# 2013 AIME II Problems/Problem 8

## Problem 8

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

## Solution

$[asy] import olympiad; import math; pair A,B,C,D,E,F; B=origin; C=(10,0); D=(12,-5); E=(10,-10); F=(0,-10); A=(-2, -5); draw(A--B);draw(B--C);draw(C--D);draw(D--E);draw(E--F);draw(F--A); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); draw(circumcircle(A, D, F)); label("A",A,W);label("B",B,NW);label("C",C,NE);label("D",D,W);label("E",E,SE);label("F",F,SW); [/asy]$

### Solution 1

Let us call the hexagon $ABCDEF$, where $AB=CD=DE=AF=22$, and $BC=EF=20$. We can just consider one half of the hexagon, $ABCD$, to make matters simpler. Draw a line from the center of the circle, $O$, to the midpoint of $BC$, $E$. Now, draw a line from $O$ to the midpoint of $AB$, $F$. Clearly, $\angle BEO=90^{\circ}$, because $BO=CO$, and $\angle BFO=90^{\circ}$, for similar reasons. Also notice that $\angle AOE=90^{\circ}$. Let us call $\angle BOF=\theta$. Therefore, $\angle AOB=2\theta$, and so $\angle BOE=90-2\theta$. Let us label the radius of the circle $r$. This means $$\sin{\theta}=\frac{BF}{r}=\frac{11}{r}$$ $$\sin{(90-2\theta)}=\frac{BE}{r}=\frac{10}{r}$$ Now we can use simple trigonometry to solve for $r$. Recall that $\sin{(90-\alpha)}=\cos(\alpha)$: That means $\sin{(90-2\theta)}=\cos{2\theta}=\frac{10}{r}$. Recall that $\cos{2\alpha}=1-2\sin^2{\alpha}$: That means $\cos{2\theta}=1-2\sin^2{\theta}=\frac{10}{r}$. Let $\sin{\theta}=x$. Substitute to get $x=\frac{11}{r}$ and $1-2x^2=\frac{10}{r}$ Now substitute the first equation into the second equation: $1-2\left(\frac{11}{r}\right)^2=\frac{10}{r}$ Multiplying both sides by $r^2$ and reordering gives us the quadratic $$r^2-10r-242=0$$ Using the quadratic equation to solve, we get that $r=5+\sqrt{267}$ (because $5-\sqrt{267}$ gives a negative value), so the answer is $5+267=\boxed{272}$.

### Solution 2

Using the trapezoid $ABCD$ mentioned above, draw an altitude of the trapezoid passing through point $B$ onto $AD$ at point $J$. Now, we can use the pythagorean theorem: $(22^2-(r-10)^2)+10^2=r^2$. Expanding and combining like terms gives us the quadratic $$r^2-10r-242=0$$ and solving for $r$ gives $r=5+\sqrt{267}$. So the solution is $5+267=\boxed{272}$.

### Solution 3

Join the diameter of the circle $AD$ and let the length be $d$. By Ptolemy's Theorem on trapezoid $ADEF$, $(AD)(EF) + (AF)(DE) = (AE)(DF)$. Since it is an isosceles trapezoid, both diagonals are equal. Let them be equal to $x$ each. Then

$$20d + 22^2 = x^2$$

Since $\angle AED$ is subtended by the diameter, it is right. Hence by the Pythagorean Theorem with right $\triangle AED$:

$$(AE)^2 + (ED)^2 = (AD)^2$$ $$x^2 + 22^2 = d^2$$

From the above equations, we have: $$x^2 = d^2 - 22^2 = 20d + 22^2$$ $$d^2 - 20d = 2\times22^2$$ $$d^2 - 20d + 100 = 968+100 = 1068$$ $$(d-10) = \sqrt{1068}$$ $$d = \sqrt{1068} + 10 = 2\times(\sqrt{267}+5)$$

Since the radius is half the diameter, it is $\sqrt{267}+5$, so the answer is $5+267 \Rightarrow \boxed{272}$.