Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

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 $B X$ .

Your answer

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

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 $B X<C X$ ).

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.