New answers tagged

0 votes

How to find the coordinate if angle and distance between the point and one point is given

Comment: Point $A(x, y, z)$ is on the base of a cone with following specifications: 1- It's tip is on the origin and it's base is on a plane parallel with YZ plane. 2- It's altitude is $h=5\cos 30^o=2....
user avatar
  • 8,491
0 votes

Find a synthetic solution: $T(24^{\circ}, 26^{\circ}, 28^{\circ}, 10^{\circ}, 22^{\circ})$

I think this is not possible with only compass and straight edge, unless I am missing something. Are you given anything to start with or should you construct the topmost figure without any given ...
user avatar
  • 1,318
0 votes

How to name a triangle on the basis of its points and Which side of a triangle should be taken as the first, second and third sides for theorem?

Let's say there is a polygon. We usually begin with a vertex named $A$, and then go clockwise or anticlockwise, naming the vertices $B, C$ and etc. If there is a quadrilateral $ACBD$, it is not very ...
user avatar
0 votes

Show that the curvature of a parametric curve is invariant under rigid motions.

Let $f = T_q \circ L_A$, $A \in \mathit{O}(3)$, $q \in \mathbb{R}^3$ be a rigid motion. Define $\beta(s) = A\alpha(s) + q$, then $\beta''(s) = A\alpha''(s)$ ($A$ is a constant) and so $k_\beta(s) = \...
user avatar
2 votes
Accepted

Find a synthetic solution: $T(12^{\circ}, 24^{\circ}, 54^{\circ}, 18^{\circ}, 30^{\circ})$

Hint: Triangle ABC is isosceles and $\angle BAC=72^o$. As can be seen in figure the bisector of angle ACB meets side AB at E . We have $\angle ECD=\angle ACD=18^o$. Triangle ACE is isosceles(why ?), ...
user avatar
  • 8,491
2 votes

Number of chords to be drawn to return the starting point

Method using modular arithmetic Notice that $\triangle AOB$ is isosceles and therefore $\angle AOB = 105^\circ$. Each successive chord adds $105^\circ$ to this angle so we want to know when this would ...
user avatar
0 votes

Find the volume of the pyramid below

You can find the result by subtracting from the volume of the cube the volumes of other five pyramids, whose volume is easy to find because their bases lie on the faces of the cube: pyramid $HIJG$ (...
user avatar
4 votes

How many lattices does it take to cover a regular $n$-gon?

$\newcommand\ZZ{\mathbb Z}\newcommand\cl{\overline{\mathbb F_p}}$This solution will show that the claimed answer, i.e. $n/6$ when $6\mid n$, $\lceil n/4\rceil$ when $6\nmid n$ but $2\mid n$, and $\...
user avatar
1 vote

Converting a line to a circle

So you have a line segment between the points: $$ P_1=(384,360)\\ P_2=(1152,360) $$ hence having length $1152-384=768$. You want this to be transformed to become circumference of a circle, hence ...
user avatar
  • 17.5k
2 votes
Accepted

Prove that $\max _{M} f-\min _{M} f$ is invariant with the choice of a $K$-invariant real symplectic form $\omega$ on manifold $M$

I am not sure this is a complete solution; it might still be interesting and/or useful. Let $\omega$ and $\tilde{\omega}$ be $K$-invariant symplectic forms in the same cohomology class. I think that, ...
user avatar
2 votes

Find the volume of the pyramid below

First, find the edges: $KJ=KI=\sqrt 6, KG=3, GJ=GI=\sqrt 5, IJ=\sqrt 2$. Let $X$ be the midpoint of $IJ$. Then $\angle KXG$ is the angle between planes $KIJ$ and $GIJ$. We can calculate $KX=\sqrt{6-\...
user avatar
  • 8,417
1 vote

What does 3-regularity mean in the context of this diagram of a fullerene?

A graph is called 3-regular if every vertex (carbon atom) has 3 edges (bonds). More generally, in an n-regular graph every vertex has degree n. Another name for a 3-regular graph is a cubic graph. So ...
user avatar
0 votes

Is there any way to add spherical vectors without converting?

You can do it using formulas that are used in great-circle navigation. First, find the angle $\theta$ between the two vectors, which is equivalent to finding the distance between two points on a ...
user avatar
  • 86.6k
1 vote
Accepted

Find the measure of the smallest angle that determine the diagonals of a quadrilateral

Comment: as you see in figure if triangle is equilateral then minimum of $\theta$ is ($\theta=24^o$)(triangle DEF and quadrilateral FIJE). ($\theta=6^o$) is possible if triangle is isosceles(...
user avatar
  • 8,491
1 vote

Where does this algorithm for detecting whether a polygon's vertices are given counter clock-wise comes from?

First of all, note that $$ (x_{i+1} - x_i)(y_{i+1} + y_i) = x_{i+1} y_{i+1} + x_{i+1} y_i - x_i y_{i+1} - x_i y_i. $$ When you add this up over all the pairs of vertices $(x_i, y_i)$ and $(x_{i+1}, y_{...
user avatar
  • 86.6k
-1 votes
Accepted

Ratio between the area of a triangle and a trapezoid

This is the answer to your problem. Correct answer is $(C)=\dfrac{1}{7}$
user avatar
0 votes

Ratio between the area of a triangle and a trapezoid

Hint: $A_{ABCD}=2A_{AEB}+A_{BEC}$, $A_{BEC}=\frac{1}{3}A_{AEB}$, $A_{AEC}=\frac{1}{3}A_{AEB}$.
user avatar
  • 8,417
3 votes

Relating energy integral to the existence of a Frostman measure

If $B(x,r)$ is disjoint from $A$, $\,$ then $\nu(B(x,r)=0$. Otherwise, there is some point $y \in B(x,r) \cap A \,. \,$ Then $B(x,r) \subset B(y,2r)$, so $$(2r)^{-s} \nu(B(x,r)) \le \int_{B(y,2r)} ...
user avatar
0 votes

Constant AM,GM,HM circles for segments through a point on negative power Circles

Without loss of generality, by translational, rotational and scaling symmetries, we can assume that $O=(0,0)$, the power line makes an angle $\alpha$ with respect to the x-axis and the circle has ...
user avatar
1 vote

The points $(0, k),(0,-k),(k, 0)$, and $(-k, 0)$, where $k \neq 0$, comprise the vertices of a square. Why is the side-length not $2k$?

First, what is the error you're making? Well, I tend to doubt that you've actually drawn a picture of the square you have in mind, with the vertices carefully labelled. From what you're saying it ...
user avatar
1 vote

The points $(0, k),(0,-k),(k, 0)$, and $(-k, 0)$, where $k \neq 0$, comprise the vertices of a square. Why is the side-length not $2k$?

Trivial answer. We have a square $\rightarrow$ it has $4$ sides having equal (Euclidean) length. Then, by Pythagorean Theorem, the length of one side is given by $\sqrt{(k-0)^2+(k-0)^2}=\sqrt{2(k-0)^2}...
user avatar
  • 159
0 votes

Finding the length of one side of cyclic trapezium if given length of two adjacent side and the diagonal between them.

Draw a point $E$ on $DC$ such that $BE$ is perpendicular to $DC$. Let $EC = x$. Since this is a cyclic trapezium, its also isosceles, and hence $BC=5$ and $DE=4+x$. Hence, $$\begin{aligned}BE^2=BD^2-...
user avatar
1 vote
Accepted

Find an angle in a picture of trapezoids

Sorry that I do not yet know how to draw figure using Latex or Mathjax. I can only describe how I approach the problem. Let $AB$ be the slant edge that make angle $\theta$ with the ground. In the ...
user avatar
1 vote

Find the exact value of $\frac{BD}{AB}$

From triangle $ADC$ we get $AD = 2\cdot AC\cdot \sin(24)$. Now use the law of cosine in triangle $ABD$ to achieve (after some manipulations) $$BD^2 =2 AC^2.$$
user avatar
1 vote

prove that this expression equals the area of a specific triangle

Let $a = 2, A = \frac {\pi} 4, B = \theta$ Then, $$ \frac{b}{\sin \theta} = \frac{2}{\frac{\sqrt{2}}2} \implies b = 2\sqrt 2 \sin \theta $$ Then, $$ \sin C = \sin(A+B) = \frac {\sqrt{2}}2 (\sin \theta ...
user avatar
0 votes

Studying Euclidean geometry using hyperbolic criteria

Charles' Segal's answer gives an answer about how you could do it if you lived in 3 dimensional hyperbolic space. Here's an answer on how you could do it if you lived in a 2-dimensional hyperbolic ...
user avatar
  • 780
0 votes

How can I create a function which takes input and gives new coordinate X, Y which has rotation in degree of some angle?

If a point, (X, Y), has distance r from the origin and lie on a line at angle $\theta$ from the line y= 0, then X= r cos($\theta$) and Y= r sin($\theta$). After a rotation through 60 degrees= pi/3 ...
user avatar
1 vote

geometrical/physical interpretation of multiplication of real numbers (including negative)

It is important to understand how the "Ancient civilizations" were defining the product of two positive quantities $a$ and $b$. @Lee Mosher has given such a definition. Another one is ...
user avatar
  • 66.5k
0 votes

What's the $2$d-object that's in $1:1$ correspondence with all lines in $\Bbb R^2$?

Wanted to present a "twistorial" perspective. The oriented affine lines of $\mathbb{R}^2$ may be identified with the unit circle's tangent bundle $TS^1$: given any tangent vector $t$ to a ...
user avatar
  • 16k
1 vote
Accepted

Proving that when the normal unit vector $N$ is parallel to the position vector $\gamma(t)$ then the image $\gamma(I)$ is part of circle

Vector $N(t)$ is perpendicular to $\gamma'(t)$, hence: $$ \gamma'(t)\cdot\gamma(t)=0\ \implies {d\over dt}\big(\gamma(t)\cdot\gamma(t)\big)=0 \ \implies \|\gamma(t)\|=\text{constant}. $$
user avatar
1 vote
Accepted

Is this a valid proof for the pythagorean theorem?

Your proof is correct and is the standard one, but you're being overly clear about what you're doing (you don't need to do everything in really small steps). Indeed it does suffice to just say that if ...
user avatar
  • 3,586
6 votes
Accepted

What's the $2$d-object that's in $1:1$ correspondence with all lines in $\Bbb R^2$?

The brief answer to your question is: an open Möbius band. In more detail, let's embed $\mathbb R^2$ into $\mathbb R^3$ by the map $(x,y) \mapsto (x,y,1)$. Using this embedding, I am going to abuse ...
user avatar
  • 102k
0 votes

Equation of a cylinder with a profile / ellipsoid

Btw, if the equation is sought for finding volume of a Barrel it can be done by integration with attention to limits, resulting in the formula $$ V= \frac{\pi h }{12}\cdot (2D^2+d^2)$$ where $D$ is ...
user avatar
  • 36.9k
1 vote

Equation of a cylinder with a profile / ellipsoid

Consider the blue curve first In my notation the small radius is $r=0.01$ and the large radius is $R=0.09$ It is an arc, of radius $R$ and center at $x=R - r$ I use the parameter angle $\theta$ for ...
user avatar
  • 10.9k
1 vote

Equation of a cylinder with a profile / ellipsoid

That would probably be called a "barrel shape". The equation depends on the type of curve you choose for the barrel. If it's a circular curve, if the $y$ axis is along the axis of symmetry ...
user avatar
  • 8,200
0 votes
Accepted

The lemniscate of Euler's $\textit{elastica}$: precise determination of a characteristic constant

In the comments, @TravisWillse provides a very useful link to Djondjorov et al.'s "Explicit parameterization of Euler’s elastica." The authors express the limits of the lemniscate as $\left(\...
user avatar
1 vote
Accepted

Find the dimension of the interior bin of a rectangular storage container

We know from the volume that $lwh=240$, and that $2(l+w)=40$ from the perimeter. We can rearrange the latter as $l=20-w$. Now, we substitute this expression for $l$ into the equation for volume. We ...
user avatar
3 votes

Trapezoid height from one base to other base given tangent circle

I am using $h$ as defined by the OP. The altitude of $M$ to $DC$ is also $h$ and has base point $I$ on $DC$. As $F$ is the midpoint to $DM$ using scaling the altitude of $F$ to $DC$ is $h/2$. Thus $EF=...
user avatar
  • 1,318
4 votes
Accepted

Trapezoid height from one base to other base given tangent circle

It's easy to show that the horizontal component of $M$ is $6$, and the vertical component is $h/2$ ($h$ is the height $AH$). Then $$DM=\sqrt{6^2+\frac{h^2}4}$$ Also easy to see that the vertical ...
user avatar
  • 33.4k
0 votes

Area of Intersection: Rectangle and Two Circles

If the offset is $(A)$, then the area of the shaded region is also $(A)$. Analysis follows (no Calculus required). The picture below provides three figures. In Figure 1, for clarity, I have ...
user avatar
  • 24.7k
2 votes

If $\vec a,\vec b,\vec c$ be three vectors such that $|\vec a|=1,|\vec b|=2,|\vec c|=4$ and then find the value of $|2\vec a+3\vec b+4\vec c|$

You have $$2a + 3b +4c = 3(a+b+c) -a+c.$$ therefore $$\begin{aligned} \lVert 2a + 3b +4c \rVert^2 &= 9 \lVert a + b +c \rVert^2 + \lVert a \rVert^2 + \lVert c \rVert^2 - 6 \lVert a \rVert^2 - 6 a \...
user avatar
3 votes

If $\vec a,\vec b,\vec c$ be three vectors such that $|\vec a|=1,|\vec b|=2,|\vec c|=4$ and then find the value of $|2\vec a+3\vec b+4\vec c|$

Recall that $\|\vec x\|^2 = \vec x \cdot \vec x$. Then $$\begin{align*} \|2\vec a + 3\vec b + 4\vec c\|^2 &= (2\vec a + 3\vec b + 4\vec c) \cdot (2\vec a + 3\vec b + 4\vec c) \\ &= 4\|\vec a\|^...
user avatar
2 votes

geometrical/physical interpretation of multiplication of real numbers (including negative)

I believe the geometrical interpretation still work, if you take the signed area of the rectangle. So, let's define an "oriented" rectangle as a rectangle with a particular orientation of ...
user avatar
3 votes
Accepted

geometrical/physical interpretation of multiplication of real numbers (including negative)

There are a couple of different ways to express multiplication using geometry. One somewhat physical interpretation of multiplication involves stretching/compressing/reflecting the number line. For ...
user avatar
  • 102k
1 vote

Intersections of lines formed by the pentagon are collinear.

This is not yet the kind of proof based on Pappus and Desargues that you're after. But it might shed some light so I'm posting it now and may add more later in an edit or a separate answer. This is a ...
user avatar
  • 38.6k
1 vote
Accepted

Can we calculate the sides of a right triangle knowing only its three angles and the sides of a similar triangle?

No. You can apply an arbitrary scaling transformation to a triangle. Doing so will preserve similarity and all angles, but change lengths. Which means that you can't determine lengths if all you have ...
user avatar
  • 38.6k
0 votes

Rectangle where $E$ is a point on side and finding length of angle bisector of $\angle BED$

Let drop altitude $AF$ to $BE$. Then triangles $ADE$ and $AFE$ are congruent right triangles. Then $AF$ is 4. Using Pythagoras theorem one can find $BF=\sqrt{AB^2-AF^2}=3$. Angles $BAE$ and $AEB$ are ...
user avatar
1 vote

Rectangle where $E$ is a point on side and finding length of angle bisector of $\angle BED$

Alternative (less elegant) approach: $\underline{\text{Tools:}}$ $\displaystyle \tan (r + s) = \frac{\tan(r) + \tan(s)}{1 - [\tan(r)\tan(s)]}.$ $\displaystyle \tan (r - s) = \frac{\tan(r) - \tan(s)}{...
user avatar
  • 24.7k
1 vote

Equation of a line passing through the intersection of 2 lines

This is dual to the situation where two points $\,p_0\,$ and $\,p_1\,$ in any affine (or vector) space. determine a unique line. In this case, the set of points of the line is parametrized by the ...
user avatar
  • 30.1k
1 vote
Accepted

Rectangle where $E$ is a point on side and finding length of angle bisector of $\angle BED$

From your angles labeled $\beta$ and $\gamma$ it follows $BAE$ is an isosceles triangle. Therefore $BE=5.$ Plotting this on the coordinate plane with $B=(5,4)$ and $E=(x,0),$ then we have $$\sqrt{(5-x)...
user avatar
  • 606

Top 50 recent answers are included