AMC 12 Mock Exam 3 Problem 4

Problem 4 of 25

A carpenter builds a frame in the shape of a right triangle from three straight boards with a total length of $70$ inches. The longest board, which forms the hypotenuse, is $29$ inches long. What is the area, in square inches, of the region enclosed by the frame?

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Solution

Let the legs (the other two boards) be $a$ and $b$ . Then $a+b=70-29=41$ and $a^2+b^2=29^2=841$ . Hence $2ab=(a+b)^2-(a^2+b^2)=1681-841=840$ , and the area is $\frac{ab}2=\boxed{210}$ . (The legs are $20$ and $21$ .)

Tagged: right triangle: perimeter, hypotenuse, area

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