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 quadrilateral such that $\angle A B D=\angle B C D=90^{\circ}$ , and let $M$ be the midpoint of segment $B D$ . Suppose that $C M=2$ and $A M=3$ . Compute $A D$ .

Your answer

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

Solution

Since triangle $B C D$ is a right triangle, we have $C M=B M=D M=2$ . With $A M=3$ and $\angle A B M=90^{\circ}$ , we get $A B=\sqrt{5}$ . Now $A D^{2}=A B^{2}+B D^{2}=5+16=21$ so $A D=\sqrt{21}$ .

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