Solve the triangle $A B C$, if the triangle exists. \[ A=41.5^{\circ} \quad a=8.7 m \quad b=10.4 m \] Select the correct choice below and fill in the answer boxes within the choice. A. There is only 1 possible solution for the triangle. The measurements for the remaining angles $B$ and $C$ and side $c$ are as follows. \[ \mathrm{m} \angle \mathrm{B}= \] \[ \mathrm{m} \angle \mathrm{C}=7^{\circ} \] The length of side $\mathrm{c}=$ (Round to the nearest (Round to the nearest (Round to the nearest tenth tenth as needed.) tenth as needed.) as needed.) B. There are 2 possible solutions for the triangle. The measurements for the solution with the longer side $c$ are as follows. $\mathrm{m} \angle \mathrm{B}=$ \[ \mathrm{m} \angle \mathrm{C}= \] The length of side $\mathrm{c}=$ (Round to the nearest (Round to the nearest (Round to the nearest tenth tenth as needed.) tenth as needed.) as needed.) The measurements for the solution with the shorter side $\mathrm{c}$ are as follows. \[ \mathrm{m} \angle \mathrm{B}=\mathrm{L}^{\circ} \] $\mathrm{m} \angle \mathrm{C}=$ (Round to the nearest (Round to the nearest tenth as needed.) The length of side $\mathrm{c}=$ (Round to the nearest tenth as needed.) C. There are no possible solutions for this triangle.
Solution
Step 1 ：We are given two sides and an angle opposite to one of the sides in a triangle. This is a case of solving a triangle using the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Step 2 ：We can use this law to find the angle B first. After that, we can find angle C by subtracting the sum of angles A and B from 180 (since the sum of angles in a triangle is 180 degrees).
Step 3 ：Finally, we can find the length of side c using the Law of Sines again.
Step 4 ：\[\mathrm{m} \angle \mathrm{B}= \boxed{52.4^\circ}\]
Step 5 ：\[\mathrm{m} \angle \mathrm{C}= \boxed{2.2^\circ}\]
Step 6 ：The length of side \[\mathrm{c}= \boxed{10.5 m}\]
Step 7 ：So, there is only 1 possible solution for the triangle.
