AMC 8 2004 Problem 24

AMC8 2004 年第 24 题

四边形与多边形★★★★☆

In the figure,

ABCD

is a rectangle and

EFGH

is a parallelogram. Using the measurements given in the figure, what is the length

d

of the segment that is perpendicular to

\overline{HE}

and

\overline{FG}

如下图所示，ABCD 是个矩形，EFGH 是个平行四边形。使用图中所标线段的长度，则同时垂直于和的线段长度 d 是多少？

A.6.8
B.7.1
C.7.6
D.7.8
E.8.1

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 两种面积矩形 10×8=80。挖去四个角的直角三角形（两个 6×5、两个 3×4）得平行四边形面积

80-2\cdot\tfrac12\cdot6\cdot5-2\cdot\tfrac12\cdot3\cdot4=80-30-12=38

。底

FG

是

3\text-4\text-5

直角三角形的斜边 =5，故 5d=38，d=7.6。

Solution 1

Elementary · area two ways The rectangle is 10×8=80. Removing the four corner right triangles (two 6×5 and two 3×4) leaves the parallelogram with area

80-2\cdot\tfrac12\cdot6\cdot5-2\cdot\tfrac12\cdot3\cdot4=80-30-12=38

. Its base

FG

is the hypotenuse of a

3\text-4\text-5

triangle, so FG=5, and 5d=38 gives d=7.6.

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