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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

$upper R upper S equals 440$

$upper S upper T equals 384$

$upper T upper R equals 584$

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 $\tan W$ ?

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Explanation

Choice D is correct. The hypotenuse of triangle $R S T$ is the longest side and is across from the right angle. The longest side length given is $584$ , which is the length of side $T R$ . Therefore, the hypotenuse of triangle $R S T$ is side $T R$ , so the right angle is angle $S$ . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side, which is the side across from the angle, to the length of the adjacent side, which is the side closest to the angle that is not the hypotenuse. It follows that the opposite side of angle $T$ is side $R S$ and the adjacent side of angle $T$ is side $S T$ . Therefore, $\tan T = \frac{R S}{S T}$ . Substituting $440$ for $R S$ and $384$ for $S T$ in this equation yields $\tan T = \frac{440}{384}$ . This is equivalent to $\tan T = \frac{55}{48}$ . It’s given that triangle $R S T$ is similar to triangle $U V W$ , where $S$ corresponds to $V$ and $T$ corresponds to $W$ . It follows that $R$ corresponds to $U$ . Therefore, the hypotenuse of triangle $U V W$ is side $W U$ , which means $\tan W = \frac{U V}{V W}$ . Since the lengths of corresponding sides of similar triangles are proportional, $\frac{R S}{S T} = \frac{U V}{V W}$ . Therefore, $\tan W = \frac{U V}{V W}$ is equivalent to $\tan W = \frac{R S}{S T}$ , or $\tan W = \tan T$ . Thus, $\tan W = \frac{55}{48}$ .

Choice A is incorrect. This is the value of $\cos W$ , not $\tan W$ .

Choice B is incorrect. This is the value of $\sin W$ , not $\tan W$ .

Choice C is incorrect. This is the value of $StartFraction 1 Over tangent upper W EndFraction$ , not $\tan W$ .