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Zorluk: OrtaRight Triangle Trigonometry (SOHCAHTOA)

In right triangle JKMJKM, the right angle is at vertex KK. Leg JKJK has a length of 1616 centimeters. If sin⁡(∠M)=817\sin(\angle M) = \frac{8}{17}, what is the length, in centimeters, of leg KMKM?

  1. A
    1515
  2. 3030Cevap
  3. C
    3434
  4. D
    23532\sqrt{353}
  5. E
    13615\frac{136}{15}

Cevap

The length of leg KMKM is 3030 centimeters.
By definition of sine in a right triangle, sin⁡(∠M)=oppositehypotenuse=JKJM\sin(\angle M) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{JK}{JM}. Given sin⁡(∠M)=817\sin(\angle M) = \frac{8}{17} and JK=16JK = 16, setting 817=16JM\frac{8}{17} = \frac{16}{JM} yields hypotenuse JM=34 cmJM = 34\text{ cm}. Using the Pythagorean theorem KM=JM2−JK2=342−162=900=30 cmKM = \sqrt{JM^2 - JK^2} = \sqrt{34^2 - 16^2} = \sqrt{900} = 30\text{ cm}. Therefore, the option with value 3030 is correct.

Adım Adım Çözüm

1
Use the definition of sine to find the hypotenuse JMJM.
sin⁡(∠M)=oppositehypotenuse=JKJM  ⟹  817=16JM  ⟹  JM=34 cm\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.
2
Apply the Pythagorean theorem to solve for the missing leg KMKM.
JK2+KM2=JM2  ⟹  162+KM2=342  ⟹  256+KM2=1156  ⟹  KM2=900  ⟹  KM=30 cmJK^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.

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Right Triangle Trigonometry (SOHCAHTOA)
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