Expert Help with High School Geometry

projection of a non-zero mean Gaussian vector into a Ball
Let d denote the dimension, ${\mathbf{B}}_d$ denote the ball of radius one in ${\mathbb{R}}^d$. For $x\in <!--\; \in \; -->{\mathbb{R}}^d$ let ${\mathrm{\Pi <!--\; \Pi \; -->}}_{{\mathbf{B}}_d}(x)=\frac{x}{max\{1,\Vert <!--\; \Vert \; -->x{\Vert <!--\; \Vert \; -->}_2\}}$.
Consider a fixed vector $x\in <!--\; \in \; -->{\mathbb{R}}^d$ with $\Vert <!--\; \Vert \; -->x{\Vert <!--\; \Vert \; -->}_2=1$.
I am interested in understanding the distribution of ${\mathrm{\Pi <!--\; \Pi \; -->}}_{{\mathbf{B}}_d}(x+\mathrm{\xi <!--\; \xi \; -->})$ where $\mathrm{\xi <!--\; \xi \; -->}\sim <!--\; \sim \; -->\mathcal{N}(0,{\mathrm{\sigma <!--\; \sigma \; -->}}^2{I}_d)$ is a isotropic Gaussian vector with zero mean.
(Finding the exact distribution might be challenging, however, I am interested in any ideas how to play with this random variable.)

So I know the length L of the curve $y=\sqrt{R^2-<!--\; -\; -->x^2}$ from $x=0$ to $x=a$ where $|a|<R$ is given by:
$L={\int <!--\; \int \; -->}_0^a\frac{R}{\sqrt{R^2-<!--\; -\; -->x^2}}dx$
Now I must set up the arc length integral and simplify it so that it is in the form listed above.
$L={\int <!--\; \int \; -->}_0^a\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx$
and
$\frac{dy}{dx}=-<!--\; -\; -->\frac{x}{\sqrt{R^2-<!--\; -\; -->x^2}}$
${\left(\frac{dy}{dx}\right)}^2=\frac{x^2}{R^2-<!--\; -\; -->x^2}$
so
$L={\int <!--\; \int \; -->}_0^a\sqrt{1+\frac{x^2}{R^2-<!--\; -\; -->x^2}}dx$
I am unsure where to go from here to simplify into the first integral, any help would be greatly appreciated. Thanks

Give a direct proof that the set $\{x|{\mathrm{\Phi <!--\; \Phi \; -->}}_x(1)\downarrow <!--\; \downarrow \; -->\}$ (which is a set of program numbers that halt on input $1$) is not recursive.

Finding a volume of a region defined by $|x-<!--\; -\; -->y+z|+|y-<!--\; -\; -->z+x|+|z-<!--\; -\; -->x+y|=1$.
I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.

Distance between (4,4,2) and (5,6,4)?

Find constant k with the probability mass function of a geometric series
Ok so I'm having some trouble getting this done. I think I've got the answer but I'm not really sure on how I could verify that the result I got is actually the correct answer.
So here's what the exercise says:
$f(x)=k(\frac{1}{5}{)}^{2x+3}$
where a discrete random variable $X=0,1,2,3,....$
So I know this is a function for a geometric series given that the range for X is infinite.
Here's what I did:
1. $\underset{i=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}k(\frac{1}{5}{)}^{2x+3}=1$
2. $\frac{k}{125}\underset{i=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}(\frac{1}{5}{)}^{2x}=1$
3. $\frac{k}{125}(\frac{1}{1-<!--\; -\; -->\frac{1}{5}})=1$
4. $\frac{k}{125}(\frac{5}{4})=1$
5. $\frac{k}{100}=1$
6. $k=100$
So how do I know that is ok, I thought about using limits but I'm not sure if that is correct.

The endpoints of AB are A (-4,8) and B (12,-4). How do you find the coordinates of P if P lies on AB and is 3/8 the distance from A to B?

Average angle between two randomly chosen vectors in a unit square
Consider two randomly chosen vectors (a,b) and (c,d) within the unit square, where a,b,c, and d are chosen uniformly from [0,1]. What is the expected angle between the vectors?

Linda is making a quilt and she has determined she needs 1238 square inches of orange fabric and 1399 square inches of green. How many square yards of each material will she need? Round your answers up to the nearest quarter yard.
The orange fabric: ? square yards
The green fabric: ? square yards
3. How many total yards of fabric will she have to buy?
square yards =

Distance between (4,3,1) and (3,4,1)?

What is the distance between (7,3,-6) and (-8,1,-1)?

Geometric or binomial distribution?
A monkey is sitting at a simplified keyboard that only includes the keys "a", "b", and "c". The monkey presses the keys at random. Let X be the number of keys pressed until the money has passed all the different keys at least once. For example, if the monkey typed "accaacbcaaac.." then X would equal 7 whereas if the money typed "cbaccaabbcab.." then X would equal 3.
a.) What is the probability $X\ge <!--\; \ge \; -->10$?
b.) Prove that for an random variable Z taking values in the range {1,2,3,...}, $E(Z)=$ Summation from $i=1$ to infinity of $P(Z\ge <!--\; \ge \; -->i)$.
c.) What's the expected value of X?
First, is this a binomial distribution or a geometric distribution? I believe it is a binomial but my other friends says that it is geometric. As for the questions above, for a can I just do $1-<!--\; -\; -->P(X=9)\text{or}1-<!--\; -\; -->P(X<9)$, but I don't know how I will calculate $X<9$, I would know how to calculate $P(X=9)$, I don't know how to do b or c.

Distance between (3,2) and (5,6)?

Distance between (10,5,-2) and (12,11,5)?

Angle chasing to show three points are collinear.
Let ABC be an acute triangle with circumcenter O and let K be such that KA is tangent to the circumcircle of $\mathrm{△<!--\; △\; -->}ABC$ and $\mathrm{\angle <!--\; \angle \; -->}KCB={90}^{\circ <!--\; \circ \; -->}$. Point D lies on BC such that KD||AB. Show that DO passes through A.

Solution of triangles in Non-Euclidean geometry with restrictions
In triangle $ABC,AB>AC$. D is a point on AB such that $AD=AC$.
Prove that $\mathrm{\angle <!--\; \angle \; -->}ADC=\frac{\mathrm{\angle <!--\; \angle \; -->}B+\mathrm{\angle <!--\; \angle \; -->}C}{2}$ and $\mathrm{\angle <!--\; \angle \; -->}BCD=\frac{\mathrm{\angle <!--\; \angle \; -->}C-<!--\; -\; -->\mathrm{\angle <!--\; \angle \; -->}B}{2}$.
Solving this problem in Euclidean geometry is very easy. But how can it be solved with the following restrictions?
1) Parallel postulate (i.e. properties of parallel lines ) cannot be used.
2) Theorems proved using properties of parallel lines cannot be used.
3) The problem has to be solved the way euclidean geometry problems are solved. Cartesian Geometry cannot be used.
If not solvable, why cannot be?

You have triangle $\mathrm{△<!--\; △\; -->}ABC$ which is orthogonal $\mathrm{\angle <!--\; \angle \; -->}C={90}^o$ with circle with center O inscribed in it. If KL is the diameter, KL∥AB, $KM\perp <!--\; \perp \; -->AB$ and $LN\perp <!--\; \perp \; -->AB$. Find angle $\mathrm{\angle <!--\; \angle \; -->}MCN$.

Geometric Probability problem in 3 unknowns
Suppose we have to choose 3 numbers, a,b and c such that $a,b,c\in <!--\; \in \; -->[0,1]$. The numbers are randomply distributed in an uniform distribution between 0 and 1. Then I've been asked to find the probability of $a+b><!--\; >\; -->2c$.
I'm not being able to represent this in a geometrical way. I've tried fixing the value of c and then figuring out where a and b would lie on a line segment, but that got me nowhere.
How should I approach this particular type of problem?

Question about the angle $\mathrm{\theta <!--\; \theta \; -->}$ Green's theorem
Given $Q(x,y)=x\cdot <!--\; \cdot \; -->y^2+y\ln <!--\; \; -->(x+\sqrt{x^2+y^2})$ and $P(x,y)=\sqrt{x^2+y^2}$ calculate the integral ${\int <!--\; \int \; -->}_{C}Pdx+Qdy$ while $C=(x-<!--\; -\; -->1{)}^2+(y-<!--\; -\; -->1{)}^2=1$
why $\mathrm{\theta <!--\; \theta \; -->}$ should be until $2\mathrm{\pi <!--\; \pi \; -->}$?