Exercise 12. Let $M N P Q$ be a rhombus such that $M P=180$. Suppose there exists a square $A B C D$ such that $A B=18, A \in[M N], B \in[N P], C \in[P Q]$ and $D \in[Q M]$. Determine the area of $M N P Q$.
Solution to Exercise 12: We take coordinates such that $A=(-9,9), B=(-9,-9)$, $C=(9,-9), D=(9,9), M=(0, y), N=(-x, 0), P=(0,-y), Q=(x, 0)$. We verify that $\frac{x}{y}=\frac{9}{y-9}$, so $x=10$. The area of $M N P Q$ is $2 x y=1800$.

## High School Exercises