AMC8 2014

AMC8 2014 · Q14

AMC8 2014 · Q14. It mainly tests Pythagorean theorem, Area & perimeter.

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?

矩形 $ABCD$ 和直角三角形 $DCE$ 面积相同。它们组合形成了一个梯形，如图所示。求 $DE$。

(A) 12 12

(B) 13 13

(D) 15 15

(E) 16 16

Answer

Correct choice: (B)

正确答案：（B）

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 $\boxed{(B)}$.

$\bigtriangleup CDE$ 的面积是 $\frac{DC \cdot CE}{2}$。矩形 $ABCD$ 的面积是 $AB \cdot AD = 5 \cdot 6 = 30$，这个面积也应等于 $\bigtriangleup CDE$ 的面积。由于 $DC = 5$，则面积等于 $\frac{5 \cdot CE}{2}$。通过等式传递，有 $\frac{5 \cdot CE}{2} = 30$，解得 $CE = 12$。接着，利用勾股定理，可以判断 $\bigtriangleup CDE$ 是一个 $5-12-13$ 的直角三角形，所以 $DE = \boxed{13}$ ，即选项 $\boxed{(B)}$。

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