Expert Help with High School Geometry

I chose a random number from 1 to 6. Afterwards, I roll a die till I get a result that is even or higher than my chosen number. What is the E(x) of the number of times I throw the die?
So I thought it's Geometric distribution will "success" where "success" is to get my number. So first the probability of choosing a random number in the die is 1/6. Now I can't configure what is the probability to get something even or higher? Since I chose the number randomly. and afterthat, I just need to divide 1 by the probability I get (like the geometric E(x) formula)?

What is the distance between (-1,14,1) and (-1,4,-2)?

Geometric probabilities with rectangle
One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than 1.15?

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

Distance between (8,2) and (4,7)?

What is the distance between (9,0,1) and (1,-4,-2)?

Distance between (1,6) and (2,5)?

Finding volume of solid using shell method.
Rotating the region bounded by $y=x^3,y=0,x=2$ around the line $y=8$.
I just want to double check if my initial formula is on the right track.
$V_y={\int <!--\; \int \; -->}_0^8(8-<!--\; -\; -->y)(y^{\frac{1}{3}})dy$

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

Let ABC be a triangle such that $\mathrm{\angle <!--\; \angle \; -->}ACB$ is acute. Suppose that D is an interior point of the triangle ABC such that $\mathrm{\measuredangle <!--\; \measuredangle \; -->}ADB=\mathrm{\measuredangle <!--\; \measuredangle \; -->}ACB+\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}$ and $AC\cdot <!--\; \cdot \; -->BD=AD\cdot <!--\; \cdot \; -->BC.$.
a) Find $\frac{AB\cdot <!--\; \cdot \; -->CD}{AC\cdot <!--\; \cdot \; -->BD}.$.
(b) Show that the tangents at C to the circumcicle of the triangle ACD and the circumcircle of the triangle BCD are perpendicular.

Finding the volume of a region rotated about the y-axis.
I'm having trouble trying to find the volume of the region formed by $y=x^2-<!--\; -\; -->7x+10$ and $y=x+3$ rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up with the integral for it.
Any help showing me how to get the integral I should use would be a huge help.

Can all polygons outside of the largest inscribed rectangle in a convex polygon be concave
Let C be a convex polygon and R the largest (by area) rectangle lying within C. Does there exist a convex polygon C such that when R is removed, all remaining polygons are concave? In other words: Does there exist a polygon C such that all parts of C∖R are concave?

What is the distance between (-1,7) and (2,12)?

Distance between (9,2,0) and (3,4,1)?

Find a harmonic function on $|z|<1$ with value 1 on an arc of $|z|=1$ and zero on the rest.
Of course, if allow the arc with value 1 to be the whole $|z|=1$ circle it would be trivial.
Without that, how to construct such a function, I'm a bit lost. I suppose if I can find a holomorphic function $f(z)$ whose real part full fill that it shall work, but playing around with the normal functions such as $e^x$, $\sin <!--\; \; -->x$ etc, I can't find one with such property.

Finding the maximum possible volume of a tetrahedron
Suppose I have three vectors $\stackrel{¯<!--\; ¯\; -->}{x}=a\stackrel{^<!--\; ^\; -->}{i}+b\stackrel{^<!--\; ^\; -->}{j}+c\stackrel{^<!--\; ^\; -->}{k}$, $\stackrel{¯<!--\; ¯\; -->}{y}=b\stackrel{^<!--\; ^\; -->}{i}+c\stackrel{^<!--\; ^\; -->}{j}+a\stackrel{^<!--\; ^\; -->}{k}$, $\stackrel{¯<!--\; ¯\; -->}{z}=c\stackrel{^<!--\; ^\; -->}{i}+a\stackrel{^<!--\; ^\; -->}{j}+b\stackrel{^<!--\; ^\; -->}{k}$, where we can choose a, b, c from the set {1,2,3,....13}. We have to find the probability of the tetrahedron formed by $\stackrel{¯<!--\; ¯\; -->}{x}$, $\stackrel{¯<!--\; ¯\; -->}{y}$, $\stackrel{¯<!--\; ¯\; -->}{z}$ having maximum volume. How should I maximise the determinant for finding the maximum possible volume of the resulting tetrahedron? Can someone help me figure this out?

Probability That the Sides of an Isosceles Triangle is an Equilateral Triangle
The sides of an isosceles triangle are whole numbers, and its perimeter is 30 units. What is the probability that the triangle is equilateral?

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

Asymptotic behavior of the geometric sequence $U_{n+1}=Q{U}_n$ in a probability problem
We know the asymptotic behavior of the scalar geometric sequence $u_{n+1}=q{u}_n$ with respect to the values of q. That is, $|q|<1$ implies $\underset{n}{lim}{u}_n=0$, $q=1$ implies $u_n$ is constant, etc.
Now I am just asking myself the same question about the geometric sequence $U_{n+1}=Q{U}_n$, where Q is a $2\times <!--\; \times \; -->2$ matrix and $U_n$ is a two dimensional vector.

Is there an easier way to solve a "Find the locus" problem?
Note: I am not concerned with the accuracy of my solution so you don't need to redo any of my calculations.
Original question: Suppose ABC is an equilateral triangular lamina of side length unity, resting in two-dimensions. If A and B were constrained to move on the x- and y-axis respectively, then what is the locus of the centre of the triangle?
My solution: Let the vertices of $\mathrm{\Delta <!--\; \Delta \; -->}ABC$ be $A=(x_0,0)$, $B=(0,y_0)$ and $C=(x,y)$. Let the centre of the triangle be $D=(X,Y)$. We wish to find the locus of D under the constraints:
$\begin{array}{rl}{x}_0^2+y_0^2& =0\\ (x-<!--\; -\; -->x_0{)}^2+y^2& =0\\ x^2+(y-<!--\; -\; -->y_0{)}^2& =0\end{array}$
Since D is the centre of the triangle, we know
$X=\frac{x+x_0}{3}\text{and}Y=\frac{y+y_0}{3}$
Parametrizing: Let $\mathrm{\theta <!--\; \theta \; -->}$ be the angle that the edge AB makes with the x-axis. We use (theta) as our parameter of choice to write, $\begin{array}{rl}{x}_0& =\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}\\ y_0& =\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}\end{array}$
for $\mathrm{\theta <!--\; \theta \; -->}\in <!--\; \in \; -->[0,2\mathrm{\pi <!--\; \pi \; -->})$
This implies, $(x-<!--\; -\; -->\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}{)}^2+y^2=0$
We can parametrize this with another parameter $\mathrm{\varphi <!--\; \varphi \; -->}\in <!--\; \in \; -->[0,2\mathrm{\pi <!--\; \pi \; -->})$ as
$\begin{array}{rl}x& =\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+\cos <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}\\ y& =\sin <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}\end{array}$
Plugging these back into third equation gives
$(\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+\cos <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}{)}^2+(\sin <!--\; \; -->\mathrm{\varphi <!--\; \varphi \; -->}-<!--\; -\; -->\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}{)}^2=1$
After lots of algebraic manipulations...
$\begin{array}{rlrl}X& =\frac{1}{2}\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}\pm <!--\; \pm \; -->\frac{1}{2\sqrt{3}}\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}& Y& =\frac{1}{2}\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}\pm <!--\; \pm \; -->\frac{1}{2\sqrt{3}}\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}\end{array}$