Expert Help with High School Geometry

Square of the angle bisector in a triangle.
Prove that the square of the length, 'd' of the angle bisector AL of $\mathrm{\angle <!--\; \angle \; -->}A$ meeting BC at L in a $\mathrm{\Delta <!--\; \Delta \; -->}ABC$ is $bc(1-<!--\; -\; -->{\left(\frac{a}{b+c}\right)}^2).$

Distance between (4,3,1) and (0,6,0)?

How do you find the distance and midpoint between the two points. (4, -6) (-2, 8)?

Suppose that a quality control engineer wishes to test the durability of a child’s toy, the Widgetron. In order to do so, she repeatedly drops the Widgetrons from a height of one metre until the toy breaks on impact. The number of drops until breakage, call it y, is recorded. We will assume that the probability that the toy will not break stays constant from drop to drop, and this probability is independent of the number of times it has been dropped. If we define the random variable, Y, to represent the number of drops until the toy breaks, an appropriate probability model for the random variable Y would be a Geometric model with parameter $\mathrm{\theta <!--\; \theta \; -->}$.
How is $\mathrm{\theta <!--\; \theta \; -->}$ interpreted in the context of this model?

What is the distance between (-7,5) and (2,12)?

The distance shown on a map is 1/500 of the actual distance. If the distance shown on map is 1/5 foot, how long is the actual distance in feet?

The diameter of a Ferris wheel is 250 feet, and a ride once around took twenty minutes.
Give the speed of a rider on the wheel in feet per minute, and in MPH if there are 5280 feet in 1 mile.

Finding the volume in a cylinder that is intersected by a plane
I have a homework question that goes as follows.
Let $V=\{(x,y,z):x^2+y^2\le <!--\; \le \; -->4\text{ and }0\le <!--\; \le \; -->z\le <!--\; \le \; -->4\}$ be a cylinder and let P be the plane through (4, 0, 2), (0, 4, 2), and (-4, -4, 4). Compute the volume of C below the plane P.
So I have these points and I set up my $a(x-<!--\; -\; -->x_0)+b(y-<!--\; -\; -->y_0)+c(z-<!--\; -\; -->z_0)=0$ where $(x_0,y_0,z_0)$ is a point in the plane and ⟨a, b, c⟩ is perpendicular.
I then got two vectors that go between two points, namely:
$PQ=<(0-<!--\; -\; -->4,(4-<!--\; -\; -->0),(2,-<!--\; -\; -->2)>=<-<!--\; -\; -->4,4,0>$ and $PR=<(-<!--\; -\; -->4-<!--\; -\; -->4),(-<!--\; -\; -->4-<!--\; -\; -->0),(4,-<!--\; -\; -->2)>=<-<!--\; -\; -->8,-<!--\; -\; -->4,2>$
I then get the cross product of them to get $<8,8,48>$ which is the coefficient in my equation of the plane:
$8(x-<!--\; -\; -->4)+8(y-<!--\; -\; -->0)+48(z-<!--\; -\; -->2)=0$
I think that this is the equation of my plane. Is it?

Recursive proof of expectation of geometric distribution?
If $T\sim <!--\; \sim \; -->\mathsf{G}\mathsf{e}\mathsf{o}(p)$ ($0<p<1$) then $\mathbf{E}T=1/p$ is well known.
Suppose that I toss a sequence of coins that come up heads with probability p. On average, how many coins do I have to toss before I see a heads?

$10$points lie in a plane,of which $4$ points are collinear.Barring these $4$ no three of the $10$ points are collinear.How many distinct quadrilaterals can be drawn?

Probability of 3 numbers chosen between 0 and 10 being within 1 of each other
I was trying to find the probability that 3 real numbers uniformly chosen from 0 to 10 are within 1 of each other (the largest number minus the smallest number is at most 1).
I tried using geometric probability, and I got a region that looks like what you would get if you put one vertex of a unit cube at the origin (and orient the edges such that they line up with the positive axes), and move the corner at the origin from (0,0,0) to (9,9,9) (is this region correct?). I calculated the volume by finding the area of the one of the faces of the cube (which is 1), and multiplying it by the distance it moved (which is 9). Since there are three of these squares, the volume is 27. I then add this to the volume of the cube, which results in 28. The total volume is ${10}^3=1000$, so the probability is $28/1000=7/250$.

How do you go from the first step where the summation of $(1-<!--\; -\; -->p{)}^{k-<!--\; -\; -->1}$ transform to the fraction
$\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}p(1-<!--\; -\; -->p{)}^{k-<!--\; -\; -->1}=p\frac{1-<!--\; -\; -->(1-<!--\; -\; -->p{)}^n}{1-<!--\; -\; -->(1-<!--\; -\; -->p)}$
How to simplify the summation?

Find angle of isosceles triangle
Let ABC an isosceles triangle ($AB=AC$) and the bisector from B intersects AC at D such that $AD+BD=BC$. Find the angle A.

Distance between (-12,4) and (-10,-5)?

Distance between (0,0,8) and (3,6,2)?

Determine whether the congruence is true or false.
$$11\equiv <!--\; \equiv \; -->15mod4$$

Use the property to copy and complete the statement. Reflexive Property of congruence: ? $$\cong <!--\; \cong \; -->\stackrel{¯<!--\; ¯\; -->}{SE}$$

Are there two non-congruent quadrilaterals with same sets of sides and angles?

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