In triangle A B C , angle B is a right angle. The length of side A B is 10

Question

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

Accepted Answer

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&rsquo;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&rsquo;s given that the lengths of sides A B and B C are 10 37 and 24 37 , respectively. Substituting these values for a and b in the formula c 2 = a 2 + b 2 yields c 2 = 10 37 2 + 24 37 2 , which is equivalent to c 2 = 100 37 + 576 37 , or c 2 = 676 37 . Taking the square root of both sides of this equation yields c = &#177; 26 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 37 . Choice A is incorrect. This is the result of solving the equation c = 24 37 - 10 37 , not c 2 = 10 37 2 + 24 37 2 . Choice C is incorrect. This is the result of solving the equation c = 10 37 + 24 37 , not c 2 = 10 37 2 + 24 37 2 . Choice D is incorrect and may result from conceptual or calculation errors.

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