For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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0answers
17 views

find out the number of students as required

in a survey of 900 student in a school, it was found that 600 students liked tea, 500 liked coeffee and 125 did not liked both the drinks. find the numberb of students who didnt liked tea only.
0
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0answers
16 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
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0answers
41 views

Geometry: Math behind “vanishing points”?

There is a problem in my geometry book which tells me to follow certain steps to find the dimensions of the back of a house. I know how to follow the steps but I can't comprehend HOW the two ...
1
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1answer
518 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
14
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4answers
191 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
-1
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1answer
27 views

Problem on a parabolic girder of a railway bridge.

The girder of a railway bridge is a parabola with its vertex at the highest point, 10 m above the ends.The span is 100 m, find the height at 20 m from the midpoint. I tried to draw a rough sketch but ...
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2answers
16 views

Function that graphs repeating upper halves of circles

I'm trying to write a periodic function that repeats the upper half of a unit circle, so it would look similar to $|\cos(x)|$ but with the upper half of a circle instead. If anyone could help me out ...
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1answer
20 views

How you could you change the surface area formula for a cylinder to calculate the curved surface area of the half pipe?

Skateboarders use half pipes for doing tricks. A half-pipe is a half cylinder. Explain how you could manipulate the surface area formula for a cylinder to calculate the curved surface area of the ...
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0answers
16 views

What is the total volume of wood used for the model?

A person makes a model of a house in construction class. The block of wood for the base measures 6 inches by 4 inches, and is 4 inches tall. He used a triangular prism for the roof, whose ...
1
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1answer
41 views

Proving a relation between inradius ,circumradius and exradii in a triangle

Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof ...
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2answers
23 views

Straight line is tangent to the curve.

The straight line $y=mx+1$ is tangent to the curve $x^2+y^2-2x+4y=0$. Find the possible values of $m$. My attempt, Substitute the $y=mx+1$ into the equation $x^2+y^2-2x+4y=0$. ...
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1answer
80 views
+50

What are the negative-dimentional n-sphere and n-cube?

The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres. $$\begin{array}{ll} S_{n-1}(R) &= ...
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2answers
30 views

Get the Equation of a Plane from a Vertex and 2 Angles? [on hold]

What is the simplest way to algebraically get the equation of a Plane (ax + by + cz = d), if you only have 1 point on the plane, and 2 angles (horizontal and vertical) which define the direction the ...
5
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1answer
42 views

An algorithm for filling a moving truck

I was recently helping a friend move. I stood in the moving truck as other people brought boxes and furniture pieces from inside the house. My job was to arrange these items in an efficient way inside ...
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1answer
14 views

Finite hyperbolic geometry with ideal points

I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 ...
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0answers
9 views

Change from one cartesian co-ordinate system to another by translation and rotation.

There are two reasons for me to ask this question: I want to know if my understanding on this issue is correct. To clarify a doubt I have. I want to change the co-ordinate system of a set of ...
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0answers
17 views

Get actual object size in mm using camera

How to get actual object size in mm when we know Object distance, focal length, image width in pixels and camera angle,
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0answers
22 views

Expected value (mean) of function from polyline

Suppose we have a polyline that has such properties: It consists of n segments First segment's ends are (0, 0) and ...
0
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5answers
49 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
3
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1answer
45 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
0
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0answers
38 views

Proving perpendicular with similarities in triangles [on hold]

$\triangle ABC$ is given . In the line through the vertex $A$ and perpendicular to side $BC$, two points $A_1$ and $A_2$ are taken so that $AA_2 = AA_1 = BC$ ($A_1$ is closer the line $BC$ than ...
2
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1answer
312 views

2D calculate position of a point relative to 4 known points

I have 4 known points (a square) in 2D space: A: {x:0, y:0} B: {x:100, y:0} C: {x:100, y:100} D: {x:0, y:100} Then I have a point inside the square. I don't know its location, but I do know the ...
2
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4answers
133 views

Find a point so that the triangle is equilateral

We have O(0,0), A(3,4) and B(x,y). Find $x,y\in{R}$ so that the OAB triangle is equilateral. I tried using the fact that the median is also the altitude(height) of the equilateral triangle. I ...
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2answers
32 views

I'm searching cutting points of a central angle of a circle

I'm working in a 2D drawing software and I need to draw points of the central angle cutting the circumference. I have a circumference, I know how its center point (2D coordinates), I know the ...
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0answers
13 views

Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
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1answer
19 views

Poritive orthant and positive functional

Let $A$ be a hyperplane of $\mathbb{R}^n$, and denote by $\mathbb{R}_+^n$ the positive orthant, i.e. $$ \mathbb{R}_+^n = \{ v \in \mathbb{R}^n \;\mid\; v_i\geq0 \quad \forall i = 1 \dots n \} $$ ...
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votes
3answers
25 views

Algorithm for intersection of n circles with approximate values

I'm trying to come up with a sort of trilateration algorithm that, given n >= 3 circles, finds the point of intersection. The radii come from samplings of electromagnetic magnitudes, therefore there ...
2
votes
1answer
132 views

Second variation formula and Jacobi fields

Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation of a geodesic $\gamma$. For $i=1,2$ we define $$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$ the ...
10
votes
2answers
87 views

Finding $\lim_{x\to 0} \frac {2\sin x-\sin 2x}{x-\sin x}$ geometrically

While looking at this question, I noticed an interesting geometric interpretation of the limit the OP was trying to evaluate. His limit came to twice the value of the limit $$\lim_{x\to 0}\frac{\sin ...
3
votes
2answers
97 views

How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: $$|(\mathbf p-\mathbf r)-(\mathbf q+\mathbf r)|=d$$ Where $\mathbf p$ and $\mathbf q$ are known ...
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1answer
45 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
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votes
3answers
70 views

Prove that $\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta)$ when $\theta+\beta = 90^\circ$ [on hold]

If $\theta, \beta$ are two acute angles prove that : $$\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta) $$ when $\theta, \beta$ are complementary angles, i.e. $\theta + \beta = 90°$. My try... ...
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1answer
31 views

Given only angles and area of triangle, find side length. [on hold]

The area of a triangle is $60$ square inches. Find the length of the side included between $A = 25°$ and $C = 110°$. (Round your answer to one decimal place.)
2
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1answer
14 views

Volume of the symmetric difference between a parallelotope and its translated.

Let $A$ be a n-dimensional parallelotope and $v \in \mathbb{R}^n$ a vector. Is there a formula giving the volume of the symmetric difference $A \Delta (v+A)$?
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1answer
36 views

Peripendicular Line at distance d from point in a given direction

I have a line given by $Ax + By + C= 0$, and a point $x_0,y_0$. From that point $x_0,y_0$ in the direction of the line up to distance $d$, I want to find the equation of the line that is perpendicular ...
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0answers
14 views

How can I measure segment of a shapes perimeter?

I am creating a tracing tool and one aspect of that is that I need to determine the length of a segment of a shape's outline. Given any shape within a 100x100 grid, and two points on that shapes ...
0
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0answers
24 views

How does the steepness of lines through a hyperboloid change the further away they are from the apex?

I'm a geoscientist and am trying to figure out how the steepness of the flanks of a hyperboloid change for straight lines that cross them. The line of reference is through the apex. Basically any ...
13
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0answers
102 views

Is Tolkien's Middle Earth flat?

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean ...
2
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0answers
28 views

spherical segment volume

Suppose I have a spherical segment like the one in the picture. I want to find the infinitesimal volume of such a segment. The angle between point A and B is $d\theta$. And the radius of the sphere ...
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votes
2answers
94 views

Ayn Rand and athematics

I am an honors undergraduate in mathematics. I have taken an interest in objectivism. I came across a discovery of Ms. Ayn Rand's in mathematics: In a triangle the inscribed circle touches the ...
5
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1answer
95 views
+50

Space Geometry: lines in a plane

If $d$ and $d'$ are two intersecting lines in a plane $P$, and $D$ is a line orthogonal to both $d$ and $d'$, then any line $\delta$ in $P$ is orthogonal to $D$ as well. How could this be proven ...
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1answer
17 views

Finding grid nodes a line passes through

For 2D grid pathfinding, I want to do a quick broadphase to check if there is a direct path from the start to the target by conceptually checking all nodes touching the line segment formed by ...
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4answers
60 views

Probability question involving infinite number of vertical chords in a 1 inch circle. [on hold]

Infinite number of vertical chords drawn on a circle with a 1 inch radius. What is the probability that a randomly picked chord is shorter than the radius? The answer should be $1 - .5√ 3$ or ...
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0answers
29 views

Sharper Bound for Minkowski's Convex Body Theorem

To satisfy the conditions of Minkowski's Convex Body Theorem we need a lattice $\Lambda$ with fundamental domain $T$ and a bounded convex symmetric subset of $\mathbb{R}^n$ (call it $X$) with ...
0
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1answer
45 views

An isosceles right triangle has legs of length 10. A pin is dropped into it and lands somewhere in the triangle where all places are equally likely.

What is the probability that it does not land within 2 units of any of the sides? From my calculations, I get that the smaller triangle has side lengths of 4,4, 4 root 2 (-2 at the right angle and ...
2
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2answers
29 views

Local existence of parallel vector field

Let $M$ be a Riemannian manifold, $p\in M$ be a point in the manifold, and $\xi\in T_p M$ be a vector in its tangent space. I am wondering whether, for some small neighborhood $U\subset M$, it is ...
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3answers
84 views

Circles revolving around each other and infinities

I just watched this video, and I'm a bit perplexed. Problem: ...
1
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1answer
23 views

logarithmic spiral around cone stump

Based on the answer on my previous question I managed to come up with the following equations: $$\begin{eqnarray} k &=& 1 \\ r_\Delta &=& r_b - r_t \\ r(\theta) &=& r_t * ...
4
votes
2answers
86 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
2
votes
1answer
56 views

inscribed circle in $n$-gon

If I'm given a circle with radius $r$ and I want to create a polygon with side $n$ (say $n=5$) which can cover the circle fully, then how to prove that a regular polygon is the solution with minimum ...