Rectangle DEFA below is a 3 × 4 rectangle with DC=CB=BA=1 . What is the area of the "bat wings" (shaded region)?

AMC 8 2016 Problem 22

AMC8 2016 年第 22 题

面积与周长（综合）★★★☆☆

Rectangle DEFA below is a

3 \times 4

rectangle with DC=CB=BA=1 . What is the area of the "bat wings" (shaded region)?

wings" (shaded area) is？

A.2
B. $\displaystyle 2 \frac{1}{2}$
C.3
D. $\displaystyle 3 \frac{1}{2}$
E.4

正确答案 · Correct Answer

解析 · Solution

Solution 1

The area of trapezoid

CBFE

\displaystyle \frac{1+3}2\cdot 4=8

. Next, we find the height of each triangle to calculate their area. The two non-colored isosceles triangles are similar, and are in a

3:1

ratio by AA similarity (alternate interior and vertical angles) so the height of the larger is 3, while the height of the smaller one is 1. Thus, their areas are

\frac12

and

\frac92

. Subtracting these areas from the trapezoid, we get

8-\frac12-\frac92 =3

. Therefore, the answer to this problem is 3.

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