Question

solve the right triangle using the given information.
a = 78.9 yd, b = 40.7 yd

c = 88.8 yd
(simplify your answer. type an integer or a decimal. round to the nearest tenth if needed.)
a = 62° 42
(simplify your answers. type integers. round to the nearest ten minutes if needed.)
b = 27° 18
(simplify your answers. type integers. round to the nearest ten minutes if needed.)

Explanation:

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

The Pythagorean theorem for a right triangle is \( c = \sqrt{a^2 + b^2} \). Given \( a = 78.9 \) yd and \( b = 40.7 \) yd, we calculate \( a^2 = 78.9^2 = 6225.21 \) and \( b^2 = 40.7^2 = 1656.49 \). Then \( a^2 + b^2 = 6225.21 + 1656.49 = 7881.7 \). So \( c = \sqrt{7881.7} \approx 88.8 \) yd.

Step2: Find angle \( A \) using tangent

\( \tan(A) = \frac{a}{b} = \frac{78.9}{40.7} \approx 1.9386 \). Then \( A = \arctan(1.9386) \approx 62.7^\circ \). Convert \( 0.7^\circ \) to minutes: \( 0.7 \times 60 = 42' \), so \( A = 62^\circ 42' \).

Step3: Find angle \( B \) using tangent or complementary angles

Since \( A + B = 90^\circ \), \( B = 90^\circ - A = 90^\circ - 62^\circ 42' = 27^\circ 18' \). (Or using \( \tan(B) = \frac{b}{a} = \frac{40.7}{78.9} \approx 0.5158 \), \( B = \arctan(0.5158) \approx 27.3^\circ \), \( 0.3 \times 60 = 18' \), so \( B = 27^\circ 18' \))

Answer:

\( c \approx 88.8 \) yd, \( A = 62^\circ 42' \), \( B = 27^\circ 18' \)