Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
47,239
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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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How to prove two alternative rotation matrix (maps one vector to another) equal?
In a paper, the rotation matrix that maps a vector $g$ to another vector $e$ (assuming unit vectors) is written very neatly as (without any skew symmetric matrix operation):
$R = I_{3\times3} + 2eg^T -...
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Maximise area of rectangle with fixed perimeter
I've got a problem where a rectangle's area must be maximised given a fixed perimeter of $60$m.
Assuming a length of $x$ and height of $y$ I wrote an equation $y = 30x - x^2$ which i differentiated, ...
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Find a triangle's vertices given sides, angles, and inscribing circle
I have a triangle inscribed in a circle of radius 1 centered on the origin. The angles A B C and opposite sides a b c of the triangle are given (but variable). Given the coordinates of one vertex (0,-...
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Given a set of polygons in 2D. How to Calculate the total the area covered by the polygons.
Considering that the polygons can overlap with each other, how do I calculate the totala area covered by them?.
What I have tried : Merge the polygons into one polygon but couldn't properly sort them ...
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exponential equation for the distance between two points a distance away from the function [0,1,...,9]x
Given the element list: b = [0,1,...,9]
Then what is the exponential equation in relation to distance from two points with a value of d from the origin of the functions: $(b_n)x$ and $(b_n-1)x$.
To ...
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Fitting a convex polytope with $n$ facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
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Find area of square from inner lines at right angles
There are five lines inside a square at right angles to each other with lengths as shown in the image in the link. What is the area of the shaded part of the square?
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Explicit representations of hemispherical intersections?
Let $v_1, \dots, v_n$ be vectors of unit norm in $\mathbb{R}^d$, and let $\mathbb{S}^{d-1}$ denote the unit sphere.
I am interested in the set
$$
X_n = \Big\{\, x \in \mathbb{S}^{d - 1} : \langle x, ...
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Length Of a Tangent from contact to intercept
If I was Given a Function y and Some x value how to find the length of the tangent from the contact to x axis intercept,you can use any example except for quadratic equations .
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If $d(x,B)\leq t_1+t_2$ then $d(x,B^{t_1})\leq t_2$ when $B^{t_1}:=\{x:d(x,B)\leq t_1\}$
Let $(X,d)$ be a metric space and $t_1, t_2 > 0$ be fixed and define $B^t:=\{x: d(x, B)\leq t\}$ for any $t > 0, B\subset X$. Suppose then that $B\subset X$ and $x\in X$ are fixed such that $d(x,...
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Angle problem between points on a unit sphere
I am dealing with an applied question with space angles correction. It’s not homework so I have no clue now.
The question: Given n points are denoted by $\{V_1,V_2,\ldots,V_{n-1},V_n\}$ on the unit ...
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How to find an equation of the plane
Find an equation of the plane passing through the point $P(1, 0, 1)$ and containing the line $[x, y, z] = [0, 1, -1] +t [2, 0, 1]$
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What is the area of intersection between two circular sectors? (Where can I find more information?)
I am trying to find an expression for the area of the intersection of two circular sectors.
There are some obvious and trivial solutions for some special cases, but I'm trying to generalize.
Here is ...
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Tangent plane for sphere containing a line
Find the equation of the tangent planes to the sphere $$x^2+y^2+z^2-2x+4y-6z+10=0$$ which pass through the line $$\frac{x+3}{14}=\frac{y+1}{-3}=\frac{z-5}{4}.$$ Find also angle between these planes.
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Rotating $y=x-2\sqrt{x}+1$ by $45^\circ$ counter-clockwise
I tried to rotate the equations
$$y=x-2\sqrt{x}+1 \quad\text{and}\quad y=x+2\sqrt{x}+1$$ (because of the two possible answers) $45^\circ$ counter-clockwise by using
$$y \cos(\theta)- x \sin(\theta)=(f(...
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Aligning more than two sets of 3D points
We have a good solution for the problem of aligning two sets of 3D points with known correspondences, assuming each set has more than three points. It consists of centering the points around the ...
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Show that the image of an irreducible affine variety is irreducible [duplicate]
I would really appreciate some help on this:
Definition: Let $k$ be a field. An affine variety is a space with functions $(X, O_X)$ i.e. a topological space equipped with a sheaf of $k-algebras$, ...
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video lectures suggestion for Euclid gemeotry
I am pleased for asking a good lecture series on Euclid geometry. I need the lectures / books suggestion specifically for school students. I know there are plenty of lecture series and book which ...
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Finding parametric form of a circle that is in a plane $x-2y+z=0$ with radius 1 and center at origin
Find a parametric form of a circle that is in a plane $x-2y+z=0$ with radius 1 and center at origin
In the book they picked $\hat u =(2,1,0)$ (hat for unit vector) , did cross product with the plane ...
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How far away is the horizon as seen from top of a mountain 1 mile high?
Today I came across this question:
How far away is the horizon as seen from top of a mountain 1 mile high. (Assume the earth to be a sphere of diameter 7920 miles).
(Correct answer 89 miles)
My ...
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Algorithm to transform a polyline into an equidistant polyline
I have a number of points in 3D space. These points represent the positions (the toolpath) for an industrial robot or a CNC machine. The points are calculated by a software program. There can be up to ...
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Finding the largest square that can be inscribed inside the astroid curve $x^{2/3}+y^{2/3}=4$
Finding the largest square that can be inscribed inside the astroid curve $x^{2/3}+y^{2/3}=4$
A square is described by four vertices. There will be one vertex of the square in each quadrant. I ...
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Find the sides of an AAA triangle inscribed in a circle with known radius
I have all of the angles for a triangle. I want to know the length of each of the sides such that the three vertices will all lie on a circle with a known diameter. How do I do it?
(The application is ...
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Circle coverage, optimising circles (r) on big circle (R) [closed]
Circle coverage, optimising pattern of circles on big circle, or how to arrange small circles (diameter d) on larger circle (diameter D)?
ex. 1) If d=D then ratio is 1, but what if r is limit -> 0 ...
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Find the length of segment $GM$ in the triangle below
In a triangle $ABC$, right at $B$, $AB=a$. Through its barycenter $G$ draw $MN$ perpendicular to $AC$, such that $M$ and $N$ are on $BC$ and $AC$. If $GN=b$, calculate $GM$.
(Answer: $ \frac{a^2}{9b}$)...
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Find dot product of vector with tangent planes of a sphere using polar coordinates
Cartesian coordinates shown here are represented with
$$P(x, y, z)$$
while polar coordinates are to be in the form of
$$P(r; \theta; \phi)$$
As shown in the figure below. Note the use of semicolons to ...
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slant height and surface area of oblique prisms
I am guessing that the slant height of an oblique prism is defined as the length of each segment connecting corresponding points of the two bases rather than the distance up the middle of any of the ...
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Can you find $\theta$ without trigonometry? Using pure geometry. [closed]
I was playing around with geometry and just making random triangles and came up with this diagram. Thought of doing it geometrically. Any suggestions about how to approach it. Using trig it’s just $45^...
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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 ...
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What is the maximum number of points $p_i\in\mathbb{Z}^2$ such that no three points are co-linear and $d(p_i,p_j)\in\mathbb{Z}\forall i,j\in[k]?$
What is $k,$ the maximum number of points $p_i = (x_i, y_i)\in\mathbb{Z}^2,\ i\in \{1,\ldots,k\},\ $ such that no three points $p_{i_1},\ p_{i_2},\ p_{i_3},\ $ are co-linear, and $d(p_i,p_j):= \sqrt{ {...
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Angle chasing on cyclic quadrilaterals
$\textbf{Problem.}$ $ABCD$ is a cyclic quadrilateral and the tangent to the circle at $D$ intersect $AB$ and $BC$ at $E$ and $F$ respectively. $T$ is a point inside $\triangle ABC$ such that $TE$ is ...
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how to prove any straight line can divide trapezoid be two equal area?
Intermediate value theorem quarantee that there is infinite line can divide trapezoid to be two equal area. I confused the properties and how to generalization (without picture) ensure that the line ...
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Is there a 2D shape where any moving object inside that passes through a line and bounces off the walls ends up at a point?
For Ellipses, it is observed that any moving object that passes through the focal points will always pass through the focal point again.
What interested me, however, is the fact that a collection of ...
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Can 3 pythagorean triangles on the same circle have areas where two of the areas sum to the third?
Let’s say we have 3 pythagorean triangles (integer sides and hypotenuse) all with the same hypotenuse. Is it possible for the areas of two of those triangles to sum to the area of the third triangle? ...
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Three parallel lines and segments
On the given diagram $AD\parallel BE\parallel CF, AE=BE=8$ and $BF=CF=16$. What is the length of $AD$?
As you can see on the diagram, they have actually found the intersect of $AC$ and $DF$, let’s ...
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Describe in a trigonometric manner the radius of circle ABD.
Let $D\in BC$ in $\triangle ABC$, so that $AD$ is the bisector of $\angle BAC$. Let $r$ be the radius of the circle inscribed in $\triangle ABC$ and $R$ the radius of the circumscribed circle of $\...
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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 ...
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Equation of a circle that passes through $(4,3)$ and which, touches the $y$ axis and another given circle
Find the equation of a circle which passes through $M(4,3)$, touches the $y$ axis and touches the circle $(x-2)^2 + y^2 =1$
Now, if I suppose that the equation is looks like $$(x-p)^2 + (y-q)^2 = r^2$...
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Find the cracked area
You have a square of side 1m, consider that both triangles have vertices at the midpoint of the square, find the cracked area
$S=2 S\triangle - S_ \boxed{LMNP}=2. \frac{1}{2}.\frac{1}{2}.1 - S \boxed{...
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A ternary relation on a group
Let $G$ be a group. Consider the ternary relation $R \subset G^3$ defined by
$$R(x,y,z) \Leftrightarrow x y^{-1} z x^{-1} y z^{-1} = 1 $$
Show that $R$ is a symmetric relation, that is if $R(x,y,z)$, ...
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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 ...
