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

learn more… | top users | synonyms

0
votes
0answers
11 views

the formula for the volume

if you know that the volume of cube ($a^3$) represents the sum of the surface squares ($a^2$). Following the same logic, if we know, the area of the circular segment is equal to the area of the ...
0
votes
2answers
17 views

The angle between $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. Calculate $[u,v,w]$.

The angle between $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. The basis $(u,v,w)$ is positive, calculate $[u,v,w]$. I did the following: Take an arbitrary ...
1
vote
0answers
14 views

Intersecting lines in sectors of a circle.

Good day everyone, I'm trying to simulate a Laser Range Finder (LRF for short) in a corridor environment. I'm including a small fast sketch I did of this. I can't upload images yet, so I include just ...
0
votes
0answers
17 views

Dual plot for complex roots of quadratic equation

Real roots of quadratic equation $ x^2 - \sqrt 3 x + 1/2 =0 \tag{1} $ can be plotted on $x$- axis as its parabola intersection at $ (\sqrt 3/2 \pm 1/2,0). $ In an improvization I assign ...
1
vote
2answers
195 views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
5
votes
0answers
47 views

What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
1
vote
2answers
22 views

Find the length of angle bisector $AD$.

In $\triangle ABC$ , the internal bisector of the angle $\angle A$ meets $BC$ at $D$. If $AB=4$, $AC=3$ and $\angle A=60^{\circ}$, then the length of $AD$ is $a.)\ 2\sqrt3\\ \color{green}{b.)\ ...
2
votes
2answers
35 views

Find the plot of $y=1+\cos t$, $x=\sin^2t$.

I'm trying to find the plot for the following : $$y=1+\cos t, x=\sin^2t$$ I'm trying to get ride off variable $t$. This is what I done for some reason is incorrect : ...
0
votes
2answers
21 views

Find the area of the shaded region under given curcumstances

It is given that $ZV||XY,WZ=ZX,ZV=2a~~\text{and}~~ZX=2b.$ Find the area of the shaded region. $a.)\dfrac{4ab}{2}\\ b.)\dfrac{8ab}{3}\\ \color{green}{c.)6ab}\\ d.)3ab\\$ $\quad$ I found that ...
5
votes
3answers
65 views

Given two points, how to find a circle through them that's also tangent to the $x$-axis?

A seemingly simple geometry problem that is surprisingly difficult. I want to find the radius of a circle that is tangent to the $x$-axis, but also must contain two given points. I understand there ...
0
votes
0answers
25 views

Vertical/Horizontal stretch ratio. [on hold]

I'm trying to calculate 3 points on a grid by using 3 other points and calculating a ratio. I have the original values of all the points and was using the ratio (x1/x2) (y2/y1) for the points however ...
-1
votes
0answers
17 views

How to find the image of the line $y=ax$ from upper half plane to poincare disk?

with cayley transformation $m(z)=\frac{z-i}{z+i}$ i cant find a solution for this exercise so if you have any suggestion for the solution will be very helpful ...
0
votes
0answers
17 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
-5
votes
0answers
26 views

Prove that ABCD is cyclic if and only if it is a rectangle [on hold]

Prove that $ABCD$ is cyclic if and only if it is a rectangle, in which case its circumcenter is the point where its diagonals intersects.
1
vote
0answers
16 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
0
votes
1answer
23 views

Speed at which hands of clock approaching one another.

A clock's hour hand's length is $1$, and its minute hand's length is $r$. First I had to find the distance between the tips of the hands at 4:00. I did this using the law of cosines. This gives me ...
0
votes
4answers
35 views

For every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$

In $R^3$,show that for every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$ Generally,in $R^n$,for every ...
0
votes
0answers
12 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
-4
votes
1answer
28 views

How do you express the following equations for a circle? [on hold]

A circle of radius a is centered at a point r1. (a) Write out the algebraic equation for the circle. (b) Write out a vector equation for the same circle. (c) How would you modify (a) and (b) above ...
6
votes
4answers
66 views

Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.

$$ \left\{ \begin{aligned} c_1 & = a_2b_3-b_2a_3 \\ c_2 & = a_3b_1-b_3a_1 \\ c_3 & = a_1b_2-b_1a_2 \end{aligned} \right. $$ $c_1,c_2,c_3\in \mathbb{Z}$ is given,prove that $\exists ...
1
vote
4answers
37 views

Find the length of chord $BC$.

On a semicircle with diameter $AD$. Chord $BC$ is parallel to the diameter.Further each of the chords $AB$ and $CD$ has length of $2$ cm while $AD$ has the length $8$ cm.Find the length of $BC$. ...
-1
votes
1answer
27 views

Geometry volume and surface area [on hold]

How do I get the two sides of the top of the prism ($9$ & $9\sqrt{3}$)
0
votes
1answer
20 views

What is the optimal way to detect a collision between an AABB figure and a non-AABB figure?

Background I'm looking to do this programmatically in Java, but if desired you can post solutions in C/C++ or plain English instructions if you're not a programmer, but I would appreciate an ...
0
votes
3answers
48 views

If the surface area of a box is 32 and its volume is doubled what is the new surface area? [on hold]

Original surface area :32 Original volume: x New volume: 2x What is the new surface area? Please provide an explanation or show work, I don't know how to do it.
0
votes
1answer
24 views

How can I derive the resultant of 2 bearing/elevation pairs

Say, for example I have a gimballed camera mounted on a metal plate, which is itself fixed horizontally to a boat. I can measure the elevation and bearing of both the camera with respect to the plate ...
1
vote
2answers
24 views

What does it mean for a set of closed shapes to intersect?

To my understanding, a "shape" is a set of points in $n$-dimensional space. e.g., rectangles, triangles, lines, spheres, hyperspheres, etc.. For two (or any amount) of shapes to "intersect", the ...
2
votes
1answer
15 views

Find the three closest surrounding neighbors from a data

I have a data of coordinates $x$ and $y$ where we know the range of both variables, e.g. $(x,y)\in[0,1]^2$. So for a given any random point $\theta_0=(x_0~~y_0)^T$ in the range of $x$ and $y$ I would ...
3
votes
4answers
93 views

How to prove that a straight line is an infinite set of points?

From the basic elementary level when we start reading geometry we get this idea developed in us that a straight line is the conjuction of infinite points.but how to prove this? I mean is this an ...
1
vote
2answers
22 views

Find the ratio of curved surface area of frustum to the cone.

In the figure, there is a cone which is being cut and extracted in three segments having heights $h_1,h_2$ and $h_3$ and the radius of their bases $1$ cm, $2$cm and $3cm$, then The ratio of the ...
0
votes
1answer
17 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
10
votes
0answers
50 views

An interesting property between a hyperbola & parabola

It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, ...
1
vote
0answers
35 views

What is real $R$ so that every subset of Euclidean space with diameter one is inside a ball of radius $R$?

What is the real number $R$ so that for every $n$ every $S \subseteq \mathbb{E}^n$, for which $d(S) = \sup\{|x-x'| \mid x,x' \in S \} = 1$, is inside some closed $n$-ball of radius $R$? In particular, ...
2
votes
5answers
358 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
1
vote
0answers
20 views

4 faces sharing an edge. Which pairs of faces belong together.

I'm sorry for the somewhat vague title, but I'm not sure how to better describe the problem. I have a single solid in 3D (arbitrary, closed collection of faces). This solid internally has a part ...
1
vote
1answer
25 views

Elliptical section of a right circular cone [on hold]

A right circular cone, having cone angle $\alpha=40^o$, is thoroughly cut with a smooth plane (normal to the plane of paper as shown by the produced line AB in the diagram below) making at an acute ...
2
votes
2answers
66 views

BMO1 2006/07 Question 4 Geometry Problem

$4.$ Two touching circles $S$ and $T$ share a common tangent which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP = ...
1
vote
0answers
18 views

Hexagonal tessellation: tesselating a pizza slice shape sector of a circle with a special constraint

Consider a sector(in the shape of a pizza slice) of a disk of radius $R$ such that the sector is enclosed by two raddi and and an arc, where the arc subtends an angle $\frac{2\pi}{n}$. We tessellate ...
0
votes
1answer
12 views

Finding the coordinate at time $t$ of a line determined by the points $(x1,y1), (x2,y2)$

I have the problem here, I create a program that clipping a line with the input (x1,y1,x2,y2). but the algorithm only explain until I get ...
3
votes
0answers
23 views

How to state Pythagorean theorem in a neutral synthetic geometry?

In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence ...
0
votes
0answers
31 views

hexagonal tesselation symmetry

Consider a disk of radius R. We tessellate the surface of the disk using hexagonal tessellation where each hexagon has a circumradius r. The tessellation is made such that the center of one of the ...
0
votes
1answer
15 views

Test if a vector is pointing towards the center of an ellipse

I have an ellipse : $$x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta - a\cos t \sin\theta$$ Let's say if we have a normal vector $n$ to the ellipse, on a point $p$ ...
0
votes
0answers
22 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
0
votes
0answers
19 views

Getting the normal vector of a point on an ellipse

I have a rotated (by $\theta$) and translated (by $h,k$) ellipse given by: $$ x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta + a\cos t \sin\theta $$ By the normal ...
1
vote
0answers
15 views

Find the minimum total surface area of the cylinder in given circumstances.

Six solid hemispherical balls have to arranged one upon the other vertically .Find the minimum total surface area of the cylinder in which the hemispherical balls can be arranged, if the radii of ...
1
vote
4answers
32 views

what is the volume of cylinder if

The total surface area of a cylinder is $80\pi~\text{cm}^2$ and the difference between the height and the radius is $2~\text{cm}$. What is the volume of that cylinder? I have tried to find the ...
0
votes
2answers
60 views

Triangle whose side lengths and area are rational numbers [on hold]

Does there exist a triangle with side lengths given by rational numbers $x$, $2x$, and $y$ such that the triangle's area is also rational number?
0
votes
3answers
55 views

Prove that $\alpha$ lies between $0$ and $4$.

Let $a,b,c$ be the length of the sides of the triangle $ABC$ . Given $(a+b+c)(b+c-a)=\alpha bc$.Then Prove that the value of $\alpha$ lies in between $0$ and $4$. ...
3
votes
2answers
81 views

BMO1 2006/07 Question 2 Geometry Problem

$2.$ In the convex quadrilateral $ABCD$, points $M,N$ lie on the side $AB$ such that $AM = MN = NB$, and points $P,Q$ lie on the side $CD$ such that $CP = PQ = QD$. Prove that Area of $AMCP=$ Area of ...
3
votes
2answers
33 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
2
votes
2answers
43 views

Find the length of tangent $x$.

Two circles $C_1$ and $C_2$ of radius $2$ and $3$ respectively touch each other as shown in the figure .If $AD$ and $BD$ are tangents then the length of $BD$ is $a.)3\sqrt6\\ b.)5\sqrt6\\ ...