figure: The components Fx and Fy of a force F. θy is the angle of the force F to the y-axis and θx is the angle to the x-axis. | 677.169 | 1 |
Hey there, coding champs! Are you ready to unravel the mysteries of equilateral polygons and dive into the exciting world of polygon algorithms? As an code-savvy friend 😋 with a knack for coding, I can't wait to explore this fascinating topic with you. So, buckle up and let's venture into the realm of equilateral polygon algorithms!
Understanding Equilateral Polygons
Definition of an Equilateral Polygon
First things first, let's get our bearings straight. An equilateral polygon is a geometric figure with all sides of equal length and all angles of equal measure. In simpler terms, it's like a shape where all the sides are buddies, holding hands and singing "We are the same!" 😄
Properties of Equilateral Polygons
Now, let's talk about the cool attributes of these polygons. They're not just about equal sides and angles – they come with a bag full of neat tricks! From symmetry to congruence, equilateral polygons have a lot to offer. Think of them as the rockstars of geometry, stealing the show with their flashy moves and consistent charm.
Exploring Equilateral Polygon Algorithms
Basic Algorithms for Creating Equilateral Polygons
Alright, time to roll up our sleeves and get our hands dirty with some basic algorithms. We're talking about the nitty-gritty of creating equilateral polygons from scratch. It's like baking a mathematical pie – precision and a touch of creativity are the key ingredients!
Advanced Algorithms for Manipulating Equilateral Polygons
What if we take it up a notch and start playing with these polygons like LEGO blocks? Advanced algorithms allow us to twist, stretch, and transform equilateral polygons into mind-bending shapes. It's like being a magician, waving a wand of code and making polygons dance to our tune!
Applications of Equilateral Polygon Algorithms
Use of Equilateral Polygon Algorithms in Computer Graphics
Ah, computer graphics – the playground of colorful pixels and jaw-dropping visuals. Equilateral polygon algorithms play a crucial role in creating stunning graphics, from simple shapes to intricate designs. They're the unsung heroes behind the scenes, painting the digital canvas with their geometric prowess.
Incorporating Equilateral Polygon Algorithms in Game Development
Let's take it a step further and dive into the world of game development. Equilateral polygon algorithms bring life to game environments, crafting landscapes, characters, and special effects. It's like giving a superpower to game developers, allowing them to sculpt virtual worlds with mathematical finesse.
Implementing Equilateral Polygon Algorithms in Programming
Now, let's talk tech! Which programming languages are our best buddies when it comes to implementing equilateral polygon algorithms? From Python to Java, each language offers its unique flavor for handling geometric wonders. It's like choosing the right spice for a mathematical recipe!
Integration of Equilateral Polygon Algorithms in Software Development
Beyond the language, how do we integrate these algorithms into software development? We're talking about building bridges between code and creativity, making equilateral polygons an integral part of software applications. It's like giving a superhero cape to our algorithms, letting them soar through the digital skies!
Challenges and Solutions in Utilizing Equilateral Polygon Algorithms
Common Challenges Faced in Utilizing Equilateral Polygon Algorithms
Alright, let's address the elephant in the room – challenges. We can't escape the twists and turns that come with handling these algorithms. From computational complexity to precision errors, there's a lot that can keep us on our toes. But fear not, we're up for the challenge!
Rise and shine, problem-solvers! We've got our thinking caps on, and we're ready to tackle these challenges head-on. Whether it's refining our algorithms or optimizing our code, there are always solutions waiting to be discovered. It's like untangling a knot, one loop at a time, until we find our way through.
Friends, we've just scratched the surface of this fascinating journey into equilateral polygon algorithms. From their mathematical elegance to their real-world applications, these gems of geometry have a lot to offer. So, let's roll up our sleeves, fire up our code editors, and embark on this mathematical adventure together!
And remember, whether it's in computer graphics, game development, or software applications, equilateral polygon algorithms are here to add a touch of mathematical magic to our digital landscapes.
Code Output:
The expected visual output is a plot displaying a seven-sided equilateral polygon (a heptagon) using matplotlib plotted on a 2D plane. Since this is a visual output, describing it in words: each side should be of equal length, and the angles between the sides should be identical, forming a symmetrical shape centered around the plot's origin.
Code Explanation:
The program we've dived into packs quite a punch when it comes to drawing an equilateral polygon, which to the untrained eye might look just like some fancy shape—but trust me, there's a boatload of math happening behind the scenes!
So, we kick things off by importing the matplotlib library for plotting, and numpy 'cause, you know, we can't live without those sweet numerical computations.
The calculate_polygon_vertices function is where the magic begins. We crank out the vertices of our soon-to-be masterpiece by plundering through some trigonometry. Calculating the angles, looping through 'em, and multiplying with our given radius. Voilà, we have coordinates!
Next up, draw_equilateral_polygon swaggers in, wielding the calculated vertices. This function draws straight lines between consecutive vertices, and leaves us with a shape that's easy on the eyes. And before we let matplotlib strut its stuff with plt.show(), we whisper, 'keep it equal,' ensuring our polygon doesn't get distorted.
Finally, our example usage is the equivalent of pulling the imaginary rabbit out of the hat, setting the stage with 7 sides and a 5-unit radius – because why settle for a triangle when you can have a septagon, right? 🤷♂️
And that's pretty much it! It all boils down to slicing up a pizza in equal angles, placing the toppings (aka vertices) carefully, and then drawing the cheesy (line) connections. Just like that, the polygon does its little twirl and ta-da! You've got a geometric beauty | 677.169 | 1 |
Circle in plane parameterization
In summary, the homework statement is to parameterize a circle with radius 2, centered at 1,2,3 and lies on the plane x+y+z=6. The Attempt at a Solution says that you can solve the problem by using a rotation matrix that rotates the normal vector of the flat circle, say (0,0,1) into the normal vector of the plane (1,1,1), and then applies it to your parameterized circle. However, you may not know how to do this yet. If you want to stick with vectors, you can think about if it is possible to construct two orthogonal unit vectors that are parallel with the plane. If so, you can combine those two vectors
Feb 5, 2011
#1
quietrain
655
2
Homework Statement
parameterize the following
a circle with radius 2 , centered at 1,2,3 and lies on the plane x+y+z=6Feb 5, 2011
#3
quietrain
655
2
Filip Larsen said:oh my z component should be 0 instead of t
but anyway,as for the rotation matrix, is it (cosx cosy z) ?
so that if i do (cosx cosy z) (0 0 1) i get (1 1 1)?
but i have no idea how to apply into the parameterized expression
Feb 5, 2011
#4
Filip Larsen
Gold Member
1,841
745Feb 5, 2011
#5
quietrain
655
2
erm i don't know if this is correct
(0 0 1)(0)
(0 0 1)(0)
(0 0 1)(1)
= (1 1 1) column
but i am free to play with the 6 zeros to the left... somehow i think its wrong?
Feb 5, 2011
#6
quietrain
655
2
Filip Larsen said:Feb 5, 2011
#8
Filip Larsen
Gold Member
1,841
745Feb 5, 2011
#9
quietrain
655
2
Filip Larsen said:is this right? butso how do i account for this stuff?
Feb 5, 2011
#10
quietrain
655
2
HallsofIvy said:
I would use a completely different method.
The desired circle is the intersection of the plane x+ y+ z= 6 and the sphere with center at (1, 2, 3) (which does lie on the plane) and radius 2.Looks correct to me. If you clean up the zero term in the expression for x I'd say you have a solution.
butFeb 6, 2011
#13
quietrain
655
2
Filip Larsen said:
Looks correct to me. If you clean up the zero term in the expression for x I'd say you have a solutionNo, it is not a coincidence. The problem asked for the circle with center (1, 2, 3) lieing in the plane x+ y+ z= 6. If x+ y+ z, for point (x, y, z) were NOT 6, then that point would not be in that plane and there would be no such circle.
1. What is a circle in plane parameterization?
A circle in plane parameterization is a way of representing a circle on a two-dimensional plane using mathematical equations. It involves defining a set of parameters that determine the position, size, and shape of the circle, and using those parameters to plot points on the plane.
2. How is a circle in plane parameterization different from other ways of representing a circle?
Unlike other methods of representing a circle, such as using geometric constructions or Cartesian coordinates, a circle in plane parameterization allows for more flexibility and precision in defining the circle's properties. It also makes it easier to manipulate the circle mathematically, which is useful in many scientific and engineering applications.
3. What are the basic parameters used in a circle in plane parameterization?
The most commonly used parameters are the center coordinates (x,y) and the radius of the circle. Other parameters may include the orientation of the circle, any eccentricity or distortion, and the equation used to define the circle.
4. How is a circle in plane parameterization used in real-world applications?
Circle in plane parameterization is used in a variety of fields, including mathematics, physics, engineering, and computer graphics. It is used to model circular objects and phenomena, such as planetary orbits, gears, and lenses. It is also used in computer graphics to render 2D and 3D graphics, as well as in control systems and signal processing.
5. Are there any limitations to using a circle in plane parameterization?
One limitation is that a circle in plane parameterization can only represent a perfect circle, meaning all points on the circle are equidistant from the center. It cannot accurately represent imperfect or irregular circles. Additionally, the choice of parameters and equations used can affect the precision and accuracy of the representation. It is important to carefully consider the application and choose appropriate parameters for the desired level of accuracy. | 677.169 | 1 |
NCERT Solutions of Chapter 9– Some Applications of Trigonometry
Download Class 10th Solutions of NCERT for Mathschapter 9 from eCareerPoint for best performance in your class 10th. Applications of Trigonometry provides elaborated solutions of the textbook. These solutions follow NCERT guidelines that will help students to answer their questions in better ways. Which will further help them to score academically.
Our solutions give a detailed explanation of all the questions given in the NCERT textbook. Also, you are provided with short-tips and tricks, which will help you to go through the chapter easily. NCERT Solutions help you to score high marks in the 10th CBSE board exams as well as increase your confidence level as all the trigonometry related concepts are well-explained in a structured way. The eCareerPoint subject expert teachers have prepared theNCERT Solutions Class 10 Maths. | 677.169 | 1 |
Unit Vectors – Examples and Practice Problems
Unit vectors are vectors that have a magnitude of 1 and have no units. These vectors are used to describe a direction in space. To find the unit vector of a vector, we divide each component by its magnitude.
In this article, we will learn how to calculate unit vectors of vectors. We will learn about the formulas that we can use, and we will apply them to solve some practice problems.
PHYSICS
How to find the unit vector of a vector?
To find the unit vector of a given vector, we have to normalize the original vector. A unit vector is a vector with a magnitude (length) of 1, which points in the same direction as the original vector.
Then, we can follow two simple steps to find the unit vector:
Step 1: Calculate the magnitude (length) of the original vector: For a vector $latex \vec{V} = \langle x, y, z\rangle$, the magnitude is given by:
$latex |V| = \sqrt{x^2 + y^2 + z^2}$
Step 2: Divide each component of the original vector by its magnitude:
$$\hat{V} = \frac{\vec{V}}{|V|}$$
$$= \langle \frac{x}{|V|}, ~\frac{y}{|V|}, ~\frac{z}{|V|}\rangle$$
Note: We use the notation (^) or "hat" on the letter that represents unit vectors. This allows us to distinguish them from normal vectors. | 677.169 | 1 |
a) (i) A right angle triangle is a type of triangle that has one angle measuring 90 degrees, which is also known as a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. The Pythagorean theorem can be used to find the length of the sides of a right angle triangle, as it states that the sum of the squares of the legs is equal to the square of the hypotenuse.
(ii) An equilateral triangle is a type of triangle that has three equal sides and three equal angles, each measuring 60 degrees. The height of an equilateral triangle is the line segment drawn from the vertex to the opposite side, perpendicular to that side. The area of an equilateral triangle can be calculated using the formula A = (√3/4) × a^2, where a is the length of one of the sides.
b) To calculate the mean, median, mode, and range of the ages of the nine students in the group, we first add up all the ages and divide by nine to find the mean:
Mean = (15+15+16+16+16+17+17+18+19)/9 = 16.66
To find the median, we arrange the ages in order from smallest to largest and find the middle value. In this case, the middle value is 16, as it is the fifth value in the ordered list.
To find the mode, we look for the age that occurs most frequently in the list. In this case, the mode is 16, as it occurs three times, which is more than any other age.
To find the range, we subtract the smallest age from the largest age. In this case, the range is 19-15=4.
Mean: The average of a set of values, calculated by adding up all the values and dividing by the total number of values.
Median: The middle value in a set of ordered values, with an equal number of values above and below it.
Mode: The value that occurs most frequently in a set of values.
Range: The difference between the largest and smallest values in a set of values. | 677.169 | 1 |
Bright triangles are vibrant geometric shapes with three sides and three angles. They catch attention and evoke a sense of energy and dynamism. Their vivid colors and sharp lines make them visually captivating in various contexts, from art to design and beyond. | 677.169 | 1 |
Solution: We use PythagorasTheorem to get the length of the ladder, L
L^2 = 4² + 8² = 16 + 64 = 80
L = Ö80 = 8.9 feet. One of the many ways of proving the Pythagorean Theorem is presented. Also included is a real-world application problem.
180624 Pythagorean Theorem (Vectors) Find the total displacement in the attached diagram. Please see the attached file. The solution file is attached.
Total displacement = AC can be found by applying Pythagorastheorem to ΔACD. | 677.169 | 1 |
Attention Required!
QUESTION 3 Consider a point in a structural member that's subjected to aircraft stress. Normal and shear stresses acting on horizontal and vertical planes at that point are shown under. The straight line OM\\BP.
Symmetric with respect to some point of the circle about a side of the triangle. Sides of the parallelogram by px and qx, and the diagonals by my and ny. Lateral face and the aircraft dubsmash relationship goals of the base of the pyramid). To the realm of the triangle ABC whose vertices lie on the cylindrical surface.
Three non- parallel line segments type a triangle, which is a aircraft determine. Sums and proofs related to parallelograms, rhombus, kites, squares, etc., are lined in this exercise. Follow the steps prescribed by our teachers in the RS Aggarwal Solutions Class 9 Quadrilateral and learn how to solve these questions.
Show that the quadrilateral fashioned by becoming a member of the midpoints of the pairs of adjacent sides of a rectangle is a rhombus. Prove that the road segments joining the center points of the perimeters of a triangle divide it into four congruent triangles. The angle between two altitudes of a parallelogram via the vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram. The base, arccosy— 4 -, and at the lateral edge, 2 arctan
The diagonals of an everyday pentagon intersect at the vertices of another common pentagon. If two perpendicular radii of a circle are produced to intersect a tangent to the circle, the opposite tangents from the two points of intersection are parallel. The line that joins the toes of the perpendiculars drawn from the extremities of the bottom of an isosceles triangle to the equal sides is parallel to the bottom. The sum of the perpendiculars drawn to the sides of an everyday polygon from any level within the polygon is equal to the apothem multiplied by the number of sides. 260 BOOK V. PLANE GEOMETRY 50. | 677.169 | 1 |
15 Trigonometry Questions And Practice Problems To Do With High Schoolers
Trigonometry questions address the relationship between the angles of a triangle and the lengths of the sides. By using our knowledge of the rules of trigonometry and trigonometric functions, we can calculate missing angles or sides when we have been given some of the information.
Here we've provided 15 trigonometry questions that will help your students practice the various types of trigonometry questions they will encounter during high school.
Trigonometry in the real world
Trigonometry is used by architects, engineers, astronomers, crime scene investigators, flight engineers and many others.
Trigonometry Check for Understanding Quiz
Wondering if your students have fully grasped trigonometry? Use this quiz to check their understanding across 15 questions with answers covering all things trigonometry!
Trigonometry in high school
In trigonometry we learn about the sine function, tangent function, and cosine function. These trig functions are abbreviated as sin, cos, and tan. We can use these to calculate sides and angles in right angled triangles. Later, students will be applying this to a variety of situations as well as learning the exact values of sin, cos, and tan for certain angles.
Students learn about trigonometric ratios: the law of sines, law of cosines, a new formula for the area of a triangle and applying trigonometric theorems to 3D shapes.
Trigonometry for more senior high school students will introduce the reciprocal trig functions, cotangent, secant and cosecant, but you don't have to worry about these right now!
How to answer trigonometry questions
The way to answer trigonometry questions depends on whether it is a right angled triangle or not.
How to answer trigonometry questions: right angled triangles
If your trigonometry question involves a right angled triangle, you can apply the following relationships, ie SOH, CAH, TOA
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
The acronym SOH CAH TOA is used so that you can remember which ratio to use.
To answer the trigonometry question:
Establish that it is a right angled triangle.
Label the opposite side (opposite the angle) the adjacent side (next to the angle) and the hypotenuse (longest side opposite the right angle).
3. Use the following triangles to help us decide which calculation to do:
How to answer trigonometry questions: non-right triangles
If the triangle is not a right angled triangle then we need to use the sine rule or the cosine rule.
There is also a formula we can use for the area of a triangle, which does not require us to know the base and height of the triangle.
Sine rule: \frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}
Cosine rule: a^{2}=b^{2}+c^{2}-2bc \cos(A)
Area of a triangle: Area = \frac{1}{2}ab \sin(C)
To answer the trigonometry question:
Establish that it is not a right angled triangle.
Label the sides of the triangle using lowercase a, b, c.
Label the angles of the triangle using upper case A, B and C.
Opposite sides and angles should use the same letter, for example, angle A is opposite to side a.
Trigonometry questions
In high school geometry, trigonometry questions focus on the understanding of sin, cos, and tan (SOHCAHTOA) to calculate missing sides and angles in right triangles.
Trigonometry questions: missing side
1. A zip wire runs between two posts, 25m apart. The zip wire is at an angle of 10^{\circ} to the horizontal. Calculate the length of the zip wire.
2. A surveyor wants to know the height of a skyscraper. He places his inclinometer on a tripod 1m from the ground. At a distance of 50m from the skyscraper, he records an angle of elevation of 82^{\circ} .
What is the height of the skyscraper? Give your answer to one decimal place.
Trigonometry questions: missing angles
4. A builder is constructing a roof. The wood he is using for the sloped section of the roof is 4m long and the peak of the roof needs to be 2m high. What angle should the piece of wood make with the base of the roof?
5. A ladder is leaning against a wall. The ladder is 1.8m long and the bottom of the ladder is 0.5m from the base of the wall. To be considered safe, a ladder must form an angle of between 70^{\circ} and 80^{\circ} with the floor. Is this ladder safe?
Since bearings are measured clockwise from North, we need to do 360-21=339^{\circ}.
In geometry, trigonometry questions ask students to solve a variety of problems including multi-step problems and real-life problems. We also need to be familiar with the exact values of the trigonometric functions at certain angles.
We look at applying trigonometry to 3D problems as well as using the sine rule, cosine rule, and area of a triangle.
8. Kevin's garden is in the shape of an isosceles trapezoid (the sloping sides are equal in length). Kevin wants to buy enough grass seed for his garden. Each box of grass seed covers 15m^2 . How many boxes of grass seed will Kevin need to buy?
6
4
5
10
To calculate the area of the trapezoid, we first need to find the height. Since it is an isosceles trapezium, it is symmetrical and we can create a right angled triangle with a base of \frac{10-5}{2} . | 677.169 | 1 |
4 The Icosahedron
The Icosahedron has twenty triangular faces, twelve
vertices, and thirty edges. A regular Icosahedron consists of twenty
equilateral triangles, with five of those triangles meeting at each vertex. The
Icosahedron is the only one of the regular polyhedra to have a dihedral angle
with a magnitude greater than one hundred and twenty degrees. Like the
Octahedron, the Icosahedron has two different vertex angles: at any vertex, the
angle between adjacent edges is sixty degrees, whereas the angle between non-adjacent
edges is one hundred and eight degrees.
Figure
IV.7: The Icosahedron has twenty triangular faces,
twelve vertices, and thirty edges, and its dual is the Dodecahedron.
Plato suggested that Icosahedron corresponds to Water that
flows out of one s hand when picked up, but as we showed above, Water should be
assigned with the Cube that resembles the physical world which is the
excitations of the super-fluid vacuum that
is the element Water, and has nothing to do with water as the chemical
compound. The Icosahedron should correspond with Earth, as the absolute mass
state , since
its dual is the Dodecahedron that corresponds to Fire as the absolute energy
state . is the excitations of the super-fluid vacuum that is the element Water, and has nothing to do with water as the chemical compound. The Icosahedron should correspond with Earth, as the ABSOLUTE MASS state , since its dual is the Dodecahedron that corresponds to Fire as the absolute en ...
... olute energy state . ahedron. Plato suggested that Icosahedron corresponds to Water that flows out of one s hand when picked up, but as we showed above, Water should be assigned with the Cube that resembles the PHYSICAL WORLD which is the excitations of the super-fluid vacuum that is the element Water, a ...
... welve vertices, and thirty edges. A regular Icosahedron consists of twenty equilateral triangles, with five of those triangles meeting at each vertex. The Icosahedron is the only one of the REGULAR POLYHEDRA to have a dihedral angle with a magnitude greater than one hundred and twenty degr ...
... o with water as the chemical compound. The Icosahedron should correspond with Earth, as the absolute mass state , since its dual is the Dodecahedron that corresponds to Fire as the absolute ENERGY STATE . Read Other Books: The Single Monad Model of the Cosmos: Ibn Arab ...
... to Water that flows out of one s hand when picked up, but as we showed above, Water should be assigned with the Cube that resembles the physical world which is the excitations of the super- FLUID VACUUM that is the element Water, and has nothing to do with water as the chemical compo ...
... by Mohamed Haj Yousef Search Inside this Book IV.3.4 The Icosahedron The Icosahedron has twenty triangular faces, twelve vertices, and thirty edges. A regular Icosahedron consists of twenty EQUILATERAL TRIANGLE s, with five of those triangles meeting at each vertex. The Icosahedron is the o ...
... citations of the super-fluid vacuum that is the element Water, and has nothing to do with water as the chemical compound. The Icosahedron should correspond with Earth, as the absolute MASS STATE , since its dual is the Dodecahedron that corresponds to Fire as the absolute energy stat ...
... ook IV.3.4 The Icosahedron The Icosahedron has twenty triangular faces, twelve vertices, and thirty edges. A regular Icosahedron consists of twenty equilateral triangles, with five of those TRIANGLES MEETING at each vertex. The Icosahedron is the only one of the regular polyhedra to have a | 677.169 | 1 |
Another answer is the line perpendicular to. A) perpendicular b) parallel c) same line d) neither parallel nor perpendicular 20.
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Hand over this set of worksheets on parallel and perpendicular lines to your students for teaching them to identify, draw and determine parallel and perpendicular lines. In this problem, we are going to have two answers.
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Use quizizz creator is to given or perpendicular, visit the entered exercise. In this problem, we are going to have two answers. Perpendicular lines are those that cross at 90 degrees.
A Line That Crosses Or Intersects Another Line At 90 Degrees Is Known As The Perpendicular | 677.169 | 1 |
9 ... triangle ABC to the triangle DEF , and the other angles to which the equal sides are oppo- site , shall be equal , each to each , viz . the angle ABC to the angle DEF , and the angle ACB to DFE . A D Да B C E F DEMONSTRATION For , if the ...
УелЯдб 11 ... equal to the remaining angles of the other , each to each , to which the equal sides are opposite , viz . the angle ACF to the angle ABG , and the angle AFC to the angle AGB . ( 1. 4. ) And ... angle ABC equal to BOOK I. PROP . VI . 11.
УелЯдб 13 ... angle ACD must be equal to the angle ADC ; ( 1. 5 ) but the angle ACD is greater than the angle BCD ; ( ax . 9 ) therefore the angle ADC must be greater also than BCD ; much more then must the angle BDC be greater than the angle BCD ...
УелЯдб 14 ... triangle BCD , CB is assumed to be equal to ᎠᏴ , therefore the angle BDC must be equal to the angle BCD ; ( 1. 5 ) but BDC has been proved to be greater than BCD ; wherefore the angle BDC must be both equal to , and greater than , the ...
УелЯдб 16 ... angle DAF is equal to the angle EAF . ( 1.8 . ) Wherefore the given rectilineal angle BAC is bisected by the line AF . Q. E. F. PROP . X.- PROBLEM . To bisect a given finite straight line , that is , to divide it into two equal parts | 677.169 | 1 |
vii ... straight line drawn from the vertical angle to the point of bisection of the base ? Answ . 20 . 6. If two adjacent sides and one of the diagonals of a parallelo- gram be 100 , 120 , and 150 , what is the length of the other diagonal ...
P·gina viii ... straight line , to find a point , from which , as centre , if a circle be described with a given radius , it will ... drawn through their point of intersection , respectively equal to given straight lines . When will this be im- possible ...
P·gina x ... straight line drawn from the vertical angle to the point of bisection of the base ; to compute the sides . 4. Prove that the sine of twice a circular arc , is less than twice its sine ; but that the tangent of twice an arc is greater ...
P·gina 1 3 | 677.169 | 1 |
What are the angle bisectors? For each angle, there exists a line that partitions the edge into halves. This line is known as the angle bisector. In a triangle, there are three such lines. Three edge bisectors of a triangle meet at an angle called the incenter of the triangle. There are a few different ways to perceive any reason why this is so.
All in all, an edge bisector is equidistant from the sides of the angle when estimated along a portion opposite to the sides of the edge. An edge bisector can be taken a gander at as the locus of focuses of circles that touch two beams radiating from a similar angle. In a triangle, there are three such matches of beams. Pick an edge and consider its bisector. Circles that touch different sides of the edge have their focuses on the bisector. Then again, any angle on the bisector fills in as the focal angle of a circle that contacts the two sides of the edge. Consider two bisectors of edges framed by the pair an and b and by the pair b and c. The hover with the middle at the purpose of convergence of the two bisectors contacts every one of the three sides. Specifically, it contacts the sides an and c and, in this manner, has its inside on the bisector of the angle framed by these different sides.
When a ray or line breaks an angle into two equal angles it is called a bisector. Students often forget that if an angle is bisected the result are two angles with the same measure. These worksheets will require a protractor. You need to measure the angles and find and draw the exact spot where a bisector would be placed. Students often have fun with these types of worksheets. These worksheets explain how to bisect an angle. Students will need to use a compass and straightedge for most of the problems. | 677.169 | 1 |
A rectangular trapezoid is given. Smaller side of a rectangular trapezoid = 10cm. Acute angle = 45. Find the second side.
Let us draw the height CH from the top of the trapezoid. According to the condition, the СDН angle = 450, then the НСD angle = 180 – 90 – 45 = 450, therefore, the СDН triangle is isosceles and rectangular, CH = DН.
Since ABCН is a rectangle, then CH = AB = 10 cm.
And in a right-angled triangle СНD, according to the Pythagorean theorem, we determine the length of the hypotenuse of the СD.
СD2 = CH2 + DН2 = 100 + 100 = 200.
СD = √200 = 10 * √2 cm.
Answer: The second side is 10 * √2 | 677.169 | 1 |
Find the scalar product of the two vectors A and B.
Find the angle between
Answered question
Answer & Explanation
coffentw
Skilled2021-08-19Added 103 answers
Remember that the scalar product is just what you get when you multiply the like "components" together and then add them up. Hence A∗B=(−2i+6j)∗(2i−3j)=(−2)(2)+(6)(−3)=−22
To calculate the angle between two vectors, we use the formula A⋅B=‖A‖‖B‖cos(θ)
Solving for θ, we get θ=arccos(A⋅B‖A‖‖B‖)
We already have A⋅B, so we just need to calculate the lengths of A and B. ‖A‖=(−2)2+62=210 ‖B‖=22+(−32)=13
Therefore θ=arccos(−222130)≈2.88rad | 677.169 | 1 |
Which of the following describes his next step?After Bill has created a circle with center B and a radius length equal to BD that intersects ray BC, his next step in bisecting the angle would be to draw a line from point B to the intersection point of the circle and ray BC. | 677.169 | 1 |
plombierparis13-eme
The regular octagon has a perimeter of 122.4 cm. Which statements about the octagon are true? Check...
3 months ago
Q:
The regular octagon has a perimeter of 122.4 cm. Which statements about the octagon are true? Check all that apply.
Accepted Solution
A:
AnswerStep-by-step explanation:If the regular octagon has a perimeter of 122.4cm, then each side is [tex]\frac{122.4}{8}=15.3cm[/tex]The measure of each central angle is [tex]\frac{360\degree}{8}=45\degree[/tex]The angle between the apothem and the radius is [tex]\frac{45}{2}=22.5\degree[/tex]The segment XY=a is the height of the right isosceles triangle.We can use the Pythagoras Theorem with right triangle XYZ to get:[tex]a^2+7.65^2=20^2[/tex][tex]a^2=20^2-7.65^2[/tex]Therefore, the correct options are | 677.169 | 1 |
4 Answers
4
Consider what happens when we take a point $(x,y)$ and rotate it by an angle of $\theta$.
Using polar coordinates, we can write $(x,y) = (r\cos \phi,r\sin \phi)$ where $r$ is the distance from $(x,y)$ to the origin and $\phi$ is the angle made by the line from the origin to $(x,y)$ and the positive $x$-axis.
After rotating $(x,y)$ by an angle of $\theta$, the new point is still $r$ units away from the origin, but now, the angle made by the line from the origin to $(x',y')$ and the positive $x$-axis increases by $\theta$. Hence, $(x',y') = (r\cos(\phi+\theta),r\sin(\phi+\theta))$.
Now, let's use the cosine and sine sum of angle formulas to write $x'$ and $y'$ in terms of $x$ and $y$:
Hence, the matrix which rotates a point by an angle of $\theta$ counterclockwise is $\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}$.
Remark: To add to what David C. Ullrich suggested, notice that the columns of this matrix $\begin{pmatrix}\cos\theta \\ \sin\theta \end{pmatrix}$ and $\begin{pmatrix}-\sin\theta\\ \cos\theta\end{pmatrix}$ are precisely the result of rotating the vectors $\begin{pmatrix}1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix}0 \\ 1 \end{pmatrix}$ counterclockwise about the origin by an angle of $\theta$. This same property holds for other linear transformations. Specifically, the $j$-th column of the matrix of a transformation is the result of applying the transformation to the $j$-th basis vector.
$\begingroup$Awesome answer. For rotating curves, it would be useful to note that $(x, y)$ can be thought of as the point obtained by rotating $(x', y') $ by *negative* $\theta$. Similar expressions could thus be obtained for x and y, which can be substituted into the original curve equation to obtain the rotated curve.$\endgroup$
Say $R_\theta(x,y)$ is $(x,y)$ rotated through an angle $\theta$. So $R_\theta$ is a map from $\Bbb R^2$ to itself.
First show that $R_\theta$ is linear. Showing that $R_\theta(cv)=cR_\theta(v)$ is more or less obvious. Showing that $R_\theta(v+w)=R_\theta(v)+R_\theta(w)$ uses the "definition" of the sum of two vectors in terms of those little parallelogram things. (If you rotate a parallelogram you get a parallelogram...)
So there is a matrix $M$ such that $R_\theta$ is given by multiplication by $M$. Now given a linear transformation $T:\Bbb R^2\to\Bbb R^2$ there's a formula somewhere in the book that says what the corresponding matrix is, in terms of $T((1,0))$ and $T((0,1))$. So you need to figure out $R_\theta(1,0)$ and $R_\theta(0,1)$. Plug them into that formula and out comes the matrix you want.
Consider an orthonormal basis ${\{v_1,v_2\}}$ in the plane. Say any object (e.g. a vector, shape etc.) in the plane is defined with respect to this basis - so if we can rotate the basis vectors through the angle $\theta$ anticlockwise, this will transform any such defined objects in the same way.
Firstly, note that the operation of rotation through $\theta$ about origin $O$ (let us call this operation $T$) is a linear transformation i.e. $T(\vec{v}+\vec{w})=T(\vec{v})+T(\vec{w})$ and $T(\alpha\vec{v})=\alpha T(\vec{v})$ for $\vec{v},\vec{w}\in\mathbb{R^2}, \alpha \in \mathbb{R}$. So we have linear $T:\mathbb{R^2}\rightarrow\mathbb{R^2}$. We then know that $T$ can be represented in matrix form as $(T(v_1),T(v_2))$ where $T(v_i)$, $i\in\{1,2\}$ is the transformed column vector. Then $T((x,y)^t)=xT(v_1)+yT(v_2), (x,y)\in\mathbb{R^2}$.
Any orthonormal basis ${\{v_1,v_2\}}$ corresponds exactly to the radial vector $\vec{e_r}=(cos\phi,sin\phi)$ and tangent vector $\vec{e_\perp}=(-sin\phi,cos\phi)$ of the unit circle with centre $O$ for some particular value of $\phi$. If we want to rotate this orthonormal pair of vectors through an angle of $\theta$, we perform $T(\vec{e_r})$ and $T(\vec{e_\perp})$. Well this is $T(\vec{e_r})=(cos(\phi+\theta),sin(\phi+\theta))$ and $T(\vec{e_\perp})=(-sin(\phi+\theta),cos(\phi+\theta))$. So we get,
Finally, note that when performing a rotation, we always consider the initial orthonormal basis via which we are working to be fixed at $\theta=0$ since this is our arbitrary way of navigating the plane and we do not consider the frame to be rotated to start with i.e. ${\{v_1,v_2\}}=\{\vec{i},\vec{j}\}$ (the natural/canonical basis), and this gives us our rotation matrix in the plane. Note that if we wish to rotate through $\theta$ in the clockwise direction instead (the non-canonical direction), all we have to do is set $\theta\rightarrow-\theta$ in $T$.
N.B. I have used $T$ to denote the rotation matrix here stemming from the fact that it is a linear transformation - but we usually use $R$ for rotation, $T$ for translation and $S$ for reflection (odd one out, originates from German verb strahlen). | 677.169 | 1 |
Elements of Geometry: Containing the First Six Books of Euclid: With a ...
Produce AB till it meet the circle in E, and draw DBF perpendicular to AE.
Then, because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arch AC to the arch AD, (33. 6.); and therefore also, the angle ABC is to four times the angle ABD as the arch AC to four times the arch AD (4. 5.);
But ABD is a right angle, and therefore four times the arch AD is equal to the whole circumference ACF; therefore, the angle ABC is
E
K
H
GA
to four right angles as the arch AC to the whole circumference ACF. COR. Equal angles at the centres of different circles stand on arches which have the same ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arches AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arch HG is to the whole circumference of the circle GHK in the same ratio. There fore, &c.
DEFINITIONS.
༥.
If two straight lines intersect one another in the centre of a circle, thé arch of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arch AC is the measure of the angle ABC.
II.
If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a minute; and if a Minute be divided into 60 equal parts, each of them is called a Second, and so on. And as many degrees, minutes, seconds, &c. as are in any arch, so many degrees, minutes, seconds, &c. are said to be in the angle measured by that arch.
COR. 1. Any arch is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the number of degrees and parts of a degree in the arch, which is the measure of that angle, is to 360.
COR. 2. Hence also, the arches which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arches has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem, 1.).
The degrees, minutes, seconds, &c. contained in any arch or angle, are usually written as in this example, 49°. 36'. 24". 42′′; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds.
III.
Two angles, which are together equal to two right angles, or two arches which are together equal to a semicircle, are called the Supplements of one another.
IV.
A straight line CD drawn through C, one of the extremities, of the
The segment DA of the diameter passing through A, one extremity of the arch AC, between the sine CD and the point A, is called the Versed sine of the arch AC, or of the angle ABC.
VI.
A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, which passes through C the other extremity, is called the Tangent of the arch AC, or of the angle ABC.
COR. The tangent of half a right angle is equal to the radius.
VII.
The straight line BE, between the centre and the extremity of the tangent AE is called the Secant of the arch AC, or of the angle ABC. COR. to Def. 4, 6, 7, the sine, tangent, and secant of any angle ABC, are likewise the sine, tangent, and secant of its supplement CBF.
It is manifest, from Def. 4, that CD is the sine of the angle CBF. Let CB be produced till it meet the circle again in I; and it is also manifest, that AE is the tangent, and BE the secant, of the angle ABI, or CBF, from Def. 6, 7.
COR. to Def. 4, 5, 6, 7. The sine versed sine, tangent, and secant of an arch, which is the measure of any given angle ABC, is to the sine, versed sine, tangent and secant, of
other arch which is the meaany sure of the same angle, as the radius of the first arch is to the radius of the second.
2
B
P
N
OMD
A
Let AC, MN be measures of the angle ABC, according to Def. 1.; CB the sine, DA the versed sine, AE the tangent, and BE the secant of the arch AC, according to Def. 4, 5, 6, 7; NO the sine, OM the versed sine, MP the tangent, and BP the secant of the arch MN, according to the same definitions. Since CD, NO, AE, MP are parallel,CD: NO :: rad. CB rad. NB, and AE : MP :: rad. AB : rad. BM, also BE: BP :: AB : BM; likewise because BC: BD :: BN : BO, that is, BA: BD :: BM: BO, by conversion and alternation, AD: MOAB: MB. Hence the corollary is manifest. And therefore, if tables be constructed, exhibiting in numbers the sines, tangents, secants, and versed sines of certain angles to a given radius, they will exhibit the ratios of the sines, tangents, &c. of the same angles to any radius whatsoever.
In such tables, which are called Trigonometrical Tables, the radius is either supposed 1, or some number in the series 10, 100, 1000, &c. The use and construction of these tables are about to be explained.
VIII.
The difference between any angle and a right angle, or between any
arch and a quadrant, is called
the Complement of that angle, or of that arch. Thus, if BH be perpendicular to AB, the angle CBH is the complement of the angle ABC, and the arch HC the complement of AC; also F the complement of the obtuse angle FBC is the angle HBC, its excess above a right angle; and the complement of the arch. FC is HC,
IX.
The sine, tangent, or secant of the complement of any angle is called the Cosine, Cotangent, or Cosecant of that angle. Thus, let CL or DB, which is equal to CL, be the sine of the angle CBH; HK the tangent, and BK the secant of the same angle; CL or BD is the cosine, HK the cotangent, and BK the cosecant of the angle ABC. COR. 1. The radius is a mean proportional between the tangent and the cotangent of any angle ABC; that is, tan. ABC Xcot. ABC = R2.
For, since HK, BA are parallel, the angles HKB, ABC are equal, and KHB, BAE are right angles; therefore the triangles BAE, KHB are similar, and therefore AE is to AB, as BH or BA to HK.
COR. 2. The radius is a mean proportional between the cosine and secant of any angle ABC; or
cos. ABC X sec. ABC=R®.
Since CD, AE are parallel, BD is to BC or BA, as BA to BE.
PROP. I.
In a right angled plané triangle, as the hypotenuse to either of the sides, so the radius to the sine of the angle opposite to that side: and as either of the sides is to the other side, so is the radius to the tangent of the angle opposite to that side.
Let ABC be a right angled plane triangle, of which BC is the bypotenuse. From the centre C, with any radius CD, describe the arch DE; draw DF at right angles to CE, and from E draw EG touching the circle in E, and meeting CB in G; DF is the sine, and EG the tangent of the arch DE, or of the angle C.
1B
The two triangles DFC, BAC are equiangular, because the angles, DFC, BAC are right angles, and the angle at C is common. Therefore, CB BA :: CD : DF; but CD is the radius, and DF the sine of the angle C, (Def. 4.); therefore CB: BA:: R : sin. C.
Also, because EG touches the circle in E, CEG is a right angle,
and therefore equal to the angle
G D
FE
:
A
BAC; and since the angle at C is common to the triangles CBA, CGE, these triangles are equiangular, wherefore CA: AB :: CE EG; but CE is the radius, and EG the tangent of the angle C; therefore, CA AB:: R: tan. C.
:
COR. 1. As the radius to the secant of the angle C, so the side adjacent to that angle to the hypotenuse. For CG is the secant of Ff
COR. 2. If the analogies in this proposition, and in the above corollary be arithmetically expressed, making the radius=1, they give
A
A
COR. 3. In every triangle, if a perpendicular be drawn, from any of the angles on the opposite side, the segments of that side are to one another as the tangents of the parts into which the opposite angle is divided by the perpendicular. For, if in the triangle ABC, AD be drawn perpendicular to the base BC, each of the triangles CAD, ABD being right angled, AD: DC: R: tan. CAD, and AD DB:: R: tan. DAB; therefore, ex æquo, DC: DB:: tan. CAD : tan. BAD.
SCHOLIUM.
B
D
The proposition, just demonstrated, is most easily remembered, by stating it thus: If in a right angled triangle the hypotenuse be made the radius, the sides become the sines of the opposite angles; and if one of the sides be made the radius, the other side becomes the tangent of the opposite angle, and the hypotenuse the secant of it.
PROP. II.
The sides of a plane triangle are to one another as the sines of the opposite angles.
From A any angle in the triangle ABC, let AD be drawn perpendicular to BC. And because the triangle ABD is right angled at D, AB : AD :: R : sin. B; and for the same reason, AC: AD: :R: sin. C, and inversely, AD: AC :: sin. CR; therefore, ex æquo inversely, AB: AC: sin. C: sin. B. In the same manner, it may be demonstrated, that AB: BC sin. C: sin. A. Therefore &c. Q. E. D. | 677.169 | 1 |
Hyperboloid Of Two Sheets: Definition, Properties, And Applications
Home » Mathematics » Hyperboloid Of Two Sheets: Definition, Properties, And Applications
Discover the , , and of the hyperboloid of two sheets. Explore its , graphical representation, surface area, volume, and real-life examples.
Definition and Properties of Hyperboloid of Two Sheets
The Hyperboloid of Two Sheets is a three-dimensional geometric shape that belongs to the family of quadric surfaces. It is defined as the set of points in space that satisfy a specific . This equation can be expressed as:
(x^2 / a^2) + (y^2 / b^2) – (z^2 / c^2) = 1
where a, b, and c are positive real numbers. The variables x, y, and z represent the coordinates of any point on the surface of the hyperboloid.
Definition of Hyperboloid of Two Sheets
The Hyperboloid of Two Sheets is a doubly ruled surface, meaning that it can be generated by two families of straight lines. These lines, known as ruling lines, intersect each other in a distinctive pattern, forming a hyperbolic shape. The hyperboloid has two separate sheets that curve away from each other in opposite directions.
Equation and Graphical Representation
The equation of the Hyperboloid of Two Sheets reveals its unique . The terms (x^2 / a^2) and (y^2 / b^2) represent the horizontal sections of the hyperboloid, while the term (z^2 / c^2) determines its vertical sections. The constant values a, b, and c determine the shape and size of the hyperboloid.
Graphically, the Hyperboloid of Two Sheets appears as two symmetrical, open curves that extend infinitely in both positive and negative directions along the x, y, and z axes. The intersection of the two sheets forms a hyperbola, giving the hyperboloid its characteristic shape.
Symmetry and Asymptotes
The Hyperboloid of Two Sheets exhibits certain symmetrical properties. It possesses a center of symmetry at the origin (0, 0, 0), which means that if a point (x, y, z) lies on the hyperboloid, the point (-x, -y, -z) will also lie on it. This symmetry is evident in the equation, as the terms involving x, y, and z are squared.
Additionally, the Hyperboloid of Two Sheets has asymptotes. These are imaginary lines that approach the surface of the hyperboloid infinitely but never actually touch it. Asymptotes help define the overall shape and orientation of the hyperboloid and can be found by examining the equation's coefficients.
Types and Examples of Hyperboloid of Two Sheets
The Hyperboloid of Two Sheets is a fascinating geometric shape that can be found in various real-life examples and constructed in different ways. Let's explore some of these types and examples in more detail:
Examples in Real Life
Have you ever wondered where you can find the Hyperboloid of Two Sheets in the world around you? While not as commonly encountered as some other shapes, it does appear in a few interesting places. Here are a couple of examples:
Cooling Towers: Next time you pass by a power plant or a chemical factory, take a look at the cooling towers. These towering structures often have a hyperboloid shape, with the purpose of dissipating excess heat into the air. The distinctive shape allows for efficient airflow and heat transfer, making it an ideal design for such industrial applications.
Sculptures and Art Installations: The Hyperboloid of Two Sheets has also caught the attention of artists and architects, who have incorporated its unique form into their creations. You may come across sculptures or art installations that feature this shape, showcasing the combination of mathematical elegance and artistic expression.
Different Types and Constructions
Did you know that there are different ways to construct a Hyperboloid of Two Sheets? Let's delve into some of these types and their constructions:
Cartesian Equation: One way to represent the Hyperboloid of Two Sheets is through its Cartesian equation. In three-dimensional Cartesian coordinates, the equation of a hyperboloid of two sheets is given by:
Revolution of a Hyperbola: Another method for constructing a Hyperboloid of Two Sheets is by revolving a hyperbola around its transverse axis. By rotating the hyperbola, we create a three-dimensional shape with two connected sheets that open up in opposite directions. This construction technique provides a geometric understanding of the shape's properties and symmetry.
Ruled Surface: The Hyperboloid of Two Sheets can also be formed as a ruled surface, meaning it can be created by moving a straight line along two intersecting curves. In this case, the curves are hyperbolas, and the straight lines connecting corresponding points on the hyperbolas generate the surface. This construction method highlights the smooth and continuous nature of the hyperboloid.
In summary, the Hyperboloid of Two Sheets can be found in various real-life examples, such as cooling towers and art installations. It can also be constructed through different methods, including its Cartesian , revolution of a hyperbola, and ruled surface construction. The versatility and aesthetic appeal of this geometric shape continue to captivate both mathematicians and artists alike.
Surface Area and Volume of Hyperboloid of Two Sheets
The surface area and volume of a hyperboloid of two sheets are important mathematical properties that help us understand and analyze this geometric shape. By calculating these measurements, we can gain insights into the shape's characteristics and apply them in various fields such as architecture, engineering, and physics.
Surface Area Formula and Calculation
To determine the surface area of a hyperboloid of two sheets, we need to use a specific formula. This formula takes into account the radii of the two sheets and the distance between them. Let's explore how we can calculate the surface area surface area:
The sinh^-1 symbol represents the inverse hyperbolic sine function.
Once you have the values for a, b, and c, substitute them into the formula and calculate the surface area.
Volume Formula and Calculation
The volume of a hyperboloid of two sheets can be determined using a specific formula that considers the radii of the two sheets and the distance between them. Let's explore how we can calculate the volume volume:
Once you have the values for a, b, and c, substitute them into the formula and calculate the volume.
By calculating the surface area and volume of a hyperboloid of two sheets, we can gain a better understanding of its physical dimensions and use this information in various . These measurements play a crucial role in architectural designs, engineering calculations, and even in the field of physics. The surface area provides insights into the amount of material required to construct or cover the hyperboloid, while the volume helps us determine its capacity or displacement.
Applications and Uses of Hyperboloid of Two Sheets
Architecture and Design
The hyperboloid of two sheets has found extensive in the field of architecture and design due to its unique shape and structural properties. Architects and designers often incorporate this geometric shape into their creations to add a visually striking and aesthetically pleasing element.
One of the notable architectural of the hyperboloid of two sheets is in the design of iconic structures such as the Shukhov Tower in Moscow. This lattice hyperboloid structure, designed by Vladimir Shukhov in the 1920s, showcases the remarkable strength and stability of the hyperboloid shape. It has become an architectural landmark and an inspiration for many designers.
The hyperboloid of two sheets offers architects and designers the ability to create structures with exceptional strength-to-weight ratios. Its unique geometric allow for the creation of large and spacious interiors while minimizing the amount of material required. This makes it an ideal choice for the construction of large-span structures such as stadiums, exhibition halls, and airport terminals.
Additionally, the hyperboloid shape lends itself well to the creation of visually captivating facades and sculptures. Its curved and flowing lines can be used to create dynamic and eye-catching designs. Architects often use the hyperboloid of two sheets to add a sense of movement and elegance to their buildings, enhancing their overall aesthetic appeal.
Engineering and Physics
The hyperboloid of two sheets also finds significant in the fields of engineering and physics. Its unique geometric and mathematical equations make it a valuable tool in various engineering and scientific calculations.
In engineering, the hyperboloid of two sheets is often used in structural analysis and design. Its inherent strength and stability make it suitable for supporting heavy loads and resisting external forces. Engineers can utilize the hyperboloid shape to design efficient and robust structures such as bridges, towers, and supports for large machinery.
In physics, the hyperboloid of two sheets plays a role in understanding electromagnetic fields and wave propagation. The shape of the hyperboloid is used to model the behavior of electric and magnetic fields in certain scenarios. This helps physicists and engineers analyze and predict the behavior of electromagnetic waves, which is crucial in the development of various technologies, including wireless communication systems and radar.
Furthermore, the hyperboloid of two sheets has applications in optics and acoustics. Its unique shape and mathematical allow for the manipulation and control of light and sound waves. This is utilized in the design of reflective surfaces, focusing devices, and acoustic resonators.
Overall, the hyperboloid of two sheets has proven to be a versatile and valuable shape in the fields of architecture, design, engineering, and physics. Its distinct properties and aesthetic appeal make it an attractive choice for creating visually stunning structures, while its strength and mathematical equations make it an essential tool in various engineering and scientific applications.
Similarities and Differences with Other Geometric Shapes
When exploring the world of geometric shapes, it is fascinating to uncover the similarities and differences between them. In this section, we will focus on the Hyperboloid of Two Sheets and its relationship with other shapes, namely the Ellipsoid and the Hyperboloid of One Sheet. By understanding these connections, we can gain a deeper appreciation for the unique and characteristics of the Hyperboloid of Two Sheets.
Comparison with Ellipsoid
The Hyperboloid of Two Sheets and the Ellipsoid are two geometric shapes that share some common features, yet they also have distinct differences. Let's delve into their similarities and contrasts to grasp a better understanding:
Similarities
Curved Surfaces: Both the Hyperboloid of Two Sheets and the Ellipsoid possess curved surfaces. These surfaces contribute to their aesthetic appeal and make them visually captivating.
Three-Dimensional Nature: Both shapes exist in three dimensions, which means they occupy space and have length, width, and height. This three-dimensionality allows for a more comprehensive representation of objects and structures.
Mathematical Representations: The Hyperboloid of Two Sheets and the Ellipsoid can be defined using mathematical equations. These equations help us visualize and analyze their and behavior.
Differences
Number of Sheets: The most noticeable distinction between the two shapes is the number of sheets they possess. The Hyperboloid of Two Sheets has, as the name suggests, two separate sheets that are connected at a central axis. On the other hand, the Ellipsoid consists of a single continuous surface with no separate sheets.
Symmetry: While both shapes exhibit certain symmetrical properties, the nature of their symmetry differs. The Hyperboloid of Two Sheets showcases a bilateral symmetry, meaning it can be divided into two equal halves that mirror each other. In contrast, the Ellipsoid typically exhibits a higher degree of symmetry, known as triaxial symmetry, where it can be divided into three equal sections.
Cross Sections: When examining the cross sections of the Hyperboloid of Two Sheets and the Ellipsoid, they reveal contrasting shapes. The cross sections of the Hyperboloid of Two Sheets are hyperbolic curves, while the cross sections of the Ellipsoid are elliptical curves.
Contrasting with Hyperboloid of One Sheet
Another intriguing comparison to make is between the Hyperboloid of Two Sheets and its counterpart, the Hyperboloid of One Sheet. While both share the commonality of being hyperbolic in shape, they also possess notable differences. Let's explore these disparities:
Similarities
Hyperbolic Shape: Both the Hyperboloid of Two Sheets and the Hyperboloid of One Sheet exhibit a hyperbolic shape. This means that their surfaces curve away from a central axis.
Mathematical Equations: Like the Hyperboloid of Two Sheets, the Hyperboloid of One Sheet can be mathematically defined using equations. These equations enable us to accurately represent and manipulate these shapes in various mathematical scenarios.
Differences
Number of Sheets: As the names suggest, the key dissimilarity between the two shapes lies in the number of sheets they possess. The Hyperboloid of Two Sheets has two separate sheets, while the Hyperboloid of One Sheet consists of a single continuous surface.
Asymptotic Behavior: The Hyperboloid of Two Sheets and the Hyperboloid of One Sheet exhibit different asymptotic behavior. The Hyperboloid of Two Sheets has two distinct sets of asymptotes, one for each sheet, while the Hyperboloid of One Sheet has only one set of asymptotes.
Applications and Uses: The Hyperboloid of Two Sheets and the Hyperboloid of One Sheet find different and uses. The Hyperboloid of Two Sheets is often employed in architecture and engineering, while the Hyperboloid of One Sheet is frequently utilized in physics and optics.
By understanding the similarities and differences between the Hyperboloid of Two Sheets and other geometric shapes, we can appreciate the unique properties and applications of each. These comparisons allow us to explore the vast world of geometry and its practical implications in various fields.
Historical Significance and Discoveries related to Hyperboloid of Two Sheets
The hyperboloid of two sheets is a fascinating geometric shape that has captured the attention of mathematicians and scientists throughout history. In this section, we will explore the historical significance of this shape, the pioneers and contributors who have made important discoveries, and the milestones that have been reached in understanding and utilizing the hyperboloid of two sheets.
Pioneers and Contributors
The study of the hyperboloid of two sheets can be traced back to the ancient Greeks, who were pioneers in the field of geometry. One of the earliest contributors to the understanding of this shape was Apollonius of Perga, a Greek mathematician who lived in the third century BCE. Apollonius made significant advancements in the study of conic sections, which includes the hyperboloid of two sheets.
Another important contributor to the understanding of the hyperboloid of two sheets was the German mathematician and astronomer Johannes Kepler. In the early 17th century, Kepler made groundbreaking discoveries in the field of optics and celestial mechanics. He recognized the hyperboloid of two sheets as a mathematical representation of certain types of double refraction, a phenomenon observed in crystals.
Historical Context and Milestones
The historical context in which the hyperboloid of two sheets emerged as a subject of study is closely linked to the development of geometry and its applications. During the Renaissance period, there was a renewed interest in mathematics and geometry, leading to significant advancements in the understanding of conic sections, including the hyperboloid of two sheets.
One milestone in the history of the hyperboloid of two sheets came in the 19th century when the Russian mathematician Nikolai Lobachevsky introduced non-Euclidean geometry. Lobachevsky's work challenged the long-held belief in Euclidean geometry and paved the way for new perspectives on geometric shapes, including the hyperboloid of two sheets.
In the 20th century, the hyperboloid of two sheets found applications in various fields, including architecture and engineering. Architects such as Vladimir Shukhov and Buckminster Fuller incorporated the hyperboloid of two sheets into their designs, creating iconic structures that showcased the elegance and versatility of this shape.
Overall, the historical significance of the hyperboloid of two sheets lies in its contribution to the development of geometry, its applications in various fields, and the insights it has provided into the nature of space and form. The pioneers and contributors who have studied and advanced our understanding of this shape have left a lasting impact on the field of mathematics and beyond | 677.169 | 1 |
NRICH Angles, Polygons and Geometrical Proof - Stage 4
This list supports teaching of angles, polygons and geometrical proof at Key Stage 4 in secondary mathematics. It provides investigations, problems and games from NRICH as well as classroom activities on the STEM Learning website that compliment them.
Here are some favourite activities selected by the NRICH team.
Cyclic Quadrilaterals This activity presents students with a circle marked with 9 equally spaced points on the edge and one in the centre. The challenge is to draw as many triangles as possible and calculate the angles.
Kite in a Square This activity involves finding the area of a kite that is constructed inside a square. Topics involved are coordinates, similar figures and Pythagoras' theorem.
Geometry: Construction, Proof This excel file presents a number of problems involving geometric proof. These begin with proofs requiring understanding of similar triangles and cyclic quadrilaterals. Students are also challenged to prove that the angle between tangent and chord is equal to the angle in the alternate segment. Later problems are much more challenging.
In each case the diagram is interactive and can help to illustrate the problem.
Book 8B has a section on Geometric Proof (pdf pages 239-248) which gives clear guidance on what is expected in a sufficient proof. Problems include the sum of interior angles of polygons and circle angles. Answers are given in this Kent Maths resource so that students can assess their own progress and seek guidance.
This investigation begins with the question whether a square fits better inside a circle or a circle within a square. More able students can progress to a similar problem with a hexagon and then other polygons. The teachers notes give an example of a possible solution.
Book 6b has a section dealing with circle angles, (from pdf page 87). Students are led through an investigation which includes proofs that the angle at the centre of a circle is twice the angle at the circumference. Answers to the questions are shown (from pdf page 91) and there are four further investigations (pdf page 90) involving circle angles. | 677.169 | 1 |
Exploring the World of Concave Vs Convex
I'm excited to dive into the fascinating world of concave vs convex shapes. In this article, we will explore the definition and characteristics of both types, as well as their applications in various fields.
By understanding the differences between concave and convex shapes, we can gain a deeper appreciation for their unique properties and how they are utilized in everyday objects.
Join me on this analytical journey as we unravel the intricacies of these captivating geometric forms.
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In the fascinating journey of understanding the principles of concave vs convex shapes, one can't help but ponder over the intriguing concept of "Exploring World's Curved Mechanics."
The Definition and Characteristics of Concave Shapes
Concave shapes are defined by their inward curvature and have characteristics such as a hollow or indented appearance. In the field of optics, concave lenses and mirrors play a crucial role.
Concave lenses are widely used in various applications, including correcting vision problems like nearsightedness. They are thinner at the center and thicker at the edges, causing light rays to diverge when passing through them.
On the other hand, concave mirrors have reflective surfaces that curve inward. They have unique properties such as forming virtual images that are upright and diminished in size. These properties make concave mirrors useful in applications like reflecting telescopes and headlights of vehicles for better illumination.
Understanding the properties and applications of concave lenses and mirrors allows us to harness their benefits effectively.
The Definition and Characteristics of Convex Shapes
A convex shape is defined as having all interior angles less than 180 degrees. Convex polygons possess several key properties that make them valuable in various applications within the field of geometry. These properties include being equilateral, equiangular, and having parallel sides. Additionally, convex shapes are symmetric across their center point and exhibit rotational symmetry.
To help illustrate the characteristics of convex polygons, consider the following table:
Property
Description
Equilateral
All sides are equal in length
Equiangular
All interior angles are equal in measure
Parallel Sides
Opposite sides are parallel to each other
Rotational Symmetry
The shape remains unchanged after rotation about its center
Some geometric examples of convex shapes include squares, rectangles, triangles (when they have acute angles), and regular polygons.
Understanding the properties and characteristics of convex shapes lays a foundation for exploring the applications of concave shapes in various fields such as architecture and computer graphics.
Exploring the Applications of Concave Shapes
One of the key applications for concave shapes is in the field of architecture, where they can be used to create unique and visually striking designs. In addition to their aesthetic appeal, concave shapes have practical uses in various industries. Here are some examples:
Concave mirrors: These mirrors play a crucial role in optics by reflecting light inward. They are commonly used in telescopes, microscopes, and cameras to focus and magnify images.
Concave lenses: These lenses are frequently employed in corrective eyewear such as glasses or contact lenses. By diverging light rays before they reach the eye, concave lenses help correct nearsightedness or myopia.
Sound amplification: The shape of certain musical instruments like trumpets and French horns incorporates concavity to enhance sound projection and clarity.
Solar energy concentration: Concave reflectors can collect sunlight over a large area and concentrate it onto a smaller surface, maximizing solar energy absorption for various applications.
Understanding the applications of concave shapes provides us with valuable insights into their versatile utility.
Now, let's delve into exploring the applications of convex shapes.
Exploring the Applications of Convex Shapes
Now let's take a look at the practical uses of convex shapes.
Convex shapes find numerous applications in various fields due to their unique properties and benefits.
One notable application is in lens design for optics. Convex lenses are commonly used to correct vision problems, such as nearsightedness or farsightedness, by bending light rays towards or away from the eye.
Another important use of convex shapes is in mirrors, where they allow for reflection and focusing of light. This is crucial in telescopes, microscopes, and other optical instruments.
Additionally, convex shapes are utilized in architectural designs to increase strength and stability. The curved surfaces distribute forces evenly across structures, making them more resistant to external pressures like wind or earthquakes.
Overall, the applications and benefits of convex shapes span from science and technology to everyday objects, demonstrating their importance and versatility in our lives.
Understanding the Differences Between Concave and Convex Shapes
Let's delve into the distinctions between concave and convex shapes. These two types of shapes play a crucial role in various fields, from design to optics. To help you understand their significance better, here are some key points:
Concave Shapes:
Advantages in Design: Concave shapes can create an illusion of depth and add a sense of elegance to designs.
Disadvantages in Design: They may limit functionality due to their inward curvature, making it challenging to accommodate certain elements.
Convex Shapes:
Role in Optical Devices: Convex lenses, for example, are widely used in optical devices like cameras and microscopes to converge light rays.
Understanding the differences between concave and convex shapes is essential for achieving precision and control in various applications. While concave shapes offer aesthetic appeal with potential functional limitations, convex shapes play a critical role by manipulating light for enhanced visual experiences.
Conclusion
In conclusion, the world of concave and convex shapes is a fascinating one. Understanding their definitions, characteristics, and applications can provide valuable insights into various fields such as mathematics, physics, architecture, and design.
The key difference between concave and convex shapes lies in their curvature and how they interact with light. Concave shapes are characterized by an inward curvature while convex shapes have an outward curvature. These distinctions play a crucial role in determining the behavior and practical uses of these shapes in different contexts.
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What is a convex hull in data structure?
For a given set of points S in a plane, the convex hull is a line completely enclosing all the points within the plane such that no concavities are left. In other words, the convex hull of a set of points is the smallest convex polygon consisting of all points within the plane.
Usage of the convex hull
The convex hull is widely used in mathematics, statistics, and computational geometry. Some applications of convex hull include:
With convex cars, detecting the paths to avoid collision is easier. In such cars, a convex hull is used to compute the routes.
The convex hull property is used to detect intersections of curves such as the Bezier curve, which lies in the convex hull of its control points.
In ethology, the convex hull is used to study animal behavior. It is used to estimate the home range of an animal depending on the points where the animal has been spotted.
Convex hull can be computed using various algorithms.
Jarvis' march (gift wrap) algorithm
Jarvis' march algorithm applies to two-dimensional cases. The algorithm has O(nh) time complexity, where n is the number of points and h is the number of points on the convex hull.
Algorithm
Assume three non-linear points, as shown in the figure above. The Jarvis' march algorithm can be changed based on linearity. The complete algorithm has to deal with degenerate cases when the convex hull includes 1 or 2 vertices and with the issues involved in computer computations and input data.
The algorithm begins with i = 0 and the point P0 is on the convex hull.
It selects point Pi + 1 such that all the other points are on the right side of the line Pi Pi + 1.
The point can be located in O(n) time by comparing the polar angles of all the points with respect to point Pi as a center of the polar coordinates.
In 2D, Jarvis' algorithm is like the process of winding a string around a set of points. This approach can be applied to higher dimensions.
Pseudocode for Jarvis' wrap algorithm
Input: Number of input points n, number of points on the hull h
Output: Corner points on the convex hull
Steps:
Algorithm jarvis(Q) // Q is the point set
pointOnHull = leftmost point in Q //Which will be part of the convex hull (Q)
k := 0 //Final set size is k.
repeat
P[k] := pointOnHull //P will be a set of points that will form the convex hull.
endpoint := Q[0] //initial endpoint for an edge on the convex hull
for m from 0 to |Q| do
// endpoint == pointOnHull will rarely occur. It can happen only when m == 1 and a better endpoint has not yet been set for the loop
if (endpoint == pointOnHull) or (Q[m] is on left of line from P[k] to endpoint) then
endpoint := Q[m] // found greater left turn, update endpoint
k := k + 1
pointOnHull = endpoint
until endpoint = P[0] // wrapped around to first hull point
Graham scan algorithm
Graham scan algorithm is used to determine the convex hull of a finite set of points in a plane. It has a time complexity of O(n log n). The algorithm searches all the vertices of the convex hull located on the boundary. This algorithm makes use of a stack to detect and remove concavities across the boundary. Graham's algorithm is also known as successive local repair.
Algorithm
Graham's algorithm begins with finding the point having the lowest y-coordinate. If such a coordinate exists in more than one point for a given set of points, then the point having the lowest x-coordinate is named P and the step is performed in O(n) time, where n is the number of points
Next, the set of points are sorted in increasing order of the angle. The point P and remaining points make an angle with the x-axis. This process can be done using any sorting algorithm like heap sort, having O(n log n) time complexity. (For the sorting, it is not required to compute the angle. Any angle which is monotonic in the interval [0,π ] can be used.)
The cosine can be calculated using the dot product or the slope of the line. If there is a numeric position, the comparison function can use the sign of cross product to find the angles in the sorting algorithm. After sorting all the points in the array, the algorithm proceeds with the next step.
Now, the algorithm checks if there are two or more points at the same angle.
If so, then all the angle points except the one farthest from P are removed to obtain a new array m. If m is less than 3, then a convex hull will not be possible.
Otherwise, points P, A, B, and C are pushed to stack S.
The process is repeated for the rest (m-3) points until the orientation of the 3 points changes counterclockwise or they take a left turn.
Finally, the process will return to the point where it began. The stack will now comprise the points on the convex hull in the counterclockwise order.
Pseudocode for Graham's scan algorithm
Input: Number of input points n, number of points on the hull h
Output: Corner points on the convex hull
P = number of points
points[P+1] = the array of points
swap points[1] with the point having the lowest y-coordinate
sort points by a polar angle with points[1]
points[0] = points[P] // These points[0] need to be a sentinel point that stops the loop.
let Q = 1 // Q will denote the number of points on the convex hull.
for i = 2 to P: // Find next valid point on convex hull.
while counterclockwise(points[Q-1], points[Q], points[k]) <= 0:
if Q > 1:
Q -= 1
continue
// All points are collinear
else if k == P:
break
else
k += 1
Q += 1 // Update Q and swap points[k] to the correct place.
swap points[Q] with points[k] // When Q and k are the same, the algorithm ends up in an infinite loop.
Other convex hull algorithms
Quick hull
Quick hull algorithm is similar to the quicksort algorithm. It has a time complexity of O(n log n), which can degenerate to O(n2) in the worst case.
Kirkpatrick-Seidel algorithm
This is the first optimal output-sensitive algorithm with a time complexity of O(n log h) where n represents the number of input points and h represents the number of points in the convex hull.
Chan's algorithm
Context and Applications
The convex hull topic is significant in all courses having data structure as a subject. This includes:
Bachelors in Computer Applications
Masters in Computer Applications
Bachelors of Science in Information Technology
Masters of Science in Information Technology
Practice Problems
Q1. What is the average-case time complexity of a quick hull algorithm?
O(N)
O(N log N)
O(2N)
O(log N)
Answer: Option b
Explanation: The average time complexity of the quick hull algorithm uses the divide and conquer technique. Hence, the time complexity, in this case, will be O(n log n).
Q2. What will be the time taken to identify n points lying in a convex quadrilateral?
O(N)
O(N log N)
O(2N)
O(log N)
Answer: Option a
Explanation: Computationally, the time required to determine the n points within a convex quadrilateral is O(n).
Q3. What is the Jarvis' march algorithm also called?
Gift wrapping
Extreme points mapping
Line segment linking
Non-convex region mapping
Answer: Option a
Explanation: The Jarvis' march algorithm is also known as the gift wrapping algorithm or Jarvis' gift wrapping algorithm.
Q4. Which of the following is an application area of Graham's scan algorithm?
Numerical problem
Computational geometry
Image processing
Vertex problem
Answer: Option b
Explanation: Graham's scan algorithm is used for determining the convex hull, which is a computational geometry concept. So, Graham's scan algorithm is an example of computational geometry algorithms.
Q5. To which of the following is the quick hull algorithm similar?
Quicksort
Merge sort
Linear sort
Shell sort
Answer: Option a
Explanation: As per the average and worst-case time complexities, the quick hull algorithm is similar to the quicksort algorithm in the data structure.
Common Mistakes
The algorithm and applications of the convex hull are different for different cases (finite points set, simple polygon). So, students need to consider the special cases while identifying the convex hull for different shapes.
When there are three collinear points, the algorithm might not work correctly.
Related Concepts
Orthogonal convex hull
Convexity
Convex hull in image processing
Other types of structures (relative convex hull, conical hull, and so on.
Want more help with your computer science homework?
We've got you covered with step-by-step solutions to millions of textbook problems, subject matter experts on standby 24/7 when you're stumped, and more. | 677.169 | 1 |
Lines and Angles Introduction
Geometry is composed of two ancient Greek words: Geo and Metron. Geo means Earth and Metron means measurement. Geometry is the branch of mathematics which deals with shape, size, position, spatial relationships and properties of different figures.
The entire geometry begins with a point. A point is a dimensionless entity which specifies the location or position. It is represented using a dot symbol and its length is zero. All the shapes that we see around us consist of infinite number of points. When a point moves in such a manner that its direction remains unaltered then a straight line is obtained. In other words, a one-dimensional collection of points extending infinitely in both the directions represents a line as shown below. A line never ends.
Any two points on a line can uniquely specify it. In the fig. 1 given above, a line passing through these two points A and B is denoted as \(\overleftrightarrow{AB}\). The arrows indicate that the line \(\overleftrightarrow{AB}\) is extending infinitely in both the directions.
A portion of a line consisting of two end points is known as a line segment. Fig. 1 given above represents a line segment \(\overline{AB}\) with A and B as two end points.
Ray is defined as a line segment with only one end point. Fig. 2 represents a ray with O as its end point and one side extending infinitely.
If a ray is rotated about its end point then the measure of its rotation between the final and initial position of the ray is known as an angle. In fig. 3, \(\overrightarrow{OB}\) is the initial position of the ray and when it is rotated about its end point i.e. O, the final position is represented by ray \(\overrightarrow{OA}\). The measure of this rotation is measured in angles. The angle between the initial and final position of a ray is measured as ∠AOB. | 677.169 | 1 |
Create a rectangle with A as a vertex. Remember it must stay a rectangle, no matter where the points move to. Construct the diagonals of your rectangle and find the length of each diagonal.
Construct a rhombus with A as a vertex. Remember it must stay a rhombus no matter how the points are moved. Find the lengths of the sides of the rhombus. Draw the diagonals of the rhombus and find the angles they create with the sides of the rhombus an
Create a square with A as a vertex. It must stay a square. Construct the diagonals and measure the angles they make with the sides and each other. Measure the lengths of the sides of the square and the lengths of the diagonals. | 677.169 | 1 |
ABCD is a square with side a. With centres A, B, C and D four circles are drawn such that each circle touches externally two of the remaining three circles. Let δ be the area of the region in the interior of the square and exterior of the circles. Then the maximum value of δ is :
A
a2(1−π)
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B
a2(4−π4)
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C
a2(π−1)
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D
πa24
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Open in App
Solution
The correct option is Ba2(4−π4) Let ABCD is a square with side a.
∴A(□ABCD)=a2
Let's draw four circles with centre A,B,C and D such that each circle touches two of the remaining circles externally.
As the circles drawn, their radius will be r=a/2 and one fourth part of a circle lie in the square. | 677.169 | 1 |
What is the ratio between the angle bisector and the angle?
When an angle bisector is drawn in a triangle, the ratio of the opposite sides forming the bisected angle is equal to the ratio of the segments formed by bisector intersecting the opposite side. This ratio applies to all types of triangles and for an angle bisector drawn from any angle.
Does incenter bisect angle?
PQR . The incenter is equidistant from the sides of the triangle.
What does an angle bisector divide?
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.
What is the formula of Incentre?
Incenter of a Triangle Properties
If I is the incenter of the triangle ABC, then ∠BAI = ∠CAI, ∠BCI = ∠ACI and ∠ABI = ∠CBI (using angle bisector theorem). The sides of the triangle are tangents to the circle, and thus, EI = FI = GI = r known as the inradii of the circle or radius of incircle.
Does angle bisector divides the opposite side?
The angle bisector of a triangle divides the opposite side into two parts proportional to the other two sides of the triangle. In a triangle, if the interior point is equidistant from the two sides of a triangle, then that point lies on the angle bisector of the angle formed by the two line segments.
What is Incentre in maths?
The point of intersection of angle bisectors of the 3 angles of triangle ABC is the incenter (denoted by I). The incircle (whose center is I) touches each side of the triangle.
How Incentre is formed?
The incentre is one of the triangle's points of concurrency formed by the intersection of the triangle's three angle bisectors. These three angle bisectors are always concurrent and always meet in the triangle's interior (unlike the orthocenter which may or may not intersect in the interior).
What is incenter Theorem?
The incenter theorem is a theorem stating that the incenter is equidistant from the angle bisectors' corresponding sides of the triangle. The angle bisectors of the triangle intersect at one point inside the triangle and this point is called the incenter.
Is Incentre the Centre of circle?
The incenter of a triangle is also acknowledged as the center of a triangle's circle as the largest circle could implement inside a triangle. The circle that is inscribed in a triangle is named the incircle of a triangle. The incenter is usually denoted by the letter I.
Is the incenter and centroid the same?
Incenters is created using the angles bisectors of the triangles. Orthocenter is created using the heights(altitudes) of the triangle. Centroid is created using the medians of the triangle.
Is centroid and Incentre same?
incenter I, the point of which is equidistant from the sides of the triangle; orthocenter H, the point at which all the altitudes of the triangle intersect; centroid G, the point of intersection of the medians of the triangle.
How do you find the angle of an incenter?
Video quote: And make a perpendicular to the side of the triangle. Those are called the in radii. And each in radius is the same length it's it's a radius of the circle.
How do you find the angle bisector?
An angle bisector divides an angle into two equal parts. So, to find where the angle bisector lays, divide the number of degrees in the angle by 2. . So, the angle bisector is at the 80-degree mark of the angle.
What is Incentre and Circumcentre?
The incentre of a circle is also the centre of the circle which touches all the sides of the triangle. Circumcentre and Circumcircle: The point of intersection of the perpendicular bisectors of the sides of a triangle ABC is called its circumcentre.
Is incenter same as circumcenter?
A circle inscribed inside a triangle is called the incenter, and has a center called the incenter. A circled drawn outside a triangle is called a circumcircle, and it's center is called the circumcenter.
What is the incenter of an equilateral triangle?
The incenter is the last triangle center we will be investigating. It is the point forming the origin of a circle inscribed inside the triangle. Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.
What is incentre of a equilateral triangle?
Internal angle (degrees) 60° In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
What segments determine the incenter of a triangle?
Finding the incenter
You find a triangle's incenter at the intersection of the triangle's three angle bisectors. This location gives the incenter an interesting property: The incenter is equally far away from the triangle's three sides.
How do you construct the incentre of a triangle?
Video quote: If we go ahead and draw a ray. Through this vertex. And through this point of intersection. That's going to be the angle bisector it's gonna be cutting that angle and a half.
What are the properties of the incenter?
Properties of the incenter
The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides. See Incircle of a | 677.169 | 1 |
A man is facing north He turns 45 degree in the clockwise direction and then another 180 degree in t...........
A man is facing north. He turns 45 degree in the clockwise direction and then another 180 degree in the same direction and then 45 degree in the anticlockwise direction. Find which direction he is facing now ?
Explanation:
The man firstly faces the direction OA. On moving 45 degree clockwise, he faces the direction OB. Now again he moved 180 degree clockwise, now he will be facing OC. From here he moved 45 degree anticlockwise, Finally he is facing OD, which is South direction. | 677.169 | 1 |
Trigonometric Ratios of Complementary Angles
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Trigonometric Ratios of Complementary Angles
In Mathematics, the complementary angles are the set of two angles such that their sum is equal to 90°. For example, 30° and 60° are complementary to each other as their sum is equal to 90°. In this article, let us discuss in detail about the complementary angles and the trigonometric ratios of complementary angles with examples in a detailed way.
Complementary Angles Definition
The two angles, say ∠X and ∠Y are complementary if,
∠X + ∠Y = 90°
In such a condition ∠X is known as the complement of ∠Y and vice-versa.
In a right angle triangle, as the measure of the right angle is fixed, the remaining two angles always form the complementary as the sum of angles in a triangle is equal to 180°.
Finding Trigonometric Ratios of Complementary Angles
Assume a triangle ∆ABC, which is right-angled at B.
∠A and ∠C form a complementary pair.
⇒ ∠A + ∠C = 90°
The relationship between the acute angle and the lengths of sides of a right-angle triangle is expressed by trigonometric ratios. For the given right angle triangle, the trigonometric ratios of ∠A is given as follows:
sin A = BC/AC
cos A = AB/AC
tan A =BC/AB
csc A = 1/sin A = AC/BC
sec A =1/cos A = AC/AB
cot A = 1/tan A = AB/BC
The trigonometric ratio of the complement of ∠A. It means that the ∠C can be given as 90° – ∠A
As ∠C = 90°- A (A is used for convenience instead of ∠A ), and the side opposite to 90° – A is AB and the side adjacent to the angle 90°- A is BC as shown in the figure given above.
Therefore,
sin (90°- A) = AB/AC
cos (90°- A) = BC/AC
tan (90°- A) = AB/BC
csc (90°- A) =1/sin (90°- A) = AC/AB
sec (90°- A) = 1/cos (90°- A) = AC/BC
cot (90°- A) = 1/tan (90°- A) = BC/AB
Comparing the above set of ratios with the ratios mentioned earlier, it can be seen that;
sin (90°- A) = cos A ; cos (90°- A) = sin A
tan (90°- A) = cot A; cot (90°- A) = tan A
sec (90°- A) = csc A; csc (90°- A) = sec A
These relations are valid for all the values of A that lies between 0° and 90°.
Summary:
Sin of an angle = Cos of its complementary angle
Cos of an angle = Sin of its complementary angle
Tan of an angle = Cot of its complementary angle
Trigonometric Ratios of Complementary Angles Examples
To have a better insight on trigonometric ratios of complementary angles consider the following example.
Example:
If A, B and C are the interior angles of a right-angle triangle, right-angled at B then find the value of A, given that tan 2A = cot(A – 30°) and 2A is an acute angle.
Solution:
Using the trigonometric ratio of complementary angles,
cot (90°- A) = tan A
From this ratio, we can write the above expression as:
⇒ tan 2A = cot (90°- 2A) ….(1)
Given expression is tan 2A = cot (A – 30°) …(2)
Now, equate the equation (1) and (2), we get
cot (90°- 2A) = cot (A – 30°)
⇒ 90°- 2A = A – 30°
⇒3A = 90° + 30°
⇒3A = 120°
⇒A = 120°/ 3
⇒ A = 40°
Thus, the measure of the acute angle A can be easily calculated by making use of trigonometry ratio of complementary angles | 677.169 | 1 |
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter ϵ2{\displaystyle \epsilon _{2}}, and whose vertical sections through the center are superellipses with the squareness parameter ϵ1{\displaystyle \epsilon _{1}}. It is a generalization of an ellipsoid, which is a special case when ϵ1=ϵ2=1{\displaystyle \epsilon _{1}=\epsilon _{2}=1}.[2]
Superellipsoids have an rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates.[6] It becomes an important geometric primitive widely used in computer vision,[6][5][7] robotics,[4] and physical simulation.[8] The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape.[6] Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available.[9] This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning.[4]
The parameters ϵ1{\displaystyle \epsilon _{1}} and ϵ2{\displaystyle \epsilon _{2}} are positive real numbers that control the squareness of the shape.
The surface of the superellipsoid is defined by the equation:
f(x,y,z)=1{\displaystyle fAny "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent 2/ϵ2{\displaystyle 2/\epsilon _{2}}, scaled by a=(1−z2ϵ1)ϵ12{\displaystyle a=(1-z^{\frac {2}{\epsilon _{1}}})^{\frac {\epsilon _{1}}{2}}}, which is
Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent 2/ϵ1{\displaystyle 2/\epsilon _{1}}, stretched horizontally by a factor w that depends on the sectioning plane. Namely, if x=ucosθ{\displaystyle x=u\cos \theta } and y=usinθ{\displaystyle y=u\sin \theta }, for a given θ{\displaystyle \theta }, then the section is
In particular, if ϵ2{\displaystyle \epsilon _{2}} is 1, the horizontal cross-sections are circles, and the horizontal stretching w{\displaystyle w} of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent 2/ϵ1{\displaystyle 2/\epsilon _{1}} around the vertical axis.
The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors ax{\displaystyle a_{x}}, ay{\displaystyle a_{y}}, az{\displaystyle a_{z}}, the semi-diameters of the resulting solid. The implicit function is [2]
Similarly, the surface of the superellipsoid is defined by the equation
F(x,y,z)=1{\displaystyle FTherefore, the implicit function is also called the inside-outside function of the superellipsoid.[2]
In computer vision and robotic applications, a superellipsoid with a general pose in the 3D Euclidean space is usually of more interest.[6][5]
For a given Euclidean transformation of the superellipsoid frame g=[R∈SO(3),t∈R3]∈SE(3){\displaystyle g=[\mathbf {R} \in SO(3),\mathbf {t} \in \mathbb {R} ^{3}]\in SE(3)} relative to the world frame, the implicit function of a general posed superellipsoid surface defined the world frame is[6]
F(g−1∘(x,y,z))=1{\displaystyle F\left(g^{-1}\circ (x,y,z)\right)=1}
where ∘{\displaystyle \circ } is the transformation operation that maps the point (x,y,z)∈R3{\displaystyle (x,y,z)\in \mathbb {R} ^{3}} in the world frame into the canonical superellipsoid frame.
Recoverying the superellipsoid (or superquadrics) representation from raw data (e.g., point cloud, mesh, images, and voxels) is an important task in computer vision,[11][7][6][5] robotics,[4] and physical simulation.[8]
Traditional computational methods model the problem as a least-square problem.[11] The goal is to find out the optimal set of superellipsoid parameters θ≐[ϵ1,ϵ2,ax,ay,az,g]{\displaystyle \theta \doteq [\epsilon _{1},\epsilon _{2},a_{x},a_{y},a_{z},g]} that minimize an objective function. Other than the shape parameters, g∈{\displaystyle g\in }SE(3) is the pose of the superellipsoid frame with respect to the world coordinate.
There are two commonly used objective functions.[12] The first one is constructed directly based on the implicit function[11]
The minimization of the objective function provides a recovered superellipsoid as close as possible to all the input points {(xi,yi,zi)∈R3,i=1,2,...,N}{\displaystyle \{(x_{i},y_{i},z_{i})\in \mathbb {R} ^{3},i=1,2,...,N\}}. At the mean time, the scalar value ax,ay,az{\displaystyle a_{x},a_{y},a_{z}} is positively proportional to the volume of the superellipsoid, and thus have the effect of minimizing the volume as well.
The other objective function tries to minimized the radial distance between the points and the superellipsoid. That is[13][12]
A probabilistic method called EMS is designed to deal with noise and outliers.[6] In this method, the superellipsoid recovery is reformulated as a maximum likelihood estimation problem, and an optimization method is proposed to avoid local minima utilizing geometric similarities of the superellipsoids.
The method is further extended by modeling with nonparametric bayesian techniques to recovery multiple superellipsoids simultaneously.[14] | 677.169 | 1 |
How to show that ABC is a right triangle
We know the perimeter of the triangle is 20cm, and we also know the lengths of two of the triangle's sides, AB = 7cm and BC = 4cm. By subtracting the known lengths from the perimeter, we can calculate the length of the third side (AC): 20 - 7 - 4 = 9 As a result, the length of AC is 9. If ABC were a right angle triangle, we could apply Pythagoras' Theorem to get the length of AC. Because it is the longest length, AC is the hypotenuse of the triangle. The Theorem of Pythagoras: a2 + b2 = c242 + 72 = 16 + 49 = 65 In the case where AC = c, c2 = 65 When c = 9, c2 = 81; nevertheless, when Pythagoras is used, c2 = 65. As a result, Pythagoras' Theorem is invalid, and the triangle ABC is not a right-angle triangle | 677.169 | 1 |
What Kind of Jobs Use Trigonometry?
Trigonometry is a branch of mathematics that arose from the observation that several triangles of various sizes (some small, some large) have the same ratio between their sides, which the mathematicians linked to the triangle's slope; if the slope of angles is equal, the ratio of their sides is equal. Trigonometry is derived from the Greek words 'trigon' and 'metron' and means 'measuring the sides of a triangle.'
After the study of the above observation linked to slope, it was used to establish a theory that can help determine the length of a triangle's side if the angle of inclination and length of any one of the triangle's sides are known. This was then used to calculate the distance between far-off objects such as the sun and the moon. The lengths of the sides of a right-angled triangle have fixed relationships, as do the angles between those sides, as discovered by 3rd-century astronomers: that is, if you know the length of one side and the value of one angle, you can algorithmically measure all the other angles and lengths.
Application of trigonometry:
Thus we can see the vast usage that trigonometry has due to its widespread applications in every field covering the sun and the moon too. Hence it becomes important to have the basics of trigonometry right from the start if the students want to have a firm career, as trigonometry is involved in almost every field. Hence to enhance their understanding of trigonometric principles, students should start practicing the trigonometry worksheets. These days plenty of them are available on several online platforms. Cuemath is one such trustable platform that focuses on enhancing the necessary math skills in kids.
Trigonometry is now used in a variety of fields, including electrical circuit design, music production, video game development, describing the state of an atom, predicting the heights of ocean tides, physical sciences, land surveying, map production, medical imaging (CAT scans and ultrasound), and many others. Trigonometry is used in almost every field of life, including astronomy, engineering, oceanography, physics, and chemistry.
Even the sea captains used it for navigation, surveyors used it to map out new lands, and engineers used it as well. Engineers use trigonometry and other mathematical relationships to discover extremely close relationships between dimensions and angles. When constructing large structures, it's important to remember that the forces acting on them must be able to balance in order for the structure to stay stable.
It's almost as if humans discovered the secret to unlocking the universe when trigonometry was born nearly 2000 years ago (some say 4000 years ago). Even in the absence of any sophisticated instrument capable of measuring faraway distances while standing firmly on earth, we were only using trigonometry.
Trigonometry and other mathematical relationships are used by engineers to find very similar relationships between dimensions and angles. To ensure stability and protection, architects calculate the exact angles of intersections for their structural components. Trigonometric theorems and analyses are used in arches, domes, support beams, and suspension bridges. Archaeologists, too, use trigonometry to excavate and dig at precise angles and points.
From observing the parabolic trajectory when a simple stone is thrown to designing the most complex motions when programming video games, trigonometry aids in moving objects by tracing their movements, trajectories that make them leap and run about, and even creating new characters. To do all of this, programmers use trigonometry formulas in their coding. Thus we see how trigonometry covers all the aspects of life in one way or the other. | 677.169 | 1 |
In geometry, a ray is a line-like figure that starts at some point - called the "endpoint" or "vertex" - and continues indefinitely in one direction. It is described in formal notation using 2 points with an arrow above it pointing in the direction of the ray, for example: AB→\overrightarrow{AB}AB.
The order of the endpoints describes the direction of the ray. It cannot be inversed.
AB→≠BA→\overrightarrow{AB} \neq \overrightarrow{BA}AB=BA.
As an example, this is the ray AB→\overrightarrow{AB}AB in Ascii art: A ---- B -> | 677.169 | 1 |
Seite 17 ... Prop . 12 . PROP . VI . THEOR . 6. 1 Eu . If two angles of a triangle be equal to one another , the sides also which subtend , or are opposite to the equal angles , shall be equal to one another ... Prop . 4 . In As DBC , ACB , PROP . VI .
Seite 18 ... PROP . VII . THEOR . If two triangles have three sides of the one re- spectively equal to the three sides of the other , each to each , the triangles are equal , and the angles are equal which are opposite to the equal sides . In As CBA ...
Beliebte Passagen
Seite 60Seite 63 ; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.
Seite 23Seite 39Seite 5 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another : XVI.
Seite 79 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle. | 677.169 | 1 |
Addition and Subtraction of Vectors
Addition and Subtraction of Vectors
Addition of Vectors
Figure 1, below, shows two vectors on a plane. To add the two vectors, translate one of the vectors so that the terminal point of one vector coincides with the starting point of the second vector and the sum is a vector whose starting point is the starting point of the first vector and the terminal point is the terminal point of the second vector as shown in figure 2.
Figure 1. - Vectors in 2 Dimensions.Figure 2. - Add 2 vectors in 2 Dimensions - Parallelogram.
When the components of the two vectors are known, the sum of two vectors is found by adding corresponding components.
Relationship Between Magnitude, Direction and Components of a Vector
The relationships between the magnitude \( |\vec A| \), the direction \( \theta \), which is the angle made by the vector and the positive direction of the x-axis, and the components \( A_x \) and \( A_y \) of vector \( \vec A \) are given by
\( A_x = |\vec A| \cos \theta \)
\( A_y = |\vec A| \sin \theta \)
\( |\vec A| = \sqrt {A_x^2 + A_y^2} \)
\( \tan \theta = \dfrac{A_y}{A_x} \)
NOTE that in detrmining \( \theta \), we need to take into account the signs of the components \( A_x \) and \( A_y \) as will be shown in example 3.
Figure 4. - Magnitude, Direction and Components of a Vector.
Example 3
The magnitudes of two vectors \( \vec U \) and \( \vec V \) are equal to 5 and 8 respectively. \( \vec U \) makes an angle of 20° with the positive direction of the x-axis and \( \vec V \) makes an angle of 80° with the positive direction of the x-axis. Both angles are measured counterclockwise. Find the magnitudes and directions of vectors \( \vec U + \vec V\) and \( \vec U - \vec V\). | 677.169 | 1 |
Vector Q3: Basic Results involving parallelogram
Timothy Gan
Basic Results involving parallelogram
The points $A$ , $B$, $C$, $D$ have position vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, $\mathbf{d}$ respectively relative to an origin $O$. If $P$ divides $AB$ in the ratio $1:2$ and $Q$ divides $CD$ in the ratio $1:2$, obtain an expression for the position vector of $X$, where $X$ is the mid-point of $PQ$. If $ABCD$ is a parallelogram, show that $X$ is the point in which diagonals $AC$ and $BD$ intersect. | 677.169 | 1 |
Problem
Solution 1 (Area Addition)
The line of symmetry divides the shaded figure into two congruent triangles, each with base and height
Therefore, the area of the shaded figure is
~MRENTHUSIASM ~Wilhelm Z
Solution 2 (Area Subtraction)
To find the area of the shaded figure, we subtract the area of the smaller triangle (base and height ) from the area of the larger triangle (base and height ):
~MRENTHUSIASM ~Steven Chen (
Solution 3 (Shoelace Theorem)
The consecutive vertices of the shaded figure are and By the Shoelace Theorem, the area is
~Taco12 ~I-AM-DA-KING
Solution 4 (Pick's Theorem)
We have lattice points in the interior and lattice points on the boundary. By Pick's Theorem, the area of the shaded figure is
~danprathab | 677.169 | 1 |
a clear definition of the adjacent faces.
The Anatomy of a Cube
To understand the number of edges a cube has, it is essential to familiarize ourselves with the different components of this geometric shape:
Faces: A cube has six faces, and each face is a square. All the faces of a cube are congruent, meaning they have the same size and shape.
Edges: A cube has twelve edges. An edge is a line segment where two faces of a cube meet. Each edge of a cube is shared by two faces.
Vertices: A cube has eight vertices. A vertex is a point where three edges of a cube intersect. Each vertex of a cube is shared by three faces.
Calculating the Number of Edges in a Cube
Now that we understand the components of a cube, let's determine the number of edges it possesses. As mentioned earlier, a cube has twelve edges. To visualize this, imagine a cube and count the number of line segments where two faces meet. Each of these line segments represents an edge.
Alternatively, we can calculate the number of edges in a cube using a formula. The formula to find the number of edges in any polyhedron is:
Number of Edges = (Number of Faces * Number of Edges per Face) / 2
Applying this formula to a cube, we have:
Number of Edges = (6 * 4) / 2 = 12
Therefore, a cube has twelve edges.
Properties of Cube Edges
Now that we know a cube has twelve edges, let's explore some interesting properties of these edges:
Equal Length: All the edges of a cube have the same length. This property is a consequence of the congruent faces of a cube.
Perpendicularity: The edges of a cube are perpendicular to the faces they connect. This means that each edge forms a right angle with the adjacent faces.
Connectivity: Each edge of a cube connects two vertices and two faces. This interconnectedness is a fundamental characteristic of a cube.
Real-World Examples
Cubes are not just abstract mathematical concepts; they have numerous real-world applications. Let's explore a few examples's edges allow for rotation, enabling players to scramble and solve the puzzle by aligning the colors on each face.
2. Dice
A traditional six-sided die, commonly used in board games and gambling, is essentially a cube. Each face of the die represents a number, and the edges allow for randomization when rolled.
3. Building Blocks
Cubes are often used as building blocks in construction and architecture. They provide a stable and uniform shape that can be easily stacked to create structures such as walls, pillars, and even entire buildings.
Q&A
Q1: Can a cube have curved edges?
No, a cube cannot have curved edges. By definition, a cube has straight edges that form right angles with the faces they connect. Curved edges would deviate from the characteristics of a cube.
Q2: How many diagonals does a cube have?
A cube has four space diagonals. A space diagonal is a line segment connecting two non-adjacent vertices of a polyhedron. In the case of a cube, there are four such diagonals that pass through the center of the cube.
Q3: What is the difference between an edge and a face in a cube?
An edge is a line segment where two faces of a cube meet. It is one-dimensional. On the other hand, a face is a two-dimensional surface that forms one side of the cube. A cube has six faces, each of which is a square.
Q4: Can a cube have more than twelve edges?
No, a cube cannot have more than twelve edges. The number of edges in a cube is fixed at twelve, regardless of its size or orientation.
Q5: How do the edges of a cube contribute to its stability?
The equal length and perpendicularity of the edges in a cube contribute to its stability. These properties ensure that the forces acting on the cube are evenly distributed, making it less prone to collapsing or deforming.
Summary
In conclusion, a cube has twelve edges, which are line segments where two faces of the cube meet. These edges are equal in length, perpendicular to the faces they connect, and contribute to the stability of the cube. Understanding the properties and characteristics of cube edges is essential in various fields, from mathematics and geometry to real-world applications such as puzzles, dice, and construction. The fixed number of edges in a cube, along with its other components, makes it a fascinating and versatile geometric shape. | 677.169 | 1 |
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УелЯдб 20 ... isosceles , and any two of them have their equal , each to each ; ... since ( hyp . ) BA = AF , and that the MAB , MBA are equal to the LFA , LAF , each to each , the side MA of the △ MAB = the side LA E. 26. 1. ) of the A LAF ; and in ...
УелЯдб 27 ... isosceles triangle to cut off a trapezium , which shall have the same base as the triangle , and shall have its three remaining sides equal to each other . Let ABC be the given isosceles △ It is re- ELEMENTS OF EUCLID . 27.
УелЯдб 28 Daniel Cresswell. Let ABC be the given isosceles △ It is re- E D B quired to cut off from it a trapezium , which , having BC for its base , shall have its three remaining sides equal to one another . Bisect ( E. 9. 1. ) the ABC by BD | 677.169 | 1 |
Translations Reflections And Rotations Worksheet Answers
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Web this product contains 10 task cards that require students to perform translations, reflections and rotations of triangles and. 1 name_____ graph the image of the figure using the transformation given. Web here are the pages that are included: Web also, trace the translation, reflection, and rotation for each shape. | 677.169 | 1 |
What is RECTANGLE
What is RECTANGLE?
A rectangle is a quadrilateral with equal opposite sides and four right angles, as defined by Euclidean Plane Geometry. In other words, a rectangle is a parallelogram with a right angle (90°) or an equiangular quadrilateral (it is a quadrilateral whose all angles are equal, i.e., 360°/4 = 90°). If the length of the four sides of the rectangle is equal then it is called a square. Sometimes, a rectangle that is not a square is called an oblong. The rectangle given below has four vertices named ABCD. So, it will be denoted as ABCD.
The word rectangle is derived from the Latin word ractangulus. It is made up of two words; rectus and angulus. Rectus means an adjective, right, or proper while angulus means an angle.
Characteristics
There are certain characteristics that are required in a convex quadrilateral to make it a rectangle. It can be possible if and only if, the quadrilateral follow any one of the following characteristics:
It is made up of four sides with four vertices.
The angle of each vertex is a right angle, i.e., 90°.
Its opposing sides are equal and, as a result, parallel.
A rectangle, in other terms, is a parallelogram with four right angles.
The triangles ABD and DCA are congruent in this parallelogram ABCD.
An equilateral quadrilateral is a rectangle.
This equilateral quadrilateral has two diagonals that always intersect each other at its center point.
Properties of Rectangle
1. Duality of Rectangle-Rhombus: When the midpoints of the sides of the rectangle are joined, the rhombus is formed. Therefore, the rhombus is the dual polygon of the rectangle. The difference between a rectangle and rhombus is given below:
Rectangle
Rhombus
It has equal angles.
It has equal alternate angles.
It has equal alternate side.
It has equal sides.
It has a circumcircle due to the equal distance between its center and vertices.
It has an incircle due to the equal distance between its center and sides.
The opposite sides are bisected by two axes of symmetry.
The opposite angles are bisected by two axes of symmetry.
The length of the diagonals is equal.
The angle at which diagonals intersect is equal.
2. Symmetry:
All corners of the rectangle lie on a single circle. Therefore, it is called cyclic.
As all corner angles are equal, i.e., right angle, therefore the rectangle is called equiangular.
All corners of the rectangle are placed within the same symmetry orbit. Therefore, it is called isogonal or vertex-transitive.
The reflection symmetry and rotational symmetry of order 2 apply to the two lines of the rectangle.
3. Others: Here are some other properties of the rectangle:
As we know the sides of the rectangle meet at the right angle, hence it is rectilinear.
Two rectangles are known as incomparable if none of them fit inside the other.
Classification
Traditional Hierarchy: A rectangle is a special case of the parallelogram which has two pairs of opposite sides. In the rectangle, these pairs of opposite sides are perpendicular to each other, hence form right angles. A parallelogram is a special case of trapezium whose two opposite sides are parallel to each other and have equal length. And a trapezium is a special case of a convex quadrilateral which has at least a single set of opposite sides parallel to each other. There are two qualities of a convex quadrilateral:
Start-shaped (as the whole interior can be seen from a single point and there is no need to cross any edge for this), and
Simple (as its boundary does not cross itself).
Hence, this traditional hierarchy can easily be understood from the given figure.
Alternative Hierarchy: In De Villiers' words, any quadrilateral which has access to symmetry via each pair of contrary sides is termed as a rectangle. This definition states both rectangles, i.e., right-angled and crossed. Each of them has a symmetrical parallel axis to and at an equal distance from a pair of opposite sides. Another axis is the perpendicular that bisects those sides. But in crossed rectangle, the conditions of the first axis are not similar because it does not follow a symmetrical pattern to either bisecting side. Quadrilaterals which have two symmetrical axes, each from a set of opposite sides, relate to the larger class of quadrilaterals with having a minimum of a single symmetrical axis running from a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia.
Formulae
If the rectangle has a length of l and width of w, the following formulas apply:
Rectangle Perimeter: The perimeter of a rectangle is the total distance covered by its outside boundary. The unit of the measurement of the perimeter is length, i.e., kilometer, meter, centimeter, etc. It can be measured by multiplying the addition of length and breadth of the rectangle. This can be understood as follows: Perimeter, P = 2 (Length × Breadth)
Area of Rectangle: The total region covered by the two-dimensional shapes, i.e., the length and breadth of the rectangle are known as its area. Its dimensions are measured in square units. As a result, the rectangle's area is defined as the area enclosed by its outer perimeter. It is calculated by multiplying the length and width together. The formula of the calculation is as under: Area, A = Length × Breadth
Diagonals of Rectangle: The two diagonals of the rectangle always intersect or cross each other. These diagonals are the same length. The rectangle is divided into two right-angle triangles by these two diagonals. Hence, the length of these diagonals can easily be found by using Pythagoras Theorem. In this theorem, the diagonals are used as the hypotenuse of the right triangle to find their length. This can be understood from the given formulae: D = (L2 + W2) Here, D refers to the hypotenuse of the triangle, L stands for the length which is the base of the triangle and W is width which is the perpendicular of the triangle.
Theorems
A parallelogram whose diagonals have equal length is known as a rectangle.
Any quadrilateral that has perpendicular diagonals from the midpoints of its sides forms a rectangle.
As per the isoperimetric theorem, the area of the square is the largest among all the rectangles with the same perimeter.
For each point P in the same plane of the rectangle with vertices A, B, C, and D, the British Flag Theorem states: (AP) 2 + (CP) 2 = (BP) 2 + (DP) 2
Types Of Rectangle
Crossed Rectangles: A crossed quadrilateral is a figure with two opposite sides of a non-self-intersecting quadrilateral that has two diagonals. A crossed rectangle, like a simple rectangle, has two opposite sides that are parallel to each other, as well as two diagonals. The vertex arrangement of the crossed rectangle is similar to the simple rectangle. This rectangle contains two identical triangles that share a common vertex at the meeting point of the diagonals. But as per the geometric intersection, this vertex is not to be considered. The best example of a crossed rectangle is the bow or butterfly. They are sometimes also termed angular eight. By twisting a 3D rectangular wireframe, the shape of a bow tie can be formed. The two angles of the crossed rectangle are reflexes and two are acute. Despite the fact that the figure is not equiangular, each pair of opposite angles is equal. This crossed quadrilateral's interior angles add up to 720°. Some of the common properties of a rectangle and a crossed rectangle are as under:
The length of opposite sides is equal.
The length of the two diagonals is equal.
This rectangle contains two lines of rotational symmetry and reflectional symmetry of order 2.
Spherical Rectangle: As per the spherical geometry, a figure that has four edges with great circle arcs is known as the spherical rectangle. At the point at which these arcs bisect each other, equal angles are formed which are greater than a right angle. The length of opposite arcs is always equal in a spherical rectangle.
Elliptic Rectangle: As per the elliptic geometry, a figure in the elliptic plane having four edges that are formed with the elliptic arcs is known as the elliptic rectangle. At the point at which these arcs bisect each other, equal angles are formed which are greater than a right angle. The length of opposite arcs is always equal in an elliptic rectangle. The simplest form of elliptic geometry is spherical geometry.
Hyperbolic Rectangle: As per the hyperbolic geometry, a figure in the hyperbolic plane having four edges that are formed with the hyperbolic arcs is known as the hyperbolic rectangle. The point at which these arcs bisect each other, the equal angles are formed which are less than a right angle. The length of opposite arcs is always equal in hyperbolic rectangle.
Some Objects With Rectangular Shape
There are many things that we see in our day to day life have rectangular shape. Such things include, television, tray, notebook, table, phone, newspaper, cricket pitch, CPU, wall, etc. | 677.169 | 1 |
Common Mistakes Made in ICSE Class 10 Board Exams Chapter Reflection
A chapter from the Co-ordinate Geometry unit – Reflection deals with coordinate axes, plotting coordinates and reflecting points on both the axes and Origin. In this blog, we will note down the mistakes made by students.
Reflection is the chapter from the unit Co-ordinate Geometry in the ICSE Class 10 syllabus. It is quite an easy chapter to attempt, still, mistakes are made by students in plotting graphs. To avoid major as well as minor mistakes in this chapter, let us try to understand the areas wherein mistakes are expected and try to overcome them.
Mistake in not using the SCALE provided in the question
Whenever the graph is to be plotted, the Scale of a graph is the most important thing. We can take the scale of the graph as per the points given for plotting. But, if the scale to be taken is already provided in the question, we are NOT supposed to change the scale at our convenience. We need to maintain the same scale given in question.
Question:
Use a graph paper for this question: (Take 2 cm = 1 unit on both axes).
Also, the scale provided in the question is often not read by students in a hurry, resulting in them settling on their scale.
It is a powerful suggestion to read the question properly and NOT MAKE MISTAKE in taking SCALE.
Mistakes in REFLECTING points on X-axis, Y-axis and Origin
Most often, it is asked to reflect the plotted points on the X-axis or Y-axis or at the Origin. And, we all know that there are certain rules for reflecting the points. Let us quickly check that!
Reflect point A(3, 5) in the X-axis, Y-axis and Origin and name the images as Aʹ, Aʹʹ and Aʹʹʹ, respectively.
Reflection of point A(3, 5) in the X-axis
Reflection of point A(3, 5) in the Y-axis
Here, students plot point A(3, 5) correctly but make mistakes in reflecting the point. While reflecting a point in the X-axis, instead of changing the sign of ordinate, students change the sign of abscissa. And the same case happens while reflecting a point in the Y-axis. Hence, the plotted points are incorrect.
Reflecting a point in the origin is most of the time done correctly.
Quite strange, but some students plot the points correctly, even reflect the points correctly in both the axes and origin, but forget to label the points they plotted on the graph. This may also lead to a reduction in marks.
Mistake in NAMING the FIGURE formed in a graph
After reflecting the points in both the axes and origin, sometimes it is asked to join the points and name the figure formed.
Let us see the figure formed in the below graph.
The correct name of the figure is Octagon. But looking at the figure, students give a variety of names such as hexagon, pentagon, star and many more. All these names are incorrect. PRACTICE QUESTIONS
Question 1: Points (4, 0) and (-3, 0) are invariant points under reflection in the line L1. Points (0, -2) and (0, 3) are invariant points on reflection in L2.
Name the lines L1 and L2.
Write down the images of point P(1, 4) and q(-3, -5) on reflection in L1. Name the images as Pʹ and Qʹ respectively.
Write down the images of point P(1, 4) and q(-3, -5) on reflection in L2. Name the images as Pʹʹ and Qʹʹ respectively.
State or describe a single transformation that maps Pʹ to Pʹʹ. Detailed solution can be viewed here.
Question 2: Plot ∆ABC whose vertices are A(3, 1), B(5, 0) and C(7, 4). On the same graph, draw the image ∆AʹBʹCʹ of ∆ABC under reflection in the line x = 2.
So, beginning from the scale of the graph till the naming of the figure, every step is crucial. Hence, when attempting questions on Reflection, take care of all the minor steps correctly to get full marks. These mistakes are based on CISCE Pupil Analysis done in a year. A huge amount of quality resources is available at TopperLearning for your understanding such as Revision Note explaining steps to plot the reflection of coordinates of points, Assertion and Reasoning Questions for practice and also Video explaining the graphs in detail. | 677.169 | 1 |
NCERT Solutions for Class 9 Maths Chapter 8 Quadrilaterals Ex 8.2 helps you understand the concepts given in the chapter. All our solutions are prepared according to the new patterned NCERT Book and syllabus. This exercise explains the mid-point theorem of triangles. Solving class 9 maths NCERT solutions exercise 8.2 will help you to understand the chapter conceptually and ultimately help to ascend marks in the exams.
Class 9 NCERT Maths Solutions Chapter 8 Exercise 8.2
You can also find the solutions to other exercises of the chapter from the links given below. | 677.169 | 1 |
Cross Product Calculator
A Cross Product Calculator is a mathematical tool used to compute the cross product of two vectors in three-dimensional space.
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Vector A
X
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Vector B
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Y
Z Cross Product Calculators: A Comprehensive Guide
In the vast realm of mathematics, the cross product holds a significant place, providing invaluable insights and solutions in various fields. As we delve into the depths of this mathematical concept, we will explore its significance, applications, and how a Cross Product Calculator can be a game-changer for students, professionals, and enthusiasts alike.
Understanding the Basics of Cross Product
Before we dive into the intricacies, let's establish a foundational understanding of what a cross product is. In vector algebra, the cross product is a mathematical operation that takes two vectors as input and produces another vector perpendicular to the plane of the input vectors. Symbolized as A × B, the cross product is a versatile tool used in physics, engineering, and computer science.
Cross Product Formula
The cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \), denoted as \( \mathbf{A} \times \mathbf{B} \), is given by:
How to Do Cross Product of Two Vectors?
The cross product of two vectors, \( \mathbf{A} \) and \( \mathbf{B} \), is a mathematical operation resulting in a third vector \( \mathbf{C} \), perpendicular to the plane formed by \( \mathbf{A} \) and \( \mathbf{B} \).
Cross Product Formula
The cross product \( \mathbf{C} \) is calculated using the following formula:
Dot product vs cross product
What is Dot Product:
The dot product, also known as the scalar product or inner produces a scalar. It is denoted by a dot (\(\cdot\)) between the vectors:
What is Cross Product:
The cross product, also known as the vector results in a vector. It is denoted by a cross (\(\times\)) between the vectors:
Characteristics of Cross Product:
Geometric Interpretation: The magnitude of the cross product is related to the area of the parallelogram formed by the vectors, and the direction is perpendicular to the plane of the vectors.
The dot product yields a scalar and is commutative, while the cross product results in a vector and is non-commutative. The dot product is useful for determining similarity or projection, while the cross product is employed in scenarios requiring determination of perpendicularity and calculation of areas or torques. Understanding these distinctions is crucial in various mathematical and physical applications.
Applications in Real-world Scenarios
Physics and Engineering
In physics, the cross product finds applications in mechanics, electromagnetism, and fluid dynamics. Engineers leverage it to calculate torque, magnetic forces, and fluid flow. A Cross Product Calculator becomes indispensable in swiftly and accurately solving complex problems in these disciplines.
Computer Graphics and 3D Modeling
The world of computer graphics relies heavily on cross products. From rendering realistic 3D scenes to calculating surface normals, cross products play a pivotal role. Professionals in the gaming and animation industry benefit from the efficiency of Cross Product Calculators in their day-to-day tasks.
Robotics and Automation
In robotics, understanding the orientation of objects is crucial. Cross products assist in determining the orientation of robotic arms and calculating the necessary movements for precise automation. This application showcases the practicality of cross products in cutting-edge technologies.
The Cross Product Calculator Advantage
Instantaneous Results
Calculating cross products manually can be time-consuming and prone to errors. With a Cross Product Calculator, you obtain instantaneous results, saving valuable time and ensuring accuracy in your computations.
User-friendly Interface
Cross Product Calculators often come with intuitive interfaces, making them accessible to users with varying levels of mathematical expertise. The user-friendly design ensures that students and professionals alike can harness the power of cross products effortlessly.
Versatility in Vectors
Whether you're dealing with 2D or 3D vectors, a Cross Product Calculator adapts to your specific needs. This versatility makes it an indispensable tool across different mathematical domains.
How to Use a Cross Product Calculator Effectively
To maximize the benefits of a Cross Product Calculator, follow these steps:
Input Vector Values
Provide the respective components of the vectors A and B. Ensure accuracy in input values to guarantee precise results.
Select the Dimensionality
Choose between 2D and 3D calculations based on your problem requirements. The calculator adapts to the dimensionality of your vectors.
Interpret Results
Understand the output vector's magnitude and direction in the context of your problem. Interpretation is key to applying the results effectively.
Conclusion
In conclusion, the cross product is a mathematical gem with applications spanning various disciplines. Embracing the efficiency of a Cross Product Calculator not only simplifies complex calculations but also enhances your overall understanding of vector algebra. Whether you're a student grappling with assignments or a professional navigating real-world problems, integrating this tool into your mathematical arsenal can undoubtedly give you a competitive edge. So, unlock the power of cross products, explore their applications, and let a Cross Product Calculator be your ally in conquering mathematical challenges.
References:
Frequently Asked Questions FAQ
What is a Cross Product Calculator?
A Cross Product Calculator is an online tool used to calculate the cross product of two vectors in three-dimensional space. It provides the resulting vector perpendicular to the input vectors, crucial in various physics, engineering, and 3D graphics applications.
What is the significance of the cross product in vector calculus?
The cross product is significant in vector calculus as it gives a vector that is orthogonal (perpendicular) to the original vectors. It is used to calculate forces, moments, torque, and surface normals in three-dimensional space.
Can the calculator handle vectors in different coordinate systems?
The Cross Product Calculator is designed for vectors in three-dimensional space, and it can handle vectors represented in Cartesian coordinates (i, j, k notation) or as separate components.
What if the input vectors are collinear or parallel?
If the input vectors are collinear or parallel, the cross product will be zero, indicating that the vectors have no unique orthogonal direction between them.
Can I use the calculator for non-orthogonal vectors in higher dimensions?
The Cross Product Calculator is specifically for vectors in three-dimensional space. For higher-dimensional vectors or non-orthogonal vectors, different mathematical techniques are required.
Is the calculator applicable to both row and column vectors?
Yes, the Cross Product Calculator can handle both row vectors and column vectors. It accepts the components of the vectors in the appropriate format.
Does the calculator account for vector unit magnitudes?
Yes, the Cross Product Calculator considers the magnitudes of the input vectors when computing the cross product, ensuring the correct length of the resulting vector.
Does the calculator support vectors with more than three components?
The Cross Product Calculator is limited to vectors in three-dimensional space (3D). For vectors with more than three components or higher dimensions, specialized software or methods are needed.
Is the cross product commutative, like addition or multiplication?
No, the cross product is not commutative. The order of the input vectors affects the direction of the resulting vector. Changing the order will yield a vector with the opposite direction | 677.169 | 1 |
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The cosine is a projection of the complex number exp(−ix) (which is a point on the unit circle in the complex plane) to the real axis on the complex plane. In the following interactive figure, you can drag the point x on the real axis and observe the behaviour of the complex number exp(−ix) and the varying value of cosine(x). | 677.169 | 1 |
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...is therefore g 9. 5. equal to the angle ACE b : but the angle AEC is equal to the out- h 5. 1. ward and opposite angle BAD : and the angle ACE is equal to the alternate angle CADb: wherefore also the angle BAD is equal to the angle CAD : therefore the angle BAC is cut into...
...angle AEC is therefore equal to the angle ACE h : But the' 5. i. angle AEC is equal to the outward and opposite angle BAD; and the angle ACE is equal to the alternate angle CAD b : Wherefore also the angle BAD is equal to the angle CAD ; Therefore the angle BAC is cut mto two... | 677.169 | 1 |
Pythagorean Theorem Proofs
There are more than 50 ways to prove the Pythagorean theorem. Using the Library, web resources, and other course materials, choose a proof of the theorem that you understand and describe it to the class. Then create a real-world application problem that can be solved by using the Pythagorean theorem. Make sure to include the question, calculations, and the solution.
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Solution Summary
One of the many ways of proving the Pythagorean Theorem is presented. Also included is a real-world application problem.
Solution Preview
(a) Proof of Pythagoras Theorem by the Area Method:
Statement: In a right triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides.
Given: A right triangle ABC, with hypotenuse length c and the other two legs measuring a171157 PythagoreanTheorem There are more than 50 ways to prove the Pythagoreantheorem. Using the Library, web resources, and other course materials, choose a proof of the theorem that you understand and describe it to the class.
After you have calculated the aproximate length using the Pythagoreantheorem, use a measuring device to measure the length of the diagonal of your monitor. Was your measurement close?
Typing hint: Type Pythagoreantheorem as a^2 + b^2 = c^2. | 677.169 | 1 |
This is an ideal example of a resource which could be used in planning sessions to highlight which approaches should be used for different ability students. There are also a number of activities which could be used as extension work. Ideal to link this resource to several places in the scheme of work highlighting which activities are to be used, how they should be used and highlight the opportunities to develop fluency, problem solving...
The text is a good source of straight forward questions useful for consolidation. Page 16 has questions requiring students to identify which triangles are right angled. Students should experience problems in context. Examples begin on page 12.
This activity is to be used in key stage four to reintroduce Pythagoras' Theorem. Opportunity to bring in many mathematical topics.
Begin by asking how we can calcuate the length of the vector (4¦3) Discuss the difference between calculate and measure. Explore the link between the area of a square and the length of a side of a square. Explore different methods for finding the area of a tilted square. Emphasise that a clear description of the method of finding the area of the square is what is required.
At this point begin the tilted square activity. The aim is to re-enforce that Pythagoras' Theorem is as much to about areas of squares as it is about right-angled triangles.
Extension (p. 153 of book, p 15 of the pdf) Q5 area of isosceles triangle, Q7 proof using Pythagors' theorem, Q8 Is Pythagors' Theorem still valid for non right-angles triangles?
Further exension (Page 155) Does Pythagoras Rule work with other shapes other than squares e.g semi-circles?
Ask students to attempt to proof Pythagoras' Theorem. Give them plenty of time ty a few different approaches. Students who succeed could be given the opportunity to make a brief video for upload on to Youe Tube.
For students who require some scaffolding, begin to play the video. Stop the video after 45 seconds.
Note there are many proofs of Pythagoras Theorem so do not lead students to this particular proof unless they require help. Once they are happy with this proof challenge them to fin d a different proof. | 677.169 | 1 |
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Select the correct answer from the drop-down menu.In the figure, mBm4C and 2D is a right angle.cos B...
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Q:
Select the correct answer from the drop-down menu.In the figure, mBm4C and 2D is a right angle.cos B=sin A>ResetNextsin DO
Accepted Solution
A:
Answer:sin (A)Explanation:In the given figure the "hidden" vertex of the larger triangle is A.Since, angle D is a right angle, both triangles BDC and CDE are right triangles (and they are also similar).Since angle B and angle C have the same measure and angle D measures 90° (right angle), angle A and angle C are complementary, i.e.∠ C + ∠ A = 90°For complementary angles, the sine of one acute angle equals the cosine of the other acute angle.So, the sine of angle C is equal to the cosine of angle B, i.e. cos (B) = sin (A). | 677.169 | 1 |
1. Coordinate Systems
c. Polar Coordinates - 2D
1. 2D Polar Coordinates
Rectangular coordinates \((x,y)\) are one way to specify a point,
\(P\), in the plane, but they are not the only way. When we are
studying circles, it is useful to use
Polar Coordinates \((r,\theta)\) which
identify the point, \(P\), using
the radius \(r\) and
the polar angle \(\theta\).
Like rectangular coordinates \(P=(x,y)\), the polar coordinates can
be written as an ordered pair, \(P=(r,\theta)\). So you need to be
careful to say whether this ordered pair is rectangular or polar.
Most often, the radius measures the (positive)
distance from the origin, \(O\), to the point, \(P\), and the
polar angle measures an angle (usually
measured in radians and usually with \(0 \le \theta \lt 2\pi\))
counterclockwise from the positive \(x\)-axis to the ray
\(\overrightarrow{OP}\).
However, frequently the angle \(\theta\) is allowed to be bigger than
\(2\pi\) or is allowed to be negative, in which case it is measured
clockwise from the posiitve \(x\)-axis. Consequently, \(\theta\) is
non-unique; we can always add or subtract an arbitrary multiple of
\(2\pi\) or \(360^\circ\). Thus the point with polar coordinates
\((3,{30^\circ})\) can also be written as \((3,{390^\circ})\), or
\((3,{750^\circ})\), or \((3,{-330^\circ})\), or \((3,{-690^\circ})\)
all of which are shown in the plot with
blue angles positive and
red angles negative.
Further, once in a while, we allow \(r\) to be negative, for example
in the context of solving equations or graphing. Then \(r\) is the
negative of the distance from \(O\) to \(P\). When \(r\) is negative,
the point \((r,\theta)\) is obtained by going a distance \(|r|\) along
the ray at the angle \(\theta\pm\pi=\theta\pm180^\circ\). Think of
this as going backwards along the ray at the angle \(\theta\).
Thus the point with polar coordinates \((-3,{30^\circ})\) is actually
the point \((3,{210^\circ})\) or \((3,-150^\circ)\).
Coordinate Curves
When you hold one of the coordinates fixed and let the other one vary, the point
\(P=(r,\theta)\) traces out a coordinate curve.
The curves are named by the coordinate which is changing.
When \(\theta\) is constant, you get a radial ray (assuming \(r \gt 0\))
called an \(r\)-curve, e.g. here
is the \(r\)-curve with \(\theta=\dfrac{\pi}{3}\):
\(r\)-curve: \(\theta=\dfrac{\pi}{3}\)
When \(r\) is constant, you get a circle called a
\(\theta\)-curve, e.g. here is
the \(\theta\)-curve with \(r=2\):
\(\theta\)-curve: \(r=2\)
When you draw several coordinate curves of each type you get a
coordinate grid. Here is a polar coordinate grid: | 677.169 | 1 |
Draw a line segment AB using the straightedge.
Place the compass point on point A and draw an arc that intersects line AB.
Without changing the compass width, place the compass point on point B and draw another arc that intersects the previous arc.
Label the intersection of the two arcs as point C.
Connect points A, B, and C to form the triangle ABC.
This method ensures that all three sides of the triangle are equal in length, creating an equilateral triangle.
2. Using Side Lengths and Angles
Another method of triangle construction involves using the lengths of the sides and the measures of the angles. Here's how:
Start by drawing a line segment AB, which will serve as the base of the triangle.
Using a protractor, measure the desired angle at point A and mark it.
Measure the length of the desired side opposite to the angle and mark it on the line segment AB.
Draw an arc with the marked length as the radius, centered at point A.
Draw another arc with the marked angle as the central angle, intersecting the previous arc.
Label the intersection of the two arcs as point C.
Connect points A, B, and C to complete the triangle ABC.
This method allows for the construction of various types of triangles, depending on the given side lengths and angles.
3. Using Trigonometric Ratios
Trigonometry provides yet another method for triangle construction. By utilizing trigonometric ratios, we can determine the lengths of the sides and the measures of the angles. Here's a simplified process:
Start by drawing a line segment AB, which will serve as the base of the triangle.
Using the given angle and side lengths, apply trigonometric ratios (such as sine, cosine, or tangent) to calculate the lengths of the other sides.
Mark the calculated lengths on the line segment AB.
Connect the marked points to complete the triangle ABC.
This method is particularly useful when dealing with real-world applications that involve angles and distances.
Applications of Triangle Construction
The ability to construct triangles accurately and efficiently has numerous practical applications across various fields. Let's explore some examples:
1. Architecture and Engineering
In architecture and engineering, triangles play a crucial role in structural stability. By constructing triangles within frameworks, architects and engineers can ensure the strength and rigidity of buildings, bridges, and other structures. Triangular trusses, for example, distribute forces evenly and minimize the risk of collapse.
2. Surveying and Mapping
Surveyors and cartographers rely on triangle construction to accurately measure and map land. By using the principles of triangulation, surveyors can determine distances, angles, and elevations. This information is essential for creating accurate topographic maps, establishing property boundaries, and planning infrastructure projects.
3. Art and Design
Artists and designers often incorporate triangles into their compositions for aesthetic and symbolic purposes. Triangles can convey stability, balance, and harmony. They can also create dynamic and visually engaging arrangements. From the pyramids of Egypt to the modernist paintings of Piet Mondrian, triangles have left an indelible mark on the world of art and design.
Conclusion
Triangles are not just simple geometric shapes; they are the building blocks of our physical and visual world. By understanding the principles and methods of triangle construction, we gain a deeper appreciation for their significance and versatility. Whether you are constructing a bridge, creating a work of art, or simply solving a math problem, triangles will always be there, guiding your way.
Q&A
1. What is the sum of the angles in a triangle?
The sum of the angles in a triangle is always 180 degrees.
2. Can you construct a triangle with three right angles?
No, it is not possible to construct a triangle with three right angles. The sum of the angles in a triangle must always be 180 degrees, and three right angles would add up to 270 degrees.
3. What is the difference between an equilateral and an isosceles triangle?
An equilateral triangle has all three sides and angles equal in length and measure, while an isosceles triangle has two sides and two angles equal in length and measure.
4. How does triangle construction relate to the Pythagorean theorem?
The Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two | 677.169 | 1 |
The Circumradius of a Triangle: Exploring its Definition, Properties, and Applications
Triangles are fundamental geometric shapes that have fascinated mathematicians and scientists for centuries. One important property of a triangle is its circumradius, which plays a significant role in various mathematical and real-world applications. In this article, we will delve into the concept of the circumradius of a triangle, exploring its definition, properties, and practical uses.
What is the Circumradius of a Triangle?
The circumradius of a triangle is defined as the radius of the circle that passes through all three vertices of the triangle. In other words, it is the distance between the center of the circumcircle and any of the triangle's vertices. The circumcircle is the circle that circumscribes the triangle, touching all three vertices.
To better understand the concept, let's consider an example. Suppose we have a triangle with vertices A, B, and C. The circumradius, denoted as R, is the distance between the center of the circumcircle and any of the vertices, such as the distance between the center and vertex A.
Properties of the Circumradius
The circumradius of a triangle possesses several interesting properties that are worth exploring:
1. Relationship with the Side Lengths
The circumradius of a triangle is related to its side lengths. In particular, for a triangle with side lengths a, b, and c, the circumradius R can be calculated using the formula:
R = (a * b * c) / (4 * Area)
where Area represents the area of the triangle. This formula provides a direct relationship between the circumradius and the side lengths of a triangle.
2. Relationship with the Inradius
The inradius of a triangle, denoted as r, is the radius of the circle that is tangent to all three sides of the triangle. Interestingly, there exists a relationship between the circumradius and the inradius of a triangle. This relationship is given by the formula:
R = (a * b * c) / (4 * Area) = (abc) / (4 * Area) = (abc) / (4 * rs)
where s represents the semiperimeter of the triangle, given by the formula:
s = (a + b + c) / 2
This relationship highlights the interplay between the circumradius and the inradius of a triangle.
3. Relationship with the Orthocenter
The orthocenter of a triangle is the point of intersection of its altitudes. Interestingly, the circumradius of a triangle is related to the distance between the orthocenter and any of the triangle's vertices. This relationship is given by the formula:
where h_a, h_b, and h_c represent the lengths of the altitudes of the triangle.
Applications of the Circumradius
The circumradius of a triangle finds applications in various fields, including mathematics, physics, and engineering. Let's explore some of its practical uses:
1. Triangle Classification
The circumradius of a triangle can be used to classify triangles into different types. For example, an equilateral triangle has all three sides equal, resulting in a circumradius that is the same as its inradius. On the other hand, an isosceles triangle has two sides of equal length, resulting in a circumradius that is different from its inradius. By analyzing the circumradius, we can gain insights into the properties and characteristics of different types of triangles.
2. Geometric Constructions
The circumradius of a triangle can be used in geometric constructions. For example, given three points in a plane, we can construct a triangle by connecting these points. By finding the circumradius of this triangle, we can determine the center and radius of the circumcircle. This information can be useful in various geometric constructions and calculations.
3. Trigonometric Calculations
The circumradius of a triangle is closely related to trigonometric functions. For example, in a right-angled triangle, the circumradius is equal to half the length of the hypotenuse. This relationship can be used to simplify trigonometric calculations and solve various problems involving right-angled triangles.
4. Mesh Generation
In computational geometry and finite element analysis, mesh generation plays a crucial role. The circumradius of a triangle is often used as a criterion for generating high-quality meshes. By ensuring that the circumradius of each triangle in the mesh satisfies certain criteria, we can create meshes that are well-suited for numerical simulations and analysis.
Summary
The circumradius of a triangle is a fundamental property that provides insights into the geometric and trigonometric characteristics of triangles. It is defined as the radius of the circle that passes through all three vertices of the triangle. The circumradius is related to the side lengths, inradius, and orthocenter of a triangle. It finds applications in triangle classification, geometric constructions, trigonometric calculations, and mesh generation. Understanding the circumradius of a triangle enhances our understanding of this fundamental geometric shape and its practical implications degenerate into a straight line or geometric meaning.
3. Can the circumradius of a triangle be greater than the sum of its side lengths?
No, the circumradius of a triangle cannot be greater than the sum of its side lengths. This violates the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the length of the third side. The circumradius is related to the side lengths of a triangle and cannot exceed their sum.
4. How is the circumradius of a triangle calculated in practice?
In practice, the circumradius of a triangle can be calculated using various methods. One common approach is to use the formula R = (a * b * c) / (4 * | 677.169 | 1 |
How do you calculate area using triangulation?
How do you calculate area using triangulation?
The sum of each Y coordinate multiplied by the difference between the two adjacent X coordinates is twice the area of any closed traverse. The result will be the same as the sum of each X coordinate multiplied by the difference between the two adjacent Y coordinates.
How do you solve a triangulation method?
The algorithm proceeds as follows :
Add a point to the triangulation.
Find all existing triangles whose circumcircle contains the new point (Figure 3-2).
Delete these triangles, which creates (always) a convex cavity.
Join the new point to all the vertices on the boundary of the cavity (Figure 3-2)
How to create a triangulation matrix in 3 D?
TR = triangulation (T,x,y,z) creates a 3-D triangulation representation with the point coordinates specified as column vectors x, y, and z. Triangulation connectivity list, specified as an m -by- n matrix, where m is the number of triangles or tetrahedra, and n is the number of vertices per triangle or tetrahedron.
How to calculate the area of an equilateral triangle?
To calculate the area of an equilateral triangle you only need to have the side given: area = a² * √3 / 4. Although we didn't make a separate calculator for the equilateral triangle area, you can quickly calculate it in this triangle area calculator.
What can you do with triangulation in math?
When your data is represented using triangulation, you can perform topological and geometric queries, which you can use to develop geometric algorithms. For example, you can find the triangles or tetrahedra attached to a vertex, those that share an edge, their circumcenters, and other features.
How is Gaussian elimination used in Matrix Triangulation?
It's also Gaussian elimination as it's a method of successive elimination of variables, when with the help of elementary transformations the equation systems is reduced to a row echelon (or triangular) form, in which all other variables are placed (starting from the last). | 677.169 | 1 |
Unlock all answers in this set
How can you best describe the outside of one of these objects using polygons?
The object has __ sides, so it is _____________.
It appears to be __________ because the sides and angles appear to be congruent.
answer
6, a hexagon
regular
question
What is the sum of the interior angle measures of a regular decagon?
What is the measure of each interior angle of a regular decagon?
answer
1440
144
question
Four interior angles of a pentagon measure 156°, 72°, 98°, and 87°. What is the measure of the final interior angle?
answer
127°
question
Which polygon has an interior angle sum of 1260°?
answer
C
question
An interior angle of a regular polygon has a measure of 108°. What type of polygon is it?
The polygon is
answer
a pentagon
question
What is the measure of angle ACB?
answer
73
question
What is the value of x?
answer
72
question
Which is a correct description of the polygon?
answer
It is a convex pentagon because it has five sides and none of the sides would extend into the inside of the polygon.
question
Three interior angles of a quadrilateral measure 55°, 117°, and 120°. What is the measure of the fourth interior angle | 677.169 | 1 |
New Elementary Geometry: With Practical Applications ; a Shorter Course Upon the Basis of the Larger Work
From inside the book
Page 23 ... hypothenuse and a side of the one equal to the hypothenuse and a side of the other , each to each , the triangles are equal . Let the two right - angled A triangles ABC , BOOK I. 23.
Page 24 ... hypothenuse A C equal to D F , and the side AB equal to DE ; then will the triangle A B C be equal to the triangle DEF . B G D CE F. The two triangles are evidently equal , if the sides B C and EF are equal ( Theo . XIII . ) . If it be ...
Page 78 ... hypothenuse diminished by the square of the other side ; thus , A B2 is equivalent to B C2 — A C2 . 197. Cor . 2. The square described on D the diagonal of a square is equivalent to double the square described on a side . For let ABCD | 677.169 | 1 |
Assessment: constructing and measuring
Use this two-page worksheet to assess students on classifying and identifying shapes according to their side length (scalene, isosceles or equilateral) and their angle properties (right, acute or obtuse). Each question has an allocated mark.
We acknowledge Aboriginal and Torres Strait Islander peoples as the Traditional Custodians of Country throughout Australia and pay our respects to Elders past and present. We recognise their continuing connection and contribution to lands, waters, communities and learning. | 677.169 | 1 |
The first book of Euclid's Elements, simplified, explained and illustrated ...
Wherefore a ▲ FCG has been described, which Om ABCD, and having the FCG =
gn. 4 E.-Q. E. F.
PROP. XLIII. THEOR.
m, of which
GEN. ENUN.-The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another. PART. ENUN.-Let ABCD be a the diam. is AC; EH, FG the ms about AC, i.e. through which AC passes; and BK, KD the other ms which make up the whole figure ABCD, and B
There are other cases of this Proposition; since it is not necessary that the ms about AC should be united at K, or that the complements should be ms. The Theorem may be equally proved, when thems EH, FG are not connected
Fig. 1.
A
H
D
To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
PART. ENUN.Let AB be the gn. st. line,
c the gn. A,
D the gn.
rectilineal; then it
is required to apply to
the st. line AB a□m = ▲ с, and having an
L = 4 D.
m BEFG
CONST. and DEMONST.-Make the = ▲ c, and having the EBG = ≤ D (Prop. XLII.), and let BE be in the same st. line with AB; through a draw AH || to BG or EF (Prop. XXXI.), and meeting FG produced in H; join HB.
2. And because the st. line нF falls upon the st. line AH, EF, .. the S AHF+ HFE = 2 rt.s (Prop. XXIX.), ... the BHF + HFE are 2 rt. s; HB, FE will meet if produced towards в and E. (Prop. XXIX. Cor.) 3. Let them meet in K; through K draw KL
B
..
to EA or FH (Prop. XXXI.), and meeting HA produced in L; also produce GB to M.
4. Then HLKF is am, of which the diam. is HK; AG, ME are the ms about HK, and LB, BF are the complements: ... the complement LB = complément BF (Prop. III.) = ▲ C. (Const.)
Δ
5. Also the ABM = vertical / EBG (Prop. XV.) D. (Const.)
=
Wherefore them LB has been applied to the gn. st. line AB, whieh = the gn. ▲ c, and having the ABM = gn. ≤ D.—Q. E. D. The converse Problem may be thus effected.
GEN. ENUN.-To a given straight line to apply a triangle, which shall be equal to a given parallelogram, and having an angle equal to a given rectilineal angle.
PART. ENUN.-Let AB be the gn. st. line, CDEF the gn. m, G the gn. rectilineal . Then it is required to apply to AB a ▲ having an ▲ = L G.
m
the ABKL
=
=
CDEF, and
CONST.-Draw the diam. CF; produce EF to H, making FH = EF, and join CH. To the st. line AB apply A CEH, and having the ABK = 4 G (Prop. XLIV.); produce BK to м, making км = BK, and join AM, AK; then ▲ ABM is E the ▲ required.
K
M
H
DMONST.-For the A ABM = 2 A ABK (Prop. NN.)
= BL (Prop. XLI.) (Prop. NN.) Wherefore to the
ABM, which
G.-Q. E. F.
gn.
=
A CEH (Const.)
CEFD. (Prop. XLI.)
gn. st. line AB has been applied a ▲ m ED, and has the ABM = gn.
PROP. XLV. PROB.
GEN. ENUN.-To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. PART. ENUN.-Let ABCD be the gn. rectilineal figure, E the
gn. rectilineal; then
it is required to describe a Om ABCD, and having an L = LE.
A
F
E
D
▲ DBC, and having ≤ GHM = ≤ E (Prop. XLIV.); then the figure FKML is the m required.
7. Now, because KF is || to HG, and HG to ML, .. KF is || to ML (Prop. XXX.); and Kм is also to FL; .. KFLM is a m.
8. Hence the figure ABCD
= ▲ ADB +
FH + om HL (Const.)
=
Wherefore a □m has been described = to the gn. figure ABCD, and having the FKM =gn. E.-Q. E. F.
an
COR. From this it is manifest how to a given st. line to apply a m, which shall have = a gn. rectilineal ▲, and be = a_gn. rectilineal figure: viz. by applying to the st. line am to the first ▲ ABD, and having = the gn. 4.
an
It is also manifest how to proceed, whatever be the number of sides to the gn. rectilineal figure. For it may be divided into as many ▲3, except two, as the figure has sides; and am to each of them in succession is to be applied to the opposite side of each successivem. | 677.169 | 1 |
PHP point-in-polygon algorithm
The point-in-polygon algorithm allows you to programmatically check if a particular point is inside a polygon or outside of it. A common way to tackle the problem is to count how many times a line drawn from the point (in any direction) intersects with the polygon boundary. If the line and the polygon intersect an even number of times (or not at all), then the point is outside. If they intersect an odd number of times, the point is inside. It is true even for complex forms that have a lot of coordinates and thus create a very precise boundary.
Let's see a sample image before we get to the code.
Here, the lines drawn from point 1 intersect twice or not at all, because it is outside.
Point 2 is inside and thus the lines drawn from it intersect once or three times.
Even in special cases, such as point 3, we see that this method works: the line intersects twice and the point is therefore outside.
I use this approach in the PHP code below, which returns one of these 4 possible values:
inside if the point is inside the polygon.
outside if, you guessed it, the point is outside of the polygon.
vertex if the point sits exactly on a vertex AND $pointOnVertex = true (line 2)
boundary if the point sits on the boundary. If $pointOnVertex = false, then boundary is also returned if the point is on a vertex.
Using the point-in-polygon PHP code
Set the point(s) value(s) and an array containing your polygon vertices (in the form "Xcoordinate Ycoordinate"), then call the pointInPolygon function. The first and last polygon coordinates must be identical, to "close the loop".
As you can see in the following example, it is easy to check multiple points at once. The code also works with negative coordinates, for the polygon as well as for the points to check.
Thanks Paco!
Yes, no problem with decimals, as I wrote it initially for precise GPS coordinates. [This was mentioned in comments on our previous website, but we are starting again with a clean slate 😉 ]
Feel free to let me know if you have any questions and I'll do my best to help you,
Michael
This script has helped me a lot! So far it has been perfect and matches 100% with the function results from turf.js. The difference being, that now I can do the logic on the servers side with PHP rather than on the client side with JavaScript. Awesome!!! Very helpful.
I think the first coordinate set in a WKT or GEOJSON string is the outer or main polygon. And any subsequent coordinate sets are all the holes? That being the case, I think I just have to go through all the hole coordinate sets and if a point is inside any one of the holes, then it is not inside the
polygon.
Thank you very much for your kind comment! I'm very happy to know it helps you.
That's right, with a hole you have to run the script for both the outer polygon and hole. If it's inside the outer polygon, but not the hole, it's inside the filled part. If the script returns inside for both, then it's inside the hole.
In that case, you could of course run the script for the hole first. If it's inside the hole, no need to run it again for the rest of the polygon.
Finally, thank you so much for your donation! Very much appreciated 🙂 | 677.169 | 1 |
wherefore CA , AB , BC are equal to one another ; and the triangle ABC is therefore ( I. def . 17. ) equilateral , and it is described upon the given straight line AB : which was required to be done . Scholium . If straight lines be ...
УелЯдб 9 ... wherefore AF and BC are each of them equal to BE ; and things that are equal to the same are equal ( I. ax . 1. ) to one another ; therefore the straight line AF is equal to BC . Wherefore from the given point A a straight line AF has ...
УелЯдб 11 ... wherefore the base BC shall coincide with the base EF ; because , the point B coinciding with E , and C with F , if BC did not coincide with EF , two straight lines would enclose a space , which ( I. def . 3. cor . ) is impossible ...
УелЯдб 12 ... wherefore ( I. 4. part 3. ) their remaining angles are equal , each to each , to which the equal sides are opposite ; therefore the angle FBC is equal to the angle GCB , and ( I. 4. schol . ) the angle ABC is equal to the angle ACB ...
УелЯдб 14 ... Wherefore , if two angles , & c . Cor . Hence every equiangular triangle is also equilateral . PROP . VII . THEOR . * C UPON the same base , and on the same side of it , there cannot be two triangles that have their sides which | 677.169 | 1 |
Complementary and Supplementary Angles Worksheets
Make use of this array of free printable complementary and supplementary angles worksheets to gauge your knowledge of the concepts. It incorporates exercises like identifying complementary and supplementary angles, finding their measures by applying appropriate properties, recognizing their presence in vertical angles, linear pairs and right angles, and more.
This set of printable worksheets are tailor-made for the students of grade 7 and grade 8.
Let these pdf worksheets help your ability to find angles in complementary and supplementary pairs flourish. Find the complement or supplement of the indicated angles, or find the measures of both the angles in a pair.
Breeze through our complementary and supplementary pdfs and learn to find the measures of the angles. Set up an equation using the given expressions and find the unknown angles. More than two steps are required to solve for x. | 677.169 | 1 |
ellipse()
Category: Shape
Subcategory: 2D Primitives
Complexity: Beginner
Description
Draws an ellipse (oval) to the screen. An ellipse with equal width and height is a circle. By default, the first two parameters set the location, and the third and fourth parameters set the shape's width and height. The origin may be changed with the ellipseMode() function. | 677.169 | 1 |
Introduction:
You will be given a number of statements. You center of a circle that circumscribes a triangle is inside the triangle.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
2. An altitude subdivides a triangle into two similar triangles.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
3. (a + b)2 = a2 + b2
Is this always, sometimes or never true? ......................................................
Reasons or examples:
4. 3x2 = (3x)2
Is this always, sometimes or never true? ......................................................
Reasons or examples:
5. A shape with a finite area has a finite perimeter.
Is this always, sometimes or never true? ......................................................
Reasons or examples:
6. A shape with a finite perimeter has a finite area.
Is this always, sometimes or never true? ...................................................... | 677.169 | 1 |
User Forum
State 'T' for true and 'F' for false.
(1) In a triangle, the measure of exterior angle is equal to the sum of the measure of interior opposite angles.
(2) The sum of the measures of the three angles of a triangle is 90°.
(3) A perpendicular is always at 90° to a given line or surface. | 677.169 | 1 |
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