HMMT November

Geometry Difficulty 4.8 AIME Find the answer

Let $A B C D$ be an isosceles trapezoid with parallel bases $A B=1$ and $C D=2$ and height 1. Find the area of the region containing all points inside $A B C D$ whose projections onto the four sides of the trapezoid lie on the segments formed by $A B, B C, C D$ and $D A$ .

Your answer

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

Solution

Let $E, F$ , be the projections of $A, B$ on $C D$ . A point whose projections lie on the sides must be contained in the square $A B F E$ . Furthermore, the point must lie under the perpendicular to $A D$ at $A$ and the perpendicular to $B C$ at $B$ , which have slopes $\frac{1}{2}$ and $-\frac{1}{2}$ . The area of the desired pentagon is $1-\frac{1}{4}-\frac{1}{8}=\frac{5}{8}$

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