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?
+179 $BYX$ ~ $ABC$ by $AAA$ similarity. Because $ABC$ has $45-45-90$ angles it follows that $BYX$ has $45-45-90$ angles (and is isosceles).
The diagram below explains $45-45-90$ triangles.
Thus, we can see that $BY = \frac{12}{\sqrt{2}} = 6\sqrt{2}$.
Similarly, we can see that $BA = 18\sqrt{2}$ by triangle $ABC$.
Subtracting $BC - BY = 12\sqrt{2} = AY$.
Thus the ratio is $\frac{BY}{AY} = \frac{6\sqrt{2}}{12\sqrt{2}} = \boxed{\frac{1}{2}}$, which is our answer.
