Triangle A B C is similar to triangle D E F , where A corresponds to D and C corresponds

Question

Triangle A B C is similar to triangle D E F , where A corresponds to D and C corresponds to F . Angles C and F are right angles. If tan A = 3 and D F = 125 , what is the length of D E &#175; ?

Accepted Answer

Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle A B C is similar to triangle D E F , where A corresponds to D , so the measure of angle A is equal to the measure of angle D . Therefore, if tan A = 3 , then tan D = 3 . It's given that angles C and F are right angles, so triangles A B C and D E F are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle D is side D F . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle D is side E F . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, tan D = E F D F . If D F = 125 , the length of side E F can be found by substituting 3 for tan D and 125 for D F in the equation tan D = E F D F , which yields 3 = E F 125 . Multiplying both sides of this equation by 125 yields 125 3 = E F . Since the length of side E F is 3 times the length of side D F , it follows that triangle D E F is a special right triangle with angle measures 30 &#176; , 60 &#176; , and 90 &#176; . Therefore, the length of the hypotenuse, D E &#175; , is 2 times the length of side D F , or D E = 2 D F . Substituting 125 for D F in this equation yields D E = 2 125 , or D E = 250 . Thus, if tan A = 3 and D F = 125 , the length of D E &#175; is 250 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the length of E F &#175; , not D E &#175; .

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