Improve Your Geometry Skills with Practice Problems

Finding a triangle angle based on side length equality
Consider the triangle ABC with angle A being 70 degrees, and the side lengths satisfying:
$B{C}^2=AC(AB+AC)$
Is there any intuitive way of finding the measure of angle B?

Pythagoras: Get b when only a and angle $\mathrm{\alpha <!--\; \alpha \; -->}$ are given
Given the Pythagoras Theorem: $a^2+b^2=c^2$
Is there a way to get the value of b when we only have a value for a and the angle $\mathrm{\alpha <!--\; \alpha \; -->}$?

How do you find the distance between (2.3, 4.5) and (–3.4, –5.2)?

Distance between (8,2) and (-5,2)?

Existence of Right Angle in Hilbert Axioms
Hilbert, in Foundations of Geometry briefly mentions that the existence of right angles is a corollary to the supplementary angle theorem. (i.e. If two angles are congruent, then their supplementary angles are congruent).
How does existence of right angles follow from this?

5 independent traffic lights, how many is car expected to pass without getting stopped
There are 5 independent traffic lights, each with chance of stopping a car equal to 0.6. How many traffic lights is the car expected to pass before being stopped - what is E(X)?

I am trying to compute this integral ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}\frac{x}{1+x^6}dx$ using the residue theorem. To do so, I am integrating $f(z)=\frac{z}{1+z^6}$ in the frontier of this sector of a circle: $\{z:|z|<R,0<arg(z)<\mathrm{\pi <!--\; \pi \; -->}/3\}$.

I kwo how to deal with the integrals over the horizontal segment of the sector and over the arc of the circle. My problem is the "diagonal segment". When I parametrize it, I do not get something easy to integrate. How could I approach this?

(The path of integration was suggested in my book, so I do not think that is the problem).

What is the distance between (8,0,6) and (1,-4,5)?

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document Find the sum of a probability of dice roll that is prime.
Consider rolling n fair dice. Let p(n) be the probability that the product of the faces is prime when you roll n dice. For example, when = $\frac{1}{2}$
When
When $\frac{1}{24}$. Find the infinite sum of p(n) (when n starts at 0) (Hint: Consider differentiating both sides of the infinite geometric series: infinite sum of when )
I can differentiate the two sides of the geometric series but I'm lost regarding what to do after that. I don't fully understand the question.

Chase is making bookmarks. He wonts to make no more than 12 bookmarks and needs 4.25 inches of fabric for each bookmark. Determine the amount of fabric the needs to buy. Then interpret the solution.

Characterization of the geometric distribution
X,Y are i.i.d. random variables with mean $\mathrm{\mu <!--\; \mu \; -->}$, and taking values in {0,1,2,...}.Suppose for all $m\ge <!--\; \ge \; -->0$, $P(X=k|X+Y=m)=\frac{1}{m+1}$, $k=0,1,...m$. Find the distribution of X in terms of $\mathrm{\mu <!--\; \mu \; -->}$.

Distance between (4,4) and (-4,11)?

Finding the volume of a solid of revolution about the x-axis
Let R be the area laying beneath the curve $f(x)=8-<!--\; -\; -->x^2$ and above the line $y=7$. Find the volume of the solid of revolution which is created when R is revolved around the x-axis.
I graphed the functions and found out I have to integrate from -1 to 1. I want to use the disc method, but I don't know how to only get the discs with radii higher than $y=7$ and lower than f(x). I thought it would be logical if the radii of the discs would be $8-<!--\; -\; -->x^2-<!--\; -\; -->7=1-<!--\; -\; -->x^2$, but this approach gives me the wrong answer when i plug it into the formula.

Volume of Solid Enclosed by an Equation
I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple integral thing once and for all:
Compute the volume of the solid enclosed by
1. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1,x=0,y=o,z=0$
2. $x^2+y^2-<!--\; -\; -->2ax=0,z=0,x^2+y^2=z^2$

Volume between under parabaloid
Suppose I want to find the volume between $z=2x^2+3y^2$ and $z=4$. Is there a way finding that with a double integral? I tried to use $4-<!--\; -\; -->2x^2-<!--\; -\; -->3y^2$ inside the integral and then convert it to polar coordinates...

Distance between (8,6,2) and (3,4,1)?

Distance between (9,1) and (5,2)?

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

$\begin{array}{l}\text{ Theorem }2-<!--\; -\; -->17:\text{ Let }\left\{a_n\right\}\text{ be a bounded sequence of real numbers. Then: }\\ \begin{array}{l}\text{ (a) }lim sup{a}_n=L\text{ if and only if, for any }\mathrm{\epsilon <!--\; \epsilon \; -->}>0,\text{ there are infinitely many terms of }\\ \left\{a_n\right\}\text{ in }(L-<!--\; -\; -->\mathrm{\epsilon <!--\; \epsilon \; -->},L+\mathrm{\epsilon <!--\; \epsilon \; -->})\text{ but only finitely many terms of }\left\{a_n\right\}\text{ with }{a}_n>L+\mathrm{\epsilon <!--\; \epsilon \; -->}.\end{array}\end{array}$
Proves directly that $lim sup\{a_n+b_n\}\le <!--\; \le \; -->lim sup\{a_n\}+lim sup\{b_n\}$

Probability about a geometric distribution
If no-one obtains "head", the game continues with the same probabilities as before." If that is the case, why does that affect the probability recursively? Could someone explain why we add the case where no one wins to the probability, and why we multiply it by p?