Solve the problem.
-Given a triangle with a = 9, b = 11, $\alpha = 31 ^ { \circ }$ , what is (are) the possible length(s) of c? Round your answer to two decimal places.
A) 16.42 or 2.44
B) 16.42 or 3.41
C) 6.61
D) 14.21

Question

Quizplus · Accepted Answer

Using the Law of Sines, $ \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} $, and knowing that $ \gamma = 180^\circ - \alpha - \beta $, we can find $ \beta $ using the Law of Sines with sides a and b, then solve for $ \gamma $, and finally solve for c. However, since the given information does not directly lead to a single solution without knowing angle $ \beta $ or $ \gamma $, we must use the Law of Cosines to find c directly: $ c^2 = a^2 + b^2 - 2ab\cos(\alpha) $. This equation might yield two solutions due to the ambiguous case of the Law of Sines, where two different triangles could be formed (one acute and one obtuse), leading to two possible lengths for side c. Calculating with the given values, we find two possible lengths for c, which are approximately 16.42 and 2.44, after considering both the direct and indirect angles that could be formed with the given information.

Solve the Problem α=31∘\alpha = 31 ^ { \circ }α=31∘ , What Is (Are) the Possible Length(s) of C? Round

Solve the Problem $\alpha = 31 ^ { \circ }$ , What Is (Are) the Possible Length(s) of C? Round