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# help geometry

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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?

May 1, 2022

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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 $$BY: YA = 6 \sqrt 2: 12 \sqrt 2 = \color{brown}\boxed{1 \over 2}$$

May 1, 2022