Question

a right triangle has side lengths ( ac = 7 ) inches, ( bc = 24 ) inches, and ( ab = 25 ) inches. what are the measures of the angles in triangle ( abc )? ( mangle a = 46.2^{circ}, mangle b = 43.8^{circ}, mangle c = 90^{circ} ) ( mangle a = 73.0^{circ}, mangle b = 17.0^{circ}, mangle c = 90^{circ} ) ( mangle a = 73.7^{circ}, mangle b = 16.3^{circ}, mangle c = 90^{circ} ) ( mangle a = 74.4^{circ}, mangle b = 15.6^{circ}, mangle c = 90^{circ} )

Explanation:

Step1: Use trigonometric ratios

In right - triangle \(ABC\) with \(\angle C = 90^{\circ}\), \(\sin A=\frac{BC}{AB}\), \(\sin B=\frac{AC}{AB}\).
Given \(AC = 7\) inches, \(BC = 24\) inches, \(AB = 25\) inches.
\(\sin A=\frac{BC}{AB}=\frac{24}{25}=0.96\).

Step2: Find \(\angle A\)

Using the inverse - sine function \(A=\sin^{- 1}(0.96)\approx73.7^{\circ}\).

Step3: Find \(\angle B\)

Since \(A + B+C=180^{\circ}\) and \(C = 90^{\circ}\), then \(B=180^{\circ}-(A + C)\).
\(\sin B=\frac{AC}{AB}=\frac{7}{25}=0.28\), \(B=\sin^{-1}(0.28)\approx16.3^{\circ}\)

Answer:

\(m\angle A = 73.7^{\circ},m\angle B = 16.3^{\circ},m\angle C = 90^{\circ}\) (the third option)