Use the Law of Sines to Solve, if Possible, the Triangle or Triangles in the Ambiguous Case
-$$\mathrm { A } = 85 ^ { \circ } , \mathrm { a } = 4 , \mathrm {~b} = 5$$
A) no triangle
B) $B = 44 ^ { \circ } , C = 51 ^ { \circ } , \mathrm { C } = 13$
C) $\mathrm { A } = 43 ^ { \circ } , \mathrm { C } = 52 ^ { \circ } , \mathrm { c } = 9$
D) $B = 42 ^ { \circ } , C = 53 ^ { \circ } , \mathrm { C } = 11$

Question

Quizplus · Accepted Answer

Given $A = 85^\circ$, $a = 4$, and $b = 5$, we use the Law of Sines to determine the possibility of a triangle. The Law of Sines states that $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$. For the given values, if $b > a$ and $A$ is acute, there is no ambiguity because the side opposite the given angle is longer, ensuring at least one solution. However, to determine if a triangle is possible, we must check if the angle $B$ can exist such that $B < 180^\circ - A$. In this case, since $b > a$ and $A$ is large, there is no issue of the ambiguous case (where two different triangles could be formed). The real question is whether any triangle can be formed at all. Given $A = 85^\circ$, the angle is too large for the given sides to form a triangle when considering the sine rule and the spatial constraints of triangle geometry. Specifically, for a triangle to exist, the side opposite the larger angle must be the longest, which is not the case here. Therefore, no triangle can be formed with the given measurements.

Use the Law of Sines to Solve, If Possible, the Triangle