AMC8 2016

AMC8 2016 · Q22

AMC8 2016 · Q22. It mainly tests Triangles (properties), Area & perimeter.

Rectangle DEFA below is a 3×4 rectangle with DC = CB = BA = 1. The area of the “bat wings” (the shaded area) is

下面的矩形DEFA是一个3×4的矩形，且DC = CB = BA = 1。“蝙蝠翅膀”（阴影区域）的面积是

(A) 2 2

(B) $2\frac{1}{2}$ $2\frac{1}{2}$

(D) $3\frac{1}{2}$ $3\frac{1}{2}$

(E) 4 4

Answer

Correct choice: (C)

正确答案：（C）

Solution

The area of triangle BCE is $\tfrac12(1)(4)=2$. Triangles CBH and EFH are similar. Since $CB=\tfrac13 EF$, it follows that $IH=\tfrac13 GH=\tfrac14 IG=1$. The area of triangle CBH is $\tfrac12$, so the area of triangle ECH is $2-\tfrac12=\tfrac32$. Thus the batwing's area is $3$.

三角形 BCE 的面积为 $\tfrac12(1)(4)=2$。三角形 CBH 与 EFH 相似。由于 $CB=\tfrac13 EF$，因此 $IH=\tfrac13 GH=\tfrac14 IG=1$。三角形 CBH 的面积为 $\tfrac12$，所以三角形 ECH 的面积是 $2-\tfrac12=\tfrac32$。因此“蝙蝠翼”图形的面积为 $3$。

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