Expert Help with High School Geometry

There are $25$ points on a plane of which $7$ are collinear . How many quadrilaterals can be formed from these points ?

Joint probability of geometric random variables
X and Y are geometric, independent random variables with parameters x and y respectively. Use their joint P.M.F. to compute an approximate summation for $P(X<Y)$.

EX of area of a circle
In a circle k(O,r) we have fixed point A, point B lands randomly on the circle. Find EX of the area of the triangle AOB
What I have tried: since the area of the triangle will be $\frac{1}{2}\ast <!--\; \ast \; -->r^2(sin(\mathrm{\alpha <!--\; \alpha \; -->}))$ where $\mathrm{\alpha <!--\; \alpha \; -->}$ is from 0 to 360 degrees then we have to find the integral ${\displaystyle {\int <!--\; \int \; -->}_0^{2\mathrm{\pi <!--\; \pi \; -->}}\frac{1}{2}\ast <!--\; \ast \; -->\mathrm{\alpha <!--\; \alpha \; -->}\ast <!--\; \ast \; -->r^2(sin(\mathrm{\alpha <!--\; \alpha \; -->}))d\mathrm{\alpha <!--\; \alpha \; -->}}$.

Geometric probability - line segment
On the segment [0,1] two numbers X, Y are being randomly selected. Find the probability that $X+Y\le <!--\; \le \; -->1$ and $X-Y>0.1$.

Apparent paradox in Geometric probability problem
Suppose we have N particles in a box. We can consider these particles to be distinguishable, and move about the box randomly, like molecules of gas in a box. Suppose, the volume of this box is V.
I've been asked to find the probability that at a certain time, a small sub-volume $V_0$ does not have a single particle inside it.

Given two iid geometric random variables with $p=\frac{1}{5}$, what is the probability that variates from them equate?

Why is the angle exactly 1/2 of one of the angles in the isosceles triangle
I'm sure there's a simple theorem that answers my question, but I'm unsure as to how to formulate the problem (not easy to describe geometry with words): It's a isosceles triangle where you draw a normal line on one of the "hypotenuse" sides, and then you make a right angled triangle out of that. My question is why is the 30 degree angle exactly 1/2 of the 60 degree angles we find in the isosceles triangle?

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

For each of the following congruences, find all integers N, with N > 1, that make the congruence true. $$a.23\equiv <!--\; \equiv \; -->13(modN).b.10\equiv <!--\; \equiv \; -->5(modN).c.6\equiv <!--\; \equiv \; -->60(modN).d.23\equiv <!--\; \equiv \; -->22(modN).$$

Probability of rolling a 1 before a 6 on a dice
what is the probability that I roll a "1" on a dice before rolling a "6". I do not know how to tackle this problem. I was thinking that this is a Geometric random variable but I do not know how to solve it.

Finding the volume of a solid enclosed by two cylinders and planes
Consider the solid E1 that is enclosed by the planes $z=0$ and $z=5$ and by the cylinders $X^2+y^2=9$ and $x^2+y^2=16$. Find the volume of E1. How to find the volume using two cylinders

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

Distance between (-65,8) and (-63,12)?

Let $a,b$ and $c$ be real numbers where $a>b$. Prove that if $ac\le <!--\; \le \; -->bc$, then $c\le <!--\; \le \; -->0$.

Distance between (-5,13,-14) and (-11,4,1)?

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

What is the negation of " $A\subseteq <!--\; \subseteq \; -->B$ "?

Is the sum of two independent geometric random variables with the same success probability a geometric random variable?

Finding the volume of a parallel piped
Given the points $P(-<!--\; -\; -->2,3,1),Q(2,-<!--\; -\; -->2,0),R(4,1,0)$.
Find the parallelepiped with edge OP,OQ,OR where O is the origin (0,0,0).
Volume for a parallelepiped is $V=A\times <!--\; \times \; -->B\cdot <!--\; \cdot \; -->C$.
So would the volume be $OP\cdot <!--\; \cdot \; -->(OQ\times <!--\; \times \; -->OR)$.
$<-<!--\; -\; -->2,3,1>\cdot <!--\; \cdot \; -->(<2,-<!--\; -\; -->2,0>\times <!--\; \times \; --><4,1,0>)$
$<2,3,1>\cdot <!--\; \cdot \; --><0,0,10>$
area parallelepiped is 10 but would this be correct?

Finding the volume of a solid bounded by a sphere and a paraboloid
I am working on a problem that requires me to find the volume of the solid bounded by the sphere $x^2+y^2+z^2=2$ and the paraboloid $x^2+y^2=z$. I know that to do this, I must use triple integration.
Thus far, I know that we need to reevaluate the given equations and put them into spherical coordinates in order to achieve our bounds. When we've gotten our bounds, it's just simple triple integration to computer the volume.
So, for our sphere, I said that $x^2+y^2+z^2=2$ is equivalent to $\mathrm{\rho <!--\; \rho \; -->}=sqrt(2)$.
Then, for our sphere, I get a little messed up. I'm trying to find $\mathrm{\varphi <!--\; \varphi \; -->}$ boundaries here, but I get the equation ${\mathrm{\rho <!--\; \rho \; -->}}^{2}si{n}^2\mathrm{\varphi <!--\; \varphi \; -->}=\mathrm{\rho <!--\; \rho \; -->}cos\mathrm{\varphi <!--\; \varphi \; -->}$. Then, the $\mathrm{\rho <!--\; \rho \; -->}$ can cancel out, making it $\mathrm{\rho <!--\; \rho \; -->}si{n}^2\mathrm{\varphi <!--\; \varphi \; -->}=cos\mathrm{\varphi <!--\; \varphi \; -->}$, but my issue is that the $\mathrm{\rho <!--\; \rho \; -->}$ doesn't cancel out completely so I can't get the bounds for $\mathrm{\varphi <!--\; \varphi \; -->}$.