In right triangle UVW, the right angle is located at vertex V. The length of hypotenuse UW… — ACT

In right triangle $UVW$ , the right angle is located at vertex $V$ . The length of hypotenuse $UW$ is $85$ centimeters. If $\sin(U) = \frac{15}{17}$ , what is the length, in centimeters, of leg $UV$ ?

$40$ Answer
B
$45$
C
$68$
D
$75$
E
$114$

Answer

The length of leg

UV

40

centimeters.

By definition,

\sin(U) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{VW}{UW}

. Given

\sin(U) = \frac{15}{17}

and

UW = 85

, the opposite leg

VW = 85 \times \frac{15}{17} = 75

cm. Using the Pythagorean theorem

UV^2 = UW^2 - VW^2 = 85^2 - 75^2 = 7225 - 5625 = 1600

, we get

UV = 40

cm. Alternatively, using

\cos(U) = \sqrt{1 - \sin^2(U)} = \frac{8}{17}

, the adjacent leg

UV = 85 \times \frac{8}{17} = 40

cm.

Step-by-Step Solution

Identify the relationship between

\sin(U)

and the sides of right triangle

UVW

\sin(U) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{VW}{UW} = \frac{15}{17}

By SOHCAHTOA, sine is the ratio of the opposite leg to the hypotenuse.

Calculate the length of the opposite leg

VW

VW = 85 \times \frac{15}{17} = 75

centimeters.

Multiply the hypotenuse length

UW = 85

by the sine ratio

\frac{15}{17}

Apply the Pythagorean theorem (

UV^2 + VW^2 = UW^2

) to solve for adjacent leg

UV

UV^2 + 75^2 = 85^2 \implies UV^2 + 5625 = 7225 \implies UV^2 = 1600 \implies UV = 40

centimeters.

The square of the hypotenuse equals the sum of the squares of the two legs.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and Pythagorean Relationship

Estimated Time:1m 30s

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