Solve the triangle using the labels as shown in the illustration (round lengths of sides to three significant digits and angles to the nearest tenth of a degree). $\angle A = 69.5$ °, $\angle B = 75.6$ °, b = 234 ft
A) a = 138 ft, c = 226 ft, $\angle C = 69.5$ °
B) a = 226 ft, c = 138 ft, $\angle C = 75.6$ °
C) a = 226 ft, c = 138 ft, $\angle C = 34.9$ °
D) a = 138 ft, c = 226 ft, $\angle C = 34.9$ °
E) a = 138 ft, c = 226 ft, $\angle C = 75.6$ °

Question

Solve the triangle using the labels as shown in the illustration (round lengths of sides to three significant digits and angles to the nearest tenth of a degree). $\angle A = 69.5$ °, $\angle B = 75.6$ °, b = 234 ft 
A) a = 138 ft, c = 226 ft, $\angle C = 69.5$ ° 
B) a = 226 ft, c = 138 ft, $\angle C = 75.6$ ° 
C) a = 226 ft, c = 138 ft, $\angle C = 34.9$ ° 
D) a = 138 ft, c = 226 ft, $\angle C = 34.9$ ° 
E) a = 138 ft, c = 226 ft, $\angle C = 75.6$ °

Quizplus · Accepted Answer

Using the law of sines, we have $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$. We know $b$ and angles $A$ and $B$, so we can use the fact that the angles in a triangle add up to 180 degrees to find angle $C$: $\angle C = 180^\circ - \angle A - \angle B = 35.9$ degrees. Then, we can use the law of sines to find the remaining sides: $\frac{\sin A}{a} = \frac{\sin 75.6^\circ}{234}$, so $a \approx 137.9$. Similarly, $\frac{\sin B}{c} = \frac{\sin 69.5^\circ}{234}$, so $c \approx 225.8$. Therefore, the best choice is C, which matches these values for $a$, $c$, and $\angle C$.

Solve the Triangle Using the Labels as Shown in the Illustration