Expert Help with High School Geometry

Name the corresponding parts in the pair of congruent triangles. Then complete the congruence statement. $$\mathrm{△<!--\; △\; -->}NPQ\cong <!--\; \cong \; -->\mathrm{△<!--\; △\; -->}DEF;\mathrm{△<!--\; △\; -->}EFD\cong <!--\; \cong \; -->$$ ___

Set of regular polygons
Defined set S of regular polygons on plane.
Known for every polygon: at least one of it's sides is parallel to x,and this side's length is rational number.
Also known- there is at least one vertex on this side that it's coordinates are rational numbers.
What is the cardinality of S?

Prove that $\frac{2022}{n}+4n$ is a perfect square iff $\frac{2022}{n}-<!--\; -\; -->8n$ is a perfect square

Geometric probability (with regards to span)
Suppose we have two non-parallel vectors in ${\mathbb{R}}^3$.
Now, if we were to randomly select another vector in ${\mathbb{R}}^3$, what is the probability that that new vector lies in the span of the first two vectors?

What is the distance between (5,-18) and (-20,-9)?

Distance between (-5,-9) and (4,7)?

What is distribution of sum of two geometric distribution?
We are tossing an unfair coin (the probability of throwing a head is p). We count the number of the first identic throws (for example, if we have HHHT..., then this number is three). What is the distribution of this random number?

What is the distance between (-1,-101) and (2,-110)?

Average area of triangle formed $\frac{1}{8}$ that of square.
A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square.

If $(n+1{)}^2$ is even then $n$ is odd
find what proof works best with this question

Find the DE that describes the change in a raindrops volume over time
A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time
Volume, $V=\frac{4}{3}\mathrm{\pi <!--\; \pi \; -->}{r}^2$ but I dont know how to find any way to relate that to an evaporation rate. This course is an introductory differential equations course and they don't give you anything else to work with.
Do they assume we covered this kind of problem in a previous physics course?
IF the question read:
A spherical raindrop evaporates at a rate proportional to its surface area as $\frac{dV}{dt}=\mathrm{\alpha <!--\; \alpha \; -->}S(t)$. Write a differential equation for the volume of the raindrop as a function of time
I might have fared a bit better.
Is there a way to derive this equation?

The minimum of two independent geometric random variables
Let $X\sim <!--\; \sim \; -->‬G(p_1)$, $Y\sim <!--\; \sim \; -->‬G(p_2)$, X and Y are independent. Prove that the minimum is also geometric, meaning: $min(X,Y)\sim <!--\; \sim \; -->G(1-<!--\; -\; -->(1-<!--\; -\; -->p_1)(1-<!--\; -\; -->p_2))$.
Instructions: first calculate the probability $P(min(X,Y)>k)$ and compare it to the parallel probability in (of?) a geometric random variable.

Probability of events for beta-geometric distribution
If we have a beta-geometric distribution X with pdf $P(X=k\mid <!--\; \mid \; -->\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})=\frac{Beta(\mathrm{\alpha <!--\; \alpha \; -->}+1,k+\mathrm{\beta <!--\; \beta \; -->})}{Beta(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})}$.

Suppose that lions have a geometric offspring distribution. $p_k=(1-<!--\; -\; -->\mathrm{\theta <!--\; \theta \; -->}){\mathrm{\theta <!--\; \theta \; -->}}^k$, $k\ge <!--\; \ge \; -->0$ with mean 1.5. If the current world population of lions is 10, what is the probability they will go extinct in 1 generation? What is the probability they will go extinct eventually?
Same two questions whose offspring distribution is Poisson with mean 0.8, and current population is 5000.

Finding Geometric Probability?
Four points are chosen at random from the interior of a circle. What is the probability that the sum of each of their distances from the center is greater than half the circumference of the circle?

A square pyramid has base sides of 9m and height of 8m

Finding volume of sphere cartesian to polar
Using cartesian coordinates the volume of a sphere can be calculated via
${\int <!--\; \int \; -->}_{-<!--\; -\; -->R}^R{\int <!--\; \int \; -->}_{-<!--\; -\; -->\sqrt{R^2-<!--\; -\; -->z^2}}^{\sqrt{R^2-<!--\; -\; -->z^2}}{\int <!--\; \int \; -->}_{-<!--\; -\; -->\sqrt{R^2-<!--\; -\; -->z^2-<!--\; -\; -->y^2}}^{\sqrt{R^2-<!--\; -\; -->z^2-<!--\; -\; -->y^2}}1\mathrm{d}\mathrm{x}\mathrm{d}\mathrm{y}\mathrm{d}\mathrm{z}$
However using a parametrisation with polar coordinates the Integral becomes:
${\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}{\int <!--\; \int \; -->}_0^R{\int <!--\; \int \; -->}_0^{\mathrm{\pi <!--\; \pi \; -->}}1R^2\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\mathrm{d}\mathrm{\theta <!--\; \theta \; -->}\mathrm{d}\mathrm{R}\mathrm{d}\mathrm{\phi <!--\; \phi \; -->}$.
Thinking closer about this I just don't see how these integrals are interchangeable. Notably substituting polar coordinates in the cartesian one I don't notice how the borders simplify.
For instance using the substitution
$\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}R\cos <!--\; \; -->(\mathrm{\phi <!--\; \phi \; -->})\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\\ R\sin <!--\; \; -->(\mathrm{\phi <!--\; \phi \; -->})\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\\ R\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\end{array}\right)$
I get for the first border:
$\sqrt{R^2(1-<!--\; -\; -->{\cos }^2<!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})-<!--\; -\; -->{\sin }^2<!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->}){\sin }^2<!--\; \; -->(\mathrm{\phi <!--\; \phi \; -->}))}$. What's that even?

Conditional probability involving a geometric random variable
Let $X_1,...$ be independent random variables with the common distribution function F, and suppose they are independent of N, a geometric random variable with parameter p. Let $M=max(X_1,...,X_N)$. Find $P(M\le <!--\; \le \; -->x)$ by conditioning on N.

Choose the letter of the word or phrase that best completes each statement: If $$\mathrm{△<!--\; △\; -->}ABC\sim <!--\; \sim \; -->\mathrm{△<!--\; △\; -->}DEF$$ and $$\mathrm{△<!--\; △\; -->}DEF\sim <!--\; \sim \; -->\mathrm{△<!--\; △\; -->}XYZ$$ then $$\mathrm{△<!--\; △\; -->}ABC\sim <!--\; \sim \; -->\mathrm{△<!--\; △\; -->}XYZ$$ by the __________. Choose from: a. scale factor, b. AA Similarity Postulate, c. SSS Similarity Theorem, d. SAS Similarity Theorem, e. Symmetric Property of Similarity, f. midsegment, g. dilation, h. enlargement, i. reduction, j. Transitive Property of Similarity.