Solve the triangle $A B C$, if the triangle exists.
\[
A=43.5^{\circ} \quad a=8.3 m \quad b=10.3 \mathrm{~m}
\]
Select the correct choice below and fill in the answer boxes within the choice.
A. There are 2 possible solutions for the triangle. The measurements for the solution with the longer side $\mathrm{c}$ are as follows.
\[
\mathrm{m} \angle \mathrm{C}=\square^{\circ}
\]
The length of side $c=$
(Round to the nearest
(Round to the nearest tenth as needed.) tenth as needed.) (Round to the nearest tenth The measurements for the solution with the shorter side $\mathrm{c}$ are as follows.
\[
\mathrm{m} \angle \mathrm{B}=
\]
\[
\mathrm{m} \angle \mathrm{C}=
\]
(Round to the nearest (Round to the nearest tenth as needed.)
The length of side $c=$
(Round to the nearest tenth as needed.)
B. There is only 1 possible solution for the triangle. The measurements for the remaining angles $B$ and $C$ and side $c$ are as follows. $\mathrm{m} \angle \mathrm{B}=$ $\mathrm{m} \angle \mathrm{C}=$ (Round to the nearest (Round to the nearest tenth as needed.) tenth as needed.) The length of side $c=$ (Round to the nearest tenth as needed.)
c. There are no possible solutions for this triangle.
Answer
Final Answer: The measurements for the remaining angles B and C and side c are as follows. \(\mathrm{m} \angle \mathrm{B}= \boxed{58.7^\circ}\) \(\mathrm{m} \angle \mathrm{C}= \boxed{77.8^\circ}\) The length of side c= \(\boxed{11.8 \mathrm{~m}}\). Therefore, there is only 1 possible solution for the triangle.
Step 1 ：We are given two sides and an included angle of a triangle. We can use the Law of Sines to find the other angles and sides. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides and angles.
Step 2 ：We can first find angle B using the formula: \[B = \sin^{-1}\left(\frac{b \cdot \sin(A)}{a}\right)\]
Step 3 ：Then, we can find angle C using the formula: \[C = 180 - A - B\]
Step 4 ：Finally, we can find side c using the formula: \[c = a \cdot \sin(C) / \sin(A)\]
Step 5 ：Substituting the given values into the formulas, we get: \[B = \sin^{-1}\left(\frac{10.3 \cdot \sin(43.5)}{8.3}\right) = 58.7^\circ\] \[C = 180 - 43.5 - 58.7 = 77.8^\circ\] \[c = 8.3 \cdot \sin(77.8) / \sin(43.5) = 11.8 m\]
Step 6 ：Final Answer: The measurements for the remaining angles B and C and side c are as follows. \(\mathrm{m} \angle \mathrm{B}= \boxed{58.7^\circ}\) \(\mathrm{m} \angle \mathrm{C}= \boxed{77.8^\circ}\) The length of side c= \(\boxed{11.8 \mathrm{~m}}\). Therefore, there is only 1 possible solution for the triangle.
