Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

Let $A B C D$ be a convex trapezoid such that $\angle B A D=\angle A D C=90^{\circ}, A B=20, A D=21$ , and $C D=28$ . Point $P \neq A$ is chosen on segment $A C$ such that $\angle B P D=90^{\circ}$ . Compute $A P$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Construct the rectangle $A B X D$ . Note that $\angle B A D=\angle B P D=\angle B X D=90^{\circ}$ so $A B X P D$ is cyclic with diameter $B D$ . By Power of a Point, we have $C X \cdot C D=C P \cdot C A$ . Note that $C X=C D-X D=C D-A B=8$ and $C A=\sqrt{A D^{2}+D C^{2}}=35$ . Therefore, $C P=\frac{C X \cdot C D}{C A}=\frac{8 \cdot 28}{35}=\frac{32}{5}$ and so $A P=A C-C P=35-\frac{32}{5}=\frac{143}{5}$

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