2011 AMC 10A Problem 9

Below is the professionally curated solution for Problem 9 of the 2011 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10A solutions, or check the answer key.

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Concepts:coordinate geometry rectangle distributive property

Difficulty rating: 900

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?

$ac+ad+bc+bd$

$ac-ad+bc-bd$

$ac+ad-bc-bd$

$-ac-ad+bc+bd$

$ac-ad-bc+bd$

Solution:

Note that the region is a rectangle with side lengths $a - (-b) = a + b$ and $d - (-c) = c + d.$ The area is then $(a + b)(c + d) =$ $ac + ad + bc + bd.$

Thus, A is the correct answer.

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