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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

In triangle $A B C$ , angle $B$ is a right angle. The length of side $A B$ is $10 StartRoot 37 EndRoot$ and the length of side $B C$ is $24 StartRoot 37 EndRoot$ . What is the length of side $A C$ ?

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Explanation

Choice B is correct. The Pythagorean theorem states that for a right triangle, $c^{2} = a^{2} + b^{2}$ , where $c$ represents the length of the hypotenuse and $a$ and $b$ represent the lengths of the legs. It’s given that in triangle $A B C$ , angle $B$ is a right angle. Therefore, triangle $A B C$ is a right triangle, where the hypotenuse is side $A C$ and the legs are sides $A B$ and $B C$ . It’s given that the lengths of sides $A B$ and $B C$ are $10 StartRoot 37 EndRoot$ and $24 StartRoot 37 EndRoot$ , respectively. Substituting these values for $a$ and $b$ in the formula $c^{2} = a^{2} + b^{2}$ yields $c^{2} = {(10 \sqrt{37})}^{2} + {(24 \sqrt{37})}^{2}$ , which is equivalent to $c^{2} = 100 (37) + 576 (37)$ , or $c squared equals 676 left parenthesis 37 right parenthesis$ . Taking the square root of both sides of this equation yields $c = \pm 26 \sqrt{37}$ . Since $c$ represents the length of the hypotenuse, side $A C$ , $c$ must be positive. Therefore, the length of side $A C$ is $26 StartRoot 37 EndRoot$ .

Choice A is incorrect. This is the result of solving the equation $c = 24 \sqrt{37} - 10 \sqrt{37}$ , not $c^{2} = {(10 \sqrt{37})}^{2} + {(24 \sqrt{37})}^{2}$ .

Choice C is incorrect. This is the result of solving the equation $c = 10 \sqrt{37} + 24 \sqrt{37}$ , not $c^{2} = {(10 \sqrt{37})}^{2} + {(24 \sqrt{37})}^{2}$ .

Choice D is incorrect and may result from conceptual or calculation errors.