AMC10 2011 A

AMC10 2011 A · Q9

AMC10 2011 A · Q9. It mainly tests Area & perimeter, Coordinate geometry.

A rectangular region is bounded by the graphs of the equations $y = a$, $y = −b$, $x = −c$, and $x = d$, where $a$, $b$, $c$, and $d$ are all positive numbers. Which of the following represents the area of this region?

一个矩形区域被方程$y = a$、$y = −b$、$x = −c$和$x = d$的图像所包围，其中$a$、$b$、$c$和$d$均为正数。以下哪项表示该区域的面积？

(A) $ac + ad + bc + bd$ $ac + ad + bc + bd$

(B) $ac −ad + bc −bd$ $ac −ad + bc −bd$

(D) −$ac −ad + bc + bd$ −$ac −ad + bc + bd$

(E) $ac −ad −bc + bd$ $ac −ad −bc + bd$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Because $a, b, c,$ and $d$ are positive numbers, $a > −b$ and $d > −c$. Therefore the height of the rectangle is $a + b$ and the width is $c + d$. The area of the region is $(a + b)(c + d) = ac + ad + bc + bd$.

因为$a, b, c,$ 和 $d$ 是正数，$a > −b$ 且 $d > −c$。因此矩形的高度是$a + b$，宽度是$c + d$。该区域的面积是$(a + b)(c + d) = ac + ad + bc + bd$。

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