Problem 280

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Problem 280: (2021 Fall AMC 12a, 21)

Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD$?

$\textbf{(A)}\: 15\qquad$ $\textbf{(B)} \: 5\sqrt{11}\qquad$ $\textbf{(C)} \: 3\sqrt{35}\qquad$ $\textbf{(D)} \: 18\qquad$ $\textbf{(E)} \: 7\sqrt{7}$

Solution

Exercise 280: Solution

Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 4$, $XY = 2$, and $YC = 3$. What is the area of $ABCD$?

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