AMC 8 2007 Problem 8

AMC8 2007 年第 8 题

四边形与多边形★★☆☆☆

In trapezoid ABCD ,

\overline{AD}

is perpendicular to

\overline{DC}

, AD = AB = 3 , and DC = 6 . In addition, E is on

\overline{DC}

, and

\overline{BE}

is parallel to

\overline{AD}

. Find the area of

\triangle BEC

在梯形 ABCD 中，线段 AD 和线段 DC 垂直，AD＝AB＝3，DC＝6。另外，E 在线段 DC 上，BE 平行于 AD。求△BEC 的面积。

A.3
B.4.5
C.6
D.9
E.18

正确答案 · Correct Answer

解析 · Solution

Solution 1

Note that 𝐴 𝐵 𝐸 𝐷 ABED is a square with side 3 3. Subtract 𝐷 𝐸 DE from 𝐷 𝐶 DC, to find that 𝐸 𝐶 ‾ EC, the base of △ 𝐵 𝐸 𝐶 △BEC, has length 3 3. The area of △ 𝐵 𝐸 𝐶 △BEC is 1 2 ⋅ 3 ⋅ 3 = 9 2 = 4.5 2 1 ⋅3⋅3= 2 9 =4.5. OR OR The area of the △ 𝐵 𝐸 𝐶 △BEC is the area of the trapezoid 𝐴 𝐵 𝐶 𝐷 ABCD minus the area of the square 𝐴 𝐵 𝐸 𝐷 ABED. The area of △ 𝐵 𝐸 𝐶 △BEC is 1 2 ( 3 + 6 ) 3 − 3 2 = 13.5 − 9 = 4.5 2 1 (3+6)3−3 2 =13.5−9=4.5. Answer: 𝐵 B .

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