2014 AMC 12B Problems/Problem 21

Problem

In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$ ?

-Insert Diagram because I don't know how-

Solution

Let $BE = x$ . Let $JA = y$ . Because $\angle FKH = \angle EJK = \angle AGJ = \angle DHG$ and $\angle FHK = \angle EKJ = \angle AJG = \angle DGH$ , $\triangle KEJ, \triangle JAG, \triangle GDH, \triangle HFK$ are all similar. Using proportions and the pythagorean theorem, we find $EK = xy$ $FK = \sqrt{1-y^2}$ $EJ = x\sqrt{1-y^2}$ Because we know that $BE+EJ+AJ = EK + FK = 1$ , we can set up a systems of equations $x + x\sqrt{1-y^2} + y = 1$ $xy + \sqrt{1-y^2} = 1$ Solving for $x$ in the second equation, we get $x= \frac{1-\sqrt{1-y^2}}{y}$ Plugging this into the first equation, we get $\frac{1-\sqrt{1-y^2}}{y} + (\sqrt{1-y^2})\frac{1-\sqrt{1-y^2}}{y} + y = 1 \implies \frac{2y^2}{y}=1 \implies y=\frac{1}{2}$ Plugging into the previous equation with $x$ , we get $x= 2\left(1-\sqrt{1-\frac{1}{4}}\right) = 2\left(\frac{2-\sqrt{3}}{2} \right) = \boxed{\textbf{(C)}\ 2-\sqrt{3}}$

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2014 AMC 12B Problems/Problem 21

Problem

Solution