R S = 20 S T = 48 T R = 52 The side lengths of right triangle R S

Question

R S = 20 S T = 48 T R = 52 The side lengths of right triangle R S T are given. Triangle R S T is similar to triangle U V W , where S corresponds to V and T corresponds to W . What is the value of tan W ?

Accepted Answer

Choice B is correct. It's given that right triangle R S T is similar to triangle U V W , where S corresponds to V and T corresponds to W . It's given that the side lengths of the right triangle R S T are R S = 20 , S T = 48 , and T R = 52 . Corresponding angles in similar triangles are equal. It follows that the measure of angle T is equal to the measure of angle W . The hypotenuse of a right triangle is the longest side. It follows that the hypotenuse of triangle R S T is side T R . The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle S is a right angle. 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 T is side S T . 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 T is side R S . 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 T = R S S T . Substituting 20 for R S and 48 for S T in this equation yields tan T = 20 48 , or tan T = 5 12 . The tangents of two acute angles with equal measures are equal. Since the measure of angle T is equal to the measure of angle W , it follows that tan T = tan W . Substituting 5 12 for tan T in this equation yields 5 12 = tan W . Therefore, the value of tan W is 5 12 . Choice A is incorrect. This is the value of sin W . Choice C is incorrect. This is the value of cos W . Choice D is incorrect. This is the value of 1 tan W .

Question r-s-20