8. In a right-angled triangle $A B C$ (with right angle at $A$ ) the bisectors of the acute angles intersect at point $P$. The distance from $P$ to the hypotenuse is $\sqrt{80000}$. What is the distance from $P$ to $A$ ?
SolUTION
400

Drop the perpendiculars from $P$ to $A B$, meeting at $M$; and from $P$ to $A C$, meeting at $N$; and from $P$ to $B C$, meeting at $L$. Triangles $B L P$ and $B M P$ are congruent, since both have a right angle and one-half of the bisected angle at $A$, and share edge $A P$.
Therefore $P M$ is $\sqrt{80000}$ in length. A similar argument is used to prove $P N$ is also $\sqrt{80000}$. The distance $A P$ is then found by Pythagoras' equation in triangle $A P M: A P=\sqrt{80000+80000}=$ $\sqrt{160000}=400$.