# 2014 AMC 8 Problems/Problem 14

## Problem

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$? $[asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label("A",A,NW); label("B",B,SW); label("C",C,S); label("D",D,N); label("E",E,S); label("5",A/2,W); label("6",(A+D)/2,N); [/asy]$ $\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad\textbf{(E) }16$

## Solution

The area of $\bigtriangleup CDE$ is $\frac{DC\cdot CE}{2}$. The area of $ABCD$ is $AB\cdot AD=5\cdot 6=30$, which also must be equal to the area of $\bigtriangleup CDE$, which, since $DC=5$, must in turn equal $\frac{5\cdot CE}{2}$. Through transitivity, then, $\frac{5\cdot CE}{2}=30$, and $CE=12$. Then, using the Pythagorean Theorem, you should be able to figure out that $\bigtriangleup CDE$ is a $5-12-13$ triangle, so $DE=\boxed{13}$, or $\boxed{(B)}$.

## Solution 2

The area of the rectangle is $5\times6=30.$ Since the parallel line pairs are identical, $DC=5$. Let $CE$ be $x$. $\dfrac{5x}{2}=30$ is the area of the right triangle. Solving for $x$, we get $x=12.$ According to the Pythagorean Theorem, we have a $5-12-13$ triangle. So, the hypotenuse $DE$ has to be $\boxed{(B)}$.

## Solution 3

This problem can be solved with the Pythagorean Theorem ( $a^2 + b^2 = c^2$). We know $AB = DC$, so $DC = 5$. $CE$ is twice the length of $AD$, so $CE = 12$. $5^2 + 12^2 = c^2$. $5^2 = 25$. $12^2 = 144$. $25 + 144 = 169$. $169$ has a square root of $13$, so the hypotenuse or $DE$ is $13$. The answer is $\boxed{(B)}$.

——MiracleMaths

## Video Solution (CREATIVE THINKING)

~Education, the Study of Everything

## Video Solution

https://youtu.be/-JsXX8WLASg ~savannahsolver

## Video Solution

~ pi_is_3.14

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 