Segment Length Probability
Source:
April 6, 2008
probabilitytrigonometryratiogeometryinequalities
Problem Statement
Triangle is a right triangle with as its right angle, m\angle ABC \equal{} 60^\circ, and AB \equal{} 10. Let be randomly chosen inside , and extend to meet at . What is the probability that ?
[asy]import math;
unitsize(4mm);
defaultpen(fontsize(8pt)+linewidth(0.7));
dotfactor=4;
pair A=(10,0);
pair C=(0,0);
pair B=(0,10.0/sqrt(3));
pair P=(2,2);
pair D=extension(A,C,B,P);draw(A--C--B--cycle);
draw(B--D);
dot(P);
label("A",A,S);
label("D",D,S);
label("C",C,S);
label("P",P,NE);
label("B",B,N);[/asy]
(A)\ \frac {2 \minus{} \sqrt2}{2} \qquad (B)\ \frac {1}{3} \qquad (C)\ \frac {3 \minus{} \sqrt3}{3} \qquad (D)\ \frac {1}{2} \qquad (E)\ \frac {5 \minus{} \sqrt5}{5}