Expert Help with High School Geometry

Negative Binomial Vs Geometric
So I am trying to get the difference between these two distributions. I think I understand them but the negative binomial has me a bit confused.
The Geometric is the probability of some amount of successes before the first failure.
The Negative binomial is the portability of some amount of successes before a specified number of failures? for example the number of successes before the 8th failure?

Qeustion about inscribed angle in a circle.

I have a circle as shown in the figure. In my text book, the angle BAD is the sum of angles ACD and angle ADC. There might be a theorem that shows this in the book but I cannot find it. Why is this true?

Finding the bounds to solve for the volume using cylindrical coordinates
My goal is to set up the triple integral that will solve for the volume inside the sphere $x^2+y^2+z^2=2z$ and above the paraboloid $x^2+y^2=z$ using cylindrical coordinates.
Upon solving, I know that the intersection of the sphere and the paraboloid is $r=1$ and the bounds for $\mathrm{\theta <!--\; \theta \; -->}$ is $0\le <!--\; \le \; -->\mathrm{\theta <!--\; \theta \; -->}\le <!--\; \le \; -->2\mathrm{\pi <!--\; \pi \; -->}$.
However, I am uncertain for the bounds of the z variable. I solve for the value of z in converted equation of the sphere $r^2+z^2=2z$. By quadratic formula, I got the value of $z=\frac{2\pm <!--\; \pm \; -->\sqrt{4-<!--\; -\; -->4r^2}}{2}$.
Should I only include $z=\frac{1}{2}(2+\sqrt{4-<!--\; -\; -->4r^2})$ as one of the bounds in the z variable? Or there will be a split integral? My yielding triple integral in this case if I only include $z=\frac{1}{2}(2+\sqrt{4-<!--\; -\; -->4r^2})$ would be ${\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}{\int <!--\; \int \; -->}_0^1{\int <!--\; \int \; -->}_{r^2}^{\frac{1}{2}(2+\sqrt{4-<!--\; -\; -->4r^2})}rdzdrd\mathrm{\theta <!--\; \theta \; -->}$

Which triangle congruence theorem is most closely related to the Leg-Angle (LA) Congruence Theorem? Explain.

The midpoint of a chord of length $2a$ is at a distance d from the midpoint of the minor arc it cuts out from the circle. Show that the diameter of the circle is $\frac{a^2+d^2}{d}$ .
I know I have to find similar triangles, I cannot see them...

Sum and average length of chords
From a specific point A on a circle's circumference, am drawing various chords. a) what is the sum of length of such chords and b) what is the average length of such chords?
Assuming a radius r, I'm trying to solve a) in the below manner:
One specific chord length $=2r\cos <!--\; \; -->(\mathrm{\Theta <!--\; \Theta \; -->})$
To get the sum, I'm integrating over $-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}/2to\mathrm{\pi <!--\; \pi \; -->}/2(2r\cos <!--\; \; -->(\mathrm{\Theta <!--\; \Theta \; -->})d\mathrm{\Theta <!--\; \Theta \; -->})$
But i don't think this is right. Please point me in the correct direction.

Maximum Number of Points That Can be Packed in a Rectangle?
Show that you can place at most 16 points in a rectangle of size $\frac{D}{2}\times <!--\; \times \; -->D$ such that no triangle formed from these points has a perimeter smaller than D.
How does one proceed to prove such a statement? I know that for the closest pair of points algorithm one can prove that a similar rectangle cannot contain more than 6 points and the proof for this can be arrived at using the intuition of placing circles as shown here. But what sort of argument can we make for triangles?

approximate probability of geometric distribution using CLT
For $i\ge 1$, let $X_i\sim G_1/2$ be distributed Geometrically with parameter 1/2. Define $Y_n=\frac{1}{\sqrt{n}}\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(X_i-<!--\; -\; -->2)$
Approximate $P(-1\le Y_n\le 2)$ with large enough n. Hint, note that $Y_n$ is not "properly" normalized.

Negative binomial distribution vs geometric distributiion
How can this question be geometric and also negative binomial. In order to join a group Peter needs 9 invitations.The probability that he receives an invitation on any day is 0.8, independent of other days.
He joins as soon as he receives his 9 th invitation. Given that he joined on the 14 th day, find the probability that he receives his first invite on the first day.

What is the distance between (5,-10) and (4,-2)?

Non-convex even-sided polygons whose vertices lie on a circle
Given $2n$ evenly spaced points on a circle, the opposite sides in the convex polygon (formed by these points) are parallel. If we remove the requirement of convexity, I get degenerate polygons that always have parallel sides. How does one show that a non-convex polygon (possibly degenerate) formed by an even number of evenly spaced points on a circle always has a pair of parallel edges?

Discrete math: proofs
For any three integers $𝒙,𝒚$, and $𝒛$, if $𝒚$ is divisible by $𝒙$ and $𝒛$ is divisible by $𝒚$, then $𝒛$ is divisible by $𝒙$.

Distance between (5,6,4) and (2,-3,1)?

Calculating The Vertex Of a Square That Circumscribed Ellipse
How can I find the vertex of a square that circumscribed the ellipse $\frac{x^2}{9}+y^2=1$?
I tried to mark the vertex at (u,v), (-u,v), (u,-v), (-u,-v) and use the equation to calculate the tangent lines to the ellipse by the vertex points, but I don't know how to continue.

One exterior angle of a regular polygon measures 36. What is the sum of the polygon's interior angle measures?

What is the probability that the quadratic equation $a{x}^2+x+1=0$ has two real roots?
A number a is chosen at random within the interval (-1,1). What is the probability that the quadratic equation $a{x}^2+x+1=0$ has two real roots?
For it to have its real roots, we must guarantee that $1-<!--\; -\; -->4a\ge <!--\; \ge \; -->0$, or $a\le <!--\; \le \; -->\frac{1}{4}$.

Hausdorff Distance Between Convex Polygons
I'm interested in calculating the Hausdorff Distance between 2 polygons (specifically quadrilaterals which are almost rectangles) defined by their vertices. They may overlap.
Recall $d_H(A,B)=max(d(A,B),d(B,A))$ where d is the Hausdorff semi-metric $d(A,B)=\underset{a\in <!--\; \in \; -->A}{sup}\underset{b\in <!--\; \in \; -->B}{inf}d(a,b)$.
Is it true that, given a finite disjoint covering of A, $\{A_i\}$, $d(A,B)=max\{d(A_i,B)\}$? A corollary of which is that $d(A,B)=d(A\setminus <!--\; \setminus \; -->B,B)$.

Derivation of a conditional probability mass function which involves Geometric and Gamma random variables
Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of X given that $N=n$ is the gamma distribution with parameters n and $\mathrm{\lambda <!--\; \lambda \; -->}.$ Find the conditional probability mass function of N given that $X=x.$.

Distance between (-12,4) and (9,3)?

Name the property of equality or congruence that justifies going from the first statement to the second statement.
5x=20
x=4