HMMT-Nov 2011 guts, Problem 19

Let $A B C D$ be a rectangle with $A B=3$ and $B C=7$. Let $W$ be a point on segment $A B$ such that $A W=1$. Let $X, Y, Z$ be points on segments $B C, C D, D A$, respectively, so that quadrilateral $W X Y Z$ is a rectangle, and $B X<X C$. Determine the length of segment $B X$.

Solution: We note that \[ \angle Y X C=90-\angle W X B=\angle X W B=90-\angle A W Z=\angle A Z W \] gives us that $X Y C \cong Z W A$ and $X Y Z \sim W X B$. Consequently, we get that $Y C=A W=1$. From $X Y Z \sim W X B$, we get that \[ \frac{B X}{B W}=\frac{C Y}{C X} \Rightarrow \frac{B X}{2}=\frac{1}{7-B X} \] from which we get \[ B X^{2}-7 B X+2=0 \Rightarrow B X=\frac{7-\sqrt{41}}{2} \] (since we have $B X<C X$ ). Guts Round $\fbox{\frac{7-\sqrt{41}}{2}}$.

Answer: \frac{7-\sqrt{41}}{2}