Problem 150

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Problem 150: (2006 AMC 12B, 23)

Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$, such that $PA=11$, $PB=7$, and $PC=6$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

$\mathrm{(A)}\ 85 \qquad$ $ \mathrm{(B)}\ 91 \qquad$ $ \mathrm{(C)}\ 108 \qquad$ $ \mathrm{(D)}\ 121 \qquad$ $ \mathrm{(E)}\ 127$

Solution

Exercise 150: Solution

Isosceles $\triangle ABC$ has a right angle at $C$. Point $P$ is inside $\triangle ABC$ such that $PA=9, PB=7, PC=4$. Legs $\overline{AC}$ and $\overline{BC}$ have length $s=\sqrt{a+b\sqrt2}$, where $a,b$ are positive integers. What is $a+b$?

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