2007 Cyprus MO/Lyceum/Problem 22

Problem

In the figure, $ABCD$ is an orthogonal trapezium with $\ang A= \ang D=90^\circ$ and bases $AB = a$ , $DC = 2a$ . If $AD = 3a$ and $M$ is the midpoint of the side $BC$ , then $AM$ equals to

Let the midpoint of $AD$ be $N$ . The length of $MN$ is the average of the bases, or $\frac{3a}{2}$ . The length of $AN$ is also $\frac{3a}{2}$ .

Since $AMN$ is a $45-45-90$ triangle, the length of $AM$ is $\frac{3a}{\sqrt{2}}$ , and the answer is $\mathrm{B}$ .