2008 Mock ARML 2 Problems/Problem 4

Problem

Equilateral triangle $ABC$ has a side length of $7$ . A ball begins at vertex $A$ , rolls through the interior of the triangle, bounces off side $BC$ , and settles at point $P$ . Given that $BP = 3$ and $CP = 5$ , find the total distance that the ball travels.

Solution

$[asy] pointpen = black; pathpen = black + linewidth(0.7); pair A=(0,0),B=(7,0),C=7*expi(pi/3); pair Q=IP(CR(B,3),CR(C,5)), P=OP(CR(B,3),CR(C,5)),D=IP(A--Q,B--C); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); D(A--D--D(MP("P",P)));D(D--D(MP("P'",Q,NE))--C--Q--B,linetype("5 5")); MP("7",(A+B)/2); MP("3",(B+Q)/2,SE); MP("5",(C+Q)/2,NE); [/asy]$

Reflect $P$ across $\overline{BC}$ to point $P'$ ; since the ball travels in a straight path, it follows that the distance the ball traveled is $AP'$ . By symmetry, $BP' = 3, CP' = 5$ . By the Law of Cosines on $\triangle BCP'$ ,