Find a hypotenuse from a leg ratio and an area

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Quantitative Reasoning

In right triangle $KLM$ , the right angle is at $L$ , and the ratio of $KL$ to $LM$ is 7 to 2. If the area of triangle $KLM$ is 252, what is the length of $KM$ ?

Select one answer.

6\sqrt{53}

3\sqrt{106}

54

6\sqrt{45}

42

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Answer: A ( $6\sqrt{53}$ )

Let the lengths of the legs be $7x$ and $2x$ , where $x$ is positive. Using the area formula for a right triangle gives $(\frac{1}{2})(7x)(2x) = 252$ , so $7x^{2} = 252$ and $x^{2} = 36$ . Thus, $x = 6$ , and the leg lengths are 42 and 12. By the Pythagorean theorem, the length of the hypotenuse is $\sqrt{42^{2} + 12^{2}}$ ${}= \sqrt{1908}$ ${}= 6\sqrt{53}$ . Therefore, the correct answer is Choice A.

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Quantitative Reasoning

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