Question

solve the right triangle using the given information.
a=76.7 yd, b=41.2 yd

c=□ yd
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
a=□°□′
(simplify your answers. type integers. round to the nearest ten minutes if needed.)
b=□°□′
(simplify your answers. type integers. round to the nearest ten minutes if needed.)

Explanation:

Step1: Find hypotenuse \( c \) using Pythagorean theorem

The Pythagorean theorem states \( c = \sqrt{a^2 + b^2} \). Given \( a = 76.7 \) yd and \( b = 41.2 \) yd, we calculate:
\( a^2 = 76.7^2 = 5882.89 \), \( b^2 = 41.2^2 = 1697.44 \)
\( c = \sqrt{5882.89 + 1697.44} = \sqrt{7580.33} \approx 87.1 \) yd

Step2: Find angle \( A \) using tangent

\( \tan(A) = \frac{a}{b} = \frac{76.7}{41.2} \approx 1.86165 \)
\( A = \arctan(1.86165) \approx 61.8^\circ \)
Convert decimal degrees to minutes: \( 0.8^\circ \times 60 = 48' \), so \( A \approx 62^\circ 50' \) (rounded to nearest ten minutes)

Step3: Find angle \( B \) using tangent

\( \tan(B) = \frac{b}{a} = \frac{41.2}{76.7} \approx 0.53716 \)
\( B = \arctan(0.53716) \approx 28.2^\circ \)
Convert decimal degrees to minutes: \( 0.2^\circ \times 60 = 12' \), so \( B \approx 28^\circ 10' \) (rounded to nearest ten minutes)

Answer:

\( c = 87.1 \) yd
\( A = 62^\circ 50' \)
\( B = 28^\circ 10' \)