2008 AMC 12B Problems/Problem 18

Problem

A pyramid has a square base $ABCD$ and vertex $E$ . The area of square $ABCD$ is $196$ , and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$ , respectively. What is the volume of the pyramid?

Let $h$ be the height of the pyramid and $a$ be the distance from $h$ to $CD$ . The side length of the base is 14. The side lengths of $\triangle ABE$ and $\triangle CDE$ are $2\cdot105\div14=15$ and $2\cdot91\div14=13$ , respectively. We have a systems of equations through the Pythagorean Theorem:

Setting them equal to each other and simplifying gives $-27+28a=225 \implies a=9$ .