# Tagged Questions

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

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### Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
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### Derive equation for shear modulus $G=E/(1+2v)$

shear modulus, G young's modulus, E and Poisson's ratio, $v$: $G=E/(1+2v)$ I have always wondered how this relation is derived, but have never found a derivation that I could follow online. I ...
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### How to find the coordinates where the altitude of a triangle intersects the base in 3 dimensions?

Assuming I know three completely random coordinates in 3d space that correspond with vertices of a triangle, how can I then find the point at which the altitude intersects the base? I know how to ...
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### Length and diameter of a spiral of nanotube

I was reading from this popular article (in french). Talking about nanotubes of carbon the author says (my translation): The diameter of the nanotube is of the order of a millionth of a millimiter....
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### A Question about Triangle Inequality [duplicate]

Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side, if it is a positive integer? I'm not sure if the word "acute" affects the problem. If not, is the ...
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### Find the area of $S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$

Let $S$ be the domain defined by $$S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$$ find the area of $S$ This is middle school problem,so I think it ...
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### Ellipse equation from center and point on ellipse [closed]

Is there a way to get the equation of an ellipse iw we know the center and one point on the ellipse ?
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### How many combinations are there for the interior angles of a triangle?

Suppose the interior angles of a triangle are all Natural numbers. How many combinations of angles are there without repeating similar triangles? So for instance, {1,1,178}, {1,2,177},...But without ...
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### How to visualize this geometric setup?

I am working on a derivation (physics) and the first part requires one to imagine a geometric setup. Since I'm not a native speaker of the language, I am having trouble in doing so. The relevant ...
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### Why should sum of coefficients of collinear position vectors be 0?

Suppose $a,b,c$ are collinear position vectors then we know that $xa+yb+zc=0$ where $x,y,z$ are scalars. But in my book something additional is also mentioned that $x+y+z=0$,why must it be so ? I ...
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### How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
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### Cutting a pie into 2 unequal peices with a single cut, minimising its length. [closed]

Suppose we have a circle with an area of 1, which we are to cut into two pieces, of area (x) and (1-x) respectively. Let x<0.5. How should we make the cut, to minimise its length? What is the ...
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### Does shearing a sphere generate an ellipsoid?

In my preferred 3D modeling software I see that however I shear a sphere, I seem to be able to make a nearly identical shape using some combination of non-uniform scaling followed by rotation. Are ...
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### What is the mathematical vernacular for $\Delta x$

I want to use the proper terminology when I discuss length scales associated with $\Delta x$, where $\Delta$ is the difference operator. In other words $\Delta x = |x_1-x_2|$. It is a measure of the ...
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### Geometry Triangle Question 3

In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? Two sides of an acute triangle are 8 and 15. How many ...
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### Change of coordinates in $R^{n}$ where the diagonal goes to $x=0$

Say I have a system of coordinates $\{y_{1},y_{2},...,y_{n}\}$. I'd like to get a new system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ where the diagonal $y_{1}=y_{2}=...=y_{n}$ is $x_{1}$, and I'd ...
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### Geometry Questions: Triangles [closed]

Can you guys please help me with these problems? In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? The ...
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### Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
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### Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
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### Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
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### Question re: derivation of formula for volume of cone

The diagram above comes from a derivation of the formula for the volume of a cone; it's one of the preliminary steps, and sadly, I'm stuck on it. What we're doing here is inscribing an "infinite" ...
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### Linear application

Let $f:\mathbb{R}^3\to\mathbb{R}^3$ be a linear application and let $\{e_1,e_2,e_3\}$ the canonical basis of $\mathbb{R}^3$. We know that $\operatorname{Im} f=\langle(1,1,3), (0,1,1)\rangle$ and that ...
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### Area of circle segment intercepted by a line

The problem I want to solve is to calculate the filled area in the following diagram - so basically the area between the two circular arcs but with the red line cutting off one side. I think I have a ...
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### Finding centers of ellipses with two points and their respective tangents

I hope you can help me with the following, probably rather complex dilemma: I generally want to find an ellipse given two points and their respective tangents in 2-D space (X and Y coordinates). Now ...
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### Sphere and tetrahedron

If we have sphere inscribed in a tetrahedron, and if the distances from the center of the sphere to the edges of the tetrahedron are equal, is it true that this tetrahedron is always regular? I'm ...
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### How did Archimedes calculate rational bounds of pi from a 96-gon?

Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final ...
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### Find the volume of a rectangular parallelpiped

Can anyone help me with this problem? I used the following drawing in solving this problem and got a wrong answer. I don't know whether it is due to my misunderstanding of the problem. Here is my ...
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### Prove concurrency in a triangle

If a circumference cuts a triangle $ABC$ at its sides $BC$, $CA$ and $AB$ at points $P, P'; Q, Q'; R, R'$; respectively (so twice on each side, and if $AP, BQ$ and $CR$ are concurrent (intersect at a ...
If I have a vector space $V^{(1)}\otimes V^{(2)}$ and I have some ray $\sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k$, is the only solution that $s_k = s$ $\forall$ $k$? All $x_k$...