In isosceles right triangle ABC, shown here, AC = BC. Point X is on side BC such that CX = 6 and XB = 12, and Y is on side AB such that XY is perpendicular to AB. What is the ratio of BY to YA?
+2668 Because \(\triangle ABC\) is isoceles, \(AB = 18 \sqrt 2 \)
Because it is an isceles right triangle, \(\angle B = 45\), meaning that \(BY = {12 \over \sqrt2} = 6 \sqrt 2 \)
We also know that \(AY = 12 \sqrt 2\), because \(BY + YA = AB\)
Thus, the ratio of \(BY: YA = 6 \sqrt 2: 12 \sqrt 2 = \color{brown}\boxed{1 \over 2} \)
