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Geometry and Trigonometry Difficulty: Hard

The length of a rectangle’s diagonal is $3 StartRoot 17 EndRoot$ , and the length of the rectangle’s shorter side is $3$ . What is the length of the rectangle’s longer side?

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Explanation

The correct answer is $12$ . The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that the length of the rectangle's diagonal is $3 StartRoot 17 EndRoot$ and the length of the rectangle's shorter side is $3$ . Thus, the length of the hypotenuse of the right triangle formed by the diagonal is $3 StartRoot 17 EndRoot$ and the length of one of the legs is $3$ . By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$ , then $a^{2} + b^{2} = c^{2}$ . Substituting $3 StartRoot 17 EndRoot$ for $c$ and $3$ for $b$ in this equation yields $a^{2} + {(3)}^{2} = {(3 \sqrt{17})}^{2}$ , or $a^{2} + 9 = 153$ . Subtracting $9$ from both sides of this equation yields $a squared equals 144$ . Taking the square root of both sides of this equation yields $a = \pm \sqrt{144}$ , or $a = \pm 12$ . Since $a$ represents a length, which must be positive, the value of $a$ is $12$ . Thus, the length of the rectangle's longer side is $12$ .