Expert Help with High School Geometry

The measure of an interior angle of a regular polygon is 160. How many sides does the polygon have?

Why is this not a geometric distribution?
A recruiting firm finds that 20% of the applications are fluent in both English and French. Applicants are selected randomly from a pool and interviewed sequentially. Find the probability that at least five applicants are interviewed before finding the first applicant who is fluent in both English and French
The answer for this question is $(0.8{)}^5$.

Inequality for convex polygons
Let P be a convex n-gon on the plane. For $k=\stackrel{¯<!--\; ¯\; -->}{1,n}$ define $a_k$ as the length of k-th side of P and $d_k$ as the length of projection of P onto the line containing k-th side of the polygon P. Prove that
$2<\frac{a_1}{d_1}+\frac{a_2}{d_2}+\dots <!--\; \dots \; -->+\frac{a_n}{d_n}\le <!--\; \le \; -->4.$
Firstly, let us prove the first inequality. Indeed, if p is the perimeter of the polygon P, then it's clear that $2d_k<p$ for all $k\in <!--\; \in \; -->\{1,2,\dots <!--\; \dots \; -->,n\}$. Hence, we obtain
$\frac{a_1}{d_1}+\frac{a_2}{d_2}+\dots <!--\; \dots \; -->+\frac{a_n}{d_n}>\frac{2(a_1+a_2+\dots <!--\; \dots \; -->+a_n)}{p}=\frac{2p}{p}=2,$
as desired.
Now, for the second part note that equality holds if, for example, P is a rectangle, so the second inequality is sharp. For polygon P denote $f(P):=\frac{a_1}{d_1}+\frac{a_2}{d_2}+\dots <!--\; \dots \; -->+\frac{a_n}{d_n}.$
Then, it can be shown that if P′ is polygon which is centrally symmetric to P, then the Minkowski sum $Q=P+P^{\prime }$ satisfy the following equality $f(Q)=f(P).$
Thus, it's sufficient to prove the inequality for Q, i. e. for centrally symmetric polygons (it's well-known that $P+P^{\prime }$ is a centrally symmetric polygon). However, it's quite unclear how to continue this approach.
So, is there any way to end this solution?

What is the Cartesian product of n polygons?
How do we calculate the Cartesian product between n polygons? Is it always a polytope? If so, can we say anything about its faces?
For example, n squares: $\{S_1:[a_1,a_2]\times <!--\; \times \; -->[b_1,b_2],\dots <!--\; \dots \; -->,S_n:[a_k,a_{k+1}]\times <!--\; \times \; -->[b_k,b_{k+1}]\}$ so, $S_1\times <!--\; \times \; -->\dots <!--\; \dots \; -->\times <!--\; \times \; -->S_n\in <!--\; \in \; -->{\mathcal{R}}^n$ forms a $n-<!--\; -\; -->$ dimensional box.

The sum of the measures of the interior angles of a convex polygon is 540. How do you classify the polygon by the number of sides?

Distance between (-2,1,-13) and (4,3,-12)?

Sum of the interior angles of a polygon is $${3060}^{\circ <!--\; \circ \; -->}$$ . How many sides does it have?

The length of each side of a square is extended 3 cm. The area of the resulting square is $$100{\text{ cm}}^2$$. Find the length of a side of the original square.

Volume of the solid
Using geometry, calculate the volume of the solid under $z=\sqrt{49-<!--\; -\; -->x^2-<!--\; -\; -->y^2}$ and over the circular disk $x^2+y^2\le <!--\; \le \; -->49$.
I am really confused for finding the limits of integration. Any help?

Why can this question be treated as a question with geometric random variable without any modifications?
In an infinite sequence of flipping a fair coin - 0.5 probability to get heads/tails. (every flip is independent from the others).What is the expected value of number of flips until we get Heads then Tails?

What is the probability of maximum of two iid geometric random variable?
- Let X,Y be independent geometric random variables, where both are having same parameter (p).
- Let $Z=max(X,Y)$.
I would like to find $P(Z=i)$ for some real values of i.
As we know for $K=min(X,Y)$, K is geometric distributed with parameter $(2p-p^2)$. Does Z also geometric distributed ?.

What is the distance between (4.7, 2.9) and (-2.6, 5.3)?

Distance between (-4,3,0) and (-1,4,-2)?

One exterior angle of a regular polygon measures 40. What is the sum of the polygon's interior angle measures? Please show working.

Finding two sides of a cuboid while knowing the volume
The volume of a cuboid is $150{\mathrm{m}}^3$, one side is 2,5m, one side is completely unknown (x) and the other one is 4 meters longer $x+4$. guessed the sides one being 6 and the other 10 but I need to figure it out with a formula.

Does $x^2\equiv <!--\; \equiv \; -->3$ (mod $q$) (where $q$ is an odd prime) have infinite solutions?

The question is as followed, prove that $\tan <!--\; \; -->{22.5}^{\circ <!--\; \circ \; -->}=\sqrt{2}-<!--\; -\; -->1$.
Would writing something like $\sqrt{3-<!--\; -\; -->2\sqrt{2}}$ $=\sqrt{2}-<!--\; -\; -->1$ be appropriate?

Probability distribution for a geometric distribution don't add up to 1
Say I'm rolling 2 dies, numbered 1 to 10. A successful outcome is considered rolling a multiple of 4. Therefore, probability of success $=0.25$ and prob of failure $=0.75$. This is an example of a geometric distribution. I can roll a maximum of 6 times. I made the prob distribution chart to represent all the different possibilities, but my values don't add up to 1. I don't understand what I'm doing wrong.
$\begin{array}{rlrlrl}\text{Roll}& \text{ Number}& & \text{Probability}& & \\ X& =1& & (0.75)ˆ0(0.25)& & 0.25\\ X& =2& & (0.75)ˆ1(0.25)& & 0.1875\\ X& =3& & (0.75)ˆ2(0.25)& & 0.140625\\ X& =4& & (0.75)ˆ3(0.25)& & 0.10546875\\ X& =5& & (0.75)ˆ4(0.25)& & 0.079101562\\ X& =6& & (0.75)ˆ5(0.25)& & 0.059326171\end{array}$

You need to construct a regular polygon. When you draw two sides, the interior angle created between them is ${120}^{\circ <!--\; \circ \; -->}$ . What will be the sum, in degrees, of the measures of the interior angles of this polygon when it is completed?

I need to find the locus of points (on an Argand diagram) such that:
(i) $\arg <!--\; \; -->(z-<!--\; -\; -->(-<!--\; -\; -->1-<!--\; -\; -->4i))+\arg <!--\; \; -->(z-<!--\; -\; -->(5+8i))=0$
(ii) $\arg <!--\; \; -->(z-<!--\; -\; -->(-<!--\; -\; -->1-<!--\; -\; -->4i))+\arg <!--\; \; -->(z-<!--\; -\; -->(5+8i))=\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}$
I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.
I am aware that $\arg <!--\; \; -->(z-<!--\; -\; -->(-<!--\; -\; -->1-<!--\; -\; -->4i))-<!--\; -\; -->\arg <!--\; \; -->(z-<!--\; -\; -->(5+8i))=\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}$ is a semicircle, and for other angles, say $\frac{\mathrm{\pi <!--\; \pi \; -->}}{3}$ or $\frac{\mathrm{\pi <!--\; \pi \; -->}}{4}$, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).
I am also interested in whether problems (i) and (ii) can be generalised to any angle between 0 to $\mathrm{\pi <!--\; \pi \; -->}$. Any help here would be greatly appreciated.