Problem:
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
Answer Choices:
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Solution:
Rectangles of the same height have areas proportional to their bases, and . Hence , so the required area is .
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We can prove that the product of the areas of the diagonally opposite rectangles is the same: If the sides are labeled and as shown, then the product of the areas of the lower left and upper right is and the product of the other areas is . Hence the area of the fourth rectangle is .
Note. For the given data, we can find the answer by factoring the areas. Since rectangles of the same height have areas proportional to their bases, the unknown area is . This factoring method will not work for all given data; e.g., if is substituted for .