Point Inside a Triangle
Source: AMC 12 2006B, Problem 23
February 17, 2006
rotationgeometrygeometric transformationdilationtrigonometryAMCAIME
Problem Statement
Isosceles has a right angle at . Point is inside , such that PA \equal{} 11, PB \equal{} 7, and PC \equal{} 6. Legs and have length s \equal{} \sqrt {a \plus{} b\sqrt {2}}, where and are positive integers. What is a \plus{} b?[asy]pointpen = black;
pathpen = linewidth(0.7);
pen f = fontsize(10);
size(5cm);
pair B = (0,sqrt(85+42*sqrt(2)));
pair A = (B.y,0);
pair C = (0,0);
pair P = IP(arc(B,7,180,360),arc(C,6,0,90));
D(A--B--C--cycle);
D(P--A);
D(P--B);
D(P--C);
MP("A",D(A),plain.E,f);
MP("B",D(B),plain.N,f);
MP("C",D(C),plain.SW,f);
MP("P",D(P),plain.NE,f);[/asy]