Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

Pentagon $S P E A K$ is inscribed in triangle $N O W$ such that $S$ and $P$ lie on segment $N O, K$ and $A$ lie on segment $N W$ , and $E$ lies on segment $O W$ . Suppose that $N S=S P=P O$ and $N K=K A=A W$ . Given that $E P=E K=5$ and $E A=E S=6$ , compute $O W$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Note that $[E S K]=[E P A]$ , since one has half the base but double the height. Since the sides are the same, we must have $\sin \angle S E K=\sin \angle P E A$ , so $\angle S E K+\angle P E A=180^{\circ}$ . Let $O W=3 x$ , so $S K=x$ and $P A=2 x$ . Then by the law of cosines $\begin{aligned} x^{2} & =61-60 \cos \angle S E K \\ 4 x^{2} & =61-60 \cos \angle P E A \end{aligned}$ Summing these two gives $5 x^{2}=122$ , since $\cos \angle S E K=-\cos \angle P E A$ . Then $x=\sqrt{\frac{122}{5}}$ , which means $3 x=\frac{3 \sqrt{610}}{5}$ .

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