In right triangle JKM, the right angle is at vertex K. Leg JK has a length of 16 centimete… — ACT

In right triangle $JKM$ , the right angle is at vertex $K$ . Leg $JK$ has a length of $16$ centimeters. If $\sin(\angle M) = \frac{8}{17}$ , what is the length, in centimeters, of leg $KM$ ?

A
$15$
$30$ Answer
C
$34$
D
$2\sqrt{353}$
E
$\frac{136}{15}$

Answer

The length of leg

KM

30

centimeters.

By definition of sine in a right triangle,

\sin(\angle M) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{JK}{JM}

. Given

\sin(\angle M) = \frac{8}{17}

and

JK = 16

, setting

\frac{8}{17} = \frac{16}{JM}

yields hypotenuse

JM = 34\text{ cm}

. Using the Pythagorean theorem

KM = \sqrt{JM^2 - JK^2} = \sqrt{34^2 - 16^2} = \sqrt{900} = 30\text{ cm}

. Therefore, the option with value

30

is correct.

Step-by-Step Solution

Use the definition of sine to find the hypotenuse

JM

\sin(\angle M) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{JK}{JM} \implies \frac{8}{17} = \frac{16}{JM} \implies JM = 34\text{ cm}

Sine is defined as the ratio of the side opposite the angle to the hypotenuse in a right triangle.

Apply the Pythagorean theorem to solve for the missing leg

KM

JK^2 + KM^2 = JM^2 \implies 16^2 + KM^2 = 34^2 \implies 256 + KM^2 = 1156 \implies KM^2 = 900 \implies KM = 30\text{ cm}

In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

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