HMMT November

Geometry Difficulty 4.6 AIME Find the answer

Isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has a point $P$ on $AB$ with $AP=11, BP=27$ , $CD=34$ , and $\angle CPD=90^{\circ}$ . Compute the height of isosceles trapezoid $ABCD$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Drop projections of $A, P, B$ onto $CD$ to get $A', P', B'$ . Since $A'B'=38$ and $CD=34$ , we get that $DA'=CB'=2$ . Thus, $P'D=9$ and $P'C=25$ . Hence, the answer is $PP'=\sqrt{P'D \cdot P'C}=15$ .

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