# Questions tagged [geometry]

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

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### Where can I learn about the "geometry of the equivalence classes of rationals"?

I was reading about the formal construction of the rational numbers as pairs of integers on wikipedia and was interested in the diagram (see below) illustrating the equivalence classes geometrically ...
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### Find distance of the line after rotation

Hi, Can anyone explain how to find the x distance and y distance which is shown in the image. Blue line is rotated 45 degree with respect to centre point. After rotation, I want to find the horizontal ...
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### Notation in surgery theory (maths)

sary Hello everyone! In reading an article on surgery theory I found an expression of the type $M \cup_{\phi} N$ where $M$ and $N$ are manifolds(or more generally topological spaces), and $\phi$ some ...
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### How to prove the triangle inequality in the case of an n-sided polygon?

Basically, I want proof for the statement that any particular side of an n-sided polygon is less than the sum of the lengths of its other sides. I tried proving it and I can successfully prove it for ...
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### Given that a circle is tangent to two perpendicular diameters of a larger circle and the larger circle itself. What is the radius of the small circle?

The actual question itself asks for the ratio between the smaller circle to the larger circle and the answer is C. $3-2\sqrt{2}$. What I do not understand is how we are meant to get the radius of the ...
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### Congruence problem $BF = FC$

We have an $ABC$ triangle in which $\angle ABD=\angle ACB,AB=12,AD=8,BC=15,DF\perp BC$.How to show that $[BF]\equiv[FC]$ I did a figure in geogebra but i don't see what triangles are similar i know ...
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### What are the properties of the configurations of a generalized "keyring"?

This is a fairly ill-defined question, based on a physical question. The design of a keyring, where a piece of metal has been molded to resemble a solid torus, in order to hold a key, has interested ...
1 vote
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### How to show that a polygon is a trapezoid?

How can I show that $ABCD$ is a trapezoid if all I know is that $AB=8$, $AD=9$, $DC=6$, $BC=16$, and $AC=12$? Can this be solved using similar triangles? I showed that $ABC$ and $DCA$ triangles are ...
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### Right triangle inside a right triangle

I have made this question but I am not able to solve it. We have a right triangle $ABC$ right angled at $B$ and $\angle BAC = 15^\circ$. From $B$, draw a segment $BD$ such that $D$ on $AC$ and then ...
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### How to find latitude and longitude values of point on surface of earth knowing the latitude and longitude values of a point nearby

Assume that there are two points $x_1$ and $x_2$ on the surface of the earth. Assume that we know the distance between both points, and we know the geo-coordinates (longitude and latitude) of $x_1$. ...
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### Proving $\cosh\alpha = \frac{\cos a}{1-\cos a}$ for a regular hyperbolic triangle with all side-lengths $\alpha$ and all angles $a$

Consider a regular triangle in the hyperbolic plane $\mathbb{H}^2$ (i.e. a triangle with all sides length $\alpha$ and all angles $a$). Prove that $\cosh\alpha = \frac{\cos a}{1-\cos a}$. I think I ...
### Is there a general geometrical connection between $f(x)$ and $f(1/x)$?
What is the relationship between graphs of a function $f(x)$ and the function $f\left(\frac{1}{x}\right)$? If I consider $f(x)=x$, $x^2$, and $x^2-x$ and then $g(x)=f(1/x)$, especially in the last ...