Problem:
Daniel finds a rectangular index card and measures its diagonal to be centimeters. Daniel then cuts out equal squares of side at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be centimeters, as shown below. What is the area of the original index card?
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Answer Choices:
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Solution:
Let and be the side lengths of the index card. Then the corners of the two cut squares are opposite vertices of a rectangle with sides and . The Pythagorean Theorem applied twice yields
Expanding the second equation and substituting the first one gives
so . Finally, the area of the rectangle is , which equals
Note: The original index card uniquely has dimensions .
The problems on this page are the property of the MAA's American Mathematics Competitions