AMC8 2004

AMC8 2004 · Q24

AMC8 2004 · Q24. It mainly tests Triangles (properties), Area & perimeter.

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 $\overline{FG}$?

在图中，$ABCD$ 是矩形，$EFGH$ 是平行四边形。使用图中给出的测量值，垂直于 $\overline{HE}$ 和 $\overline{FG}$ 的线段 $d$ 的长度是多少？

(A) 6.8 6.8

(B) 7.1 7.1

(D) 7.8 7.8

(E) 8.1 8.1

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) By the Pythagorean Theorem, $HE = 5$. Rectangle $ABCD$ has area $10 \times 8 = 80$, and the corner triangles have areas $\frac{1}{2} \times 3 \times 4 = 6$ and $\frac{1}{2} \times 6 \times 5 = 15$. So the area of $EFGH$ is $80 - (2)(6) - (2)(15) = 38$. Because the area of $EFGH$ is $EH \times d$ and $EH = 5$, $38 = 5 \times d$, so $d = 7.6$.

（C）由勾股定理，$HE = 5$。矩形 $ABCD$ 的面积为 $10 \times 8 = 80$，四角的三角形面积分别为 $\frac{1}{2} \times 3 \times 4 = 6$ 和 $\frac{1}{2} \times 6 \times 5 = 15$。因此 $EFGH$ 的面积是 $80 - (2)(6) - (2)(15) = 38$。因为 $EFGH$ 的面积为 $EH \times d$ 且 $EH = 5$，所以 $38 = 5 \times d$，从而 $d = 7.6$。

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