Expert Help with High School Geometry

Finding the volume of a ${\mathbb{R}}^3$ triangle
I have a triangle ABC defined with the points $A=(2.4,-<!--\; -\; -->5.4,6)$, $B=(0,1.1,3.2)$, $C=(-<!--\; -\; -->7.6,3,0)$.
And I'm asked to find the volume of the solid of ${\mathbb{R}}^3$ given by the points between the plane that is made by the points of the triangle and the plaze $z=0$ (The same points of the triangle with all the $z=0$)
I think I should use integrals but I don't know from which regions to do the triple integration.

Multivariable calculus double integration volume question
Use a double integral to find the volume of the solid bounded by graphs of the equations given by:
$\begin{array}{rl}z=x{y}^2,\text{ where: }& z>0\\ & x>0\\ & 5x<y<2\end{array}$
My problem is finding the limits of integration. I know my $f(x,y)$... my guess is my y integration is going from 5x going to 2. But how do I find the x limitations?

Distance between (-7,8) and (3,7)?

Finding the volume of a rotating cross section.
I have to find the volume of a bounded region as it rotates around the Y-axis. My points are $y=6;x=-<!--\; -\; -->2;x=2$; and $y=0$. What would be the steps to solving this and the solution?

Distance between (6,8,2) and (3,6,2)?

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

Joint Probability of Independent Geometric Random Variables
Let $X,Y\sim <!--\; \sim \; -->G(p)$ be independent Geometric random variables ($p\in <!--\; \in \; -->(0,1)$). Show that $P(X=Y)=p/(2-p)$.

A system used to read electric meters automatically requires the use of a 64-bit computer message. Occasionally random interference causes a digit reversal resulting in a transmission error. assume that the probability of a digit reversal for each bit is 1/2000. Let X denote the number of transmission errors per 64-bit message sent. Is X geometric?

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

Are there other similar discrete probability distributions where the probability of random variable Y taking on value n is given by e.g. the n-th term in the series expansion of a function divided by the closed form of the function?
Conversely, does this mean that every function with a series expansion has a corresponding discrete probability distribution? Which are the more commonly known functions and their corresponding probability distributions?

"Inverse" of the inscribed angle theorem.
While doing geometry problem I encountered something I would call "inverse" of inscribed angle theorem. At first I wanted to state it like this:
Given $\mathrm{△<!--\; △\; -->}$ ABC and point O such that $\mathrm{\angle <!--\; \angle \; -->}$ BOC = 2 $\mathrm{\angle <!--\; \angle \; -->}$ BAC it implies that O is the centre of the circle described on the $\mathrm{△<!--\; △\; -->}$ ABC
But of course it is easy to show that this statement doesn't hold. But on my native language forum I found that this theorem should hold:
Given $\mathrm{△<!--\; △\; -->}$ ABC and $\mathrm{\angle <!--\; \angle \; -->}$ A = $\mathrm{\alpha <!--\; \alpha \; -->}$ and point O lying on the perpendicular bisector of the segment BC with $\mathrm{\angle <!--\; \angle \; -->}$ BOC = 2 $\mathrm{\alpha <!--\; \alpha \; -->}$ and point O lying on the same side of line BC as point A then point O is the centre of the circle described on $\mathrm{△<!--\; △\; -->}$ ABC

Conditional Probability and Maximum values of random variables including a Geometric Random Variable
Let $X_1,X_2,...$ be i.i.d. random variables with the common CDF F , and suppose they are independent of N, a geometric random variable with parameter p. Let $M=max\{X_1,...,X_N\}$
(a) Find $Pr\{M\le <!--\; \le \; -->x\}$ by conditioning on N.
(b) Find $Pr\{M\le <!--\; \le \; -->x|N=1\}$.
(c) Find $PrM\le <!--\; \le \; -->x|N>1$.
Could anyone explain why in (c)
$Pr\{M\le x|N>1\}=F(x)Pr\{M\le <!--\; \le \; -->x\}$

Euclidean Version of Pappus's theorem
Let A, B, C, be points on a line l, and let A′, B′, C′ be points on a line m. Assume AC′∥A′C and B′C∥BC′. Show that AB′∥A′B.

Is there a congruence theorem that says that if three sides of two quadrilaterals are equal, then the two quadrilaterals are congruent?

Distance between (3,0) and (6,6)?

Let V be volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$.
Express V as a multiple integral, whose limits should be clearly stated. Hence calculate V.
Solution
I want to find out the limits of the multiple integral that's needed to calculate V.
I'm guessing that: $x=a-<!--\; -\; -->z$, $x=(y^2)/4a$.
$y=\pm <!--\; \pm \; -->\sqrt{4ax},z=0,z=a-<!--\; -\; -->x$
But it seems like I've used the upper plane equation twice?
Also, it would really help if we could compare our answer volumes to check that this is right from the start!

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

Distance between (17,−6) and (-1,24)?

Distance between (0,0,8) and (6,8,2)?

Two circles cut orthogonally in A and B. A diameter of one of the circles is drawn cutting the other circle in C and D. Prove that $BC\cdot <!--\; \cdot \; -->AD=AC\cdot <!--\; \cdot \; -->BD$.