Solve the triangle, if possible. Determine the number of possible solutions
\[
A=77.4^{\circ} \quad a=11.7 \quad b=9.9
\]
Select the correct choice below and fill in the answer boxes within the choice. (Round to the nearest tenth as needed.)
A. There is only 1 possible solution for the triangle. The measurements for the remaining angles $\mathrm{A}$ and $\mathrm{C}$ and side $\mathrm{C}$ are as follows. $\mathrm{m} \angle \mathrm{B}=$
\[
\mathrm{m} \angle \mathrm{C}=
\]
The length of side $c=$
B. There are 2 possible solutions for the triangle. The measurements for the solution with the longer side $\mathrm{c}$ are as follows. $\mathrm{m} \angle \mathrm{B}=\mathrm{T}^{\circ} \quad \mathrm{m} \angle \mathrm{c}=\square^{\circ} \quad$ The length of side $\mathrm{c}=$ The measurements for the solution with the shorter side $\mathrm{C}$ are as follows. $\mathrm{m} \angle \mathrm{B}=$ " $^{\circ}$ $\mathrm{m} \angle \mathrm{C}=$
The length of side $c=$
c. There are no possible solutions for the triangle.
Answer
Final Answer: There is only 1 possible solution for the triangle. The measurements for the remaining angles B and C and side C are as follows: \(\boxed{\mathrm{m} \angle \mathrm{B}= 55.7^{\circ}}\), \(\boxed{\mathrm{m} \angle \mathrm{C}= 46.9^{\circ}}\), and the length of side \(c=\boxed{8.8}\).
Step 1 ：Given the triangle with \(A = 77.4^{\circ}\), \(a = 11.7\), and \(b = 9.9\).
Step 2 ：Use the Law of Sines to find angle B: \(B = \sin^{-1}\left(\frac{b \cdot \sin(A)}{a}\right)\).
Step 3 ：Substitute the given values into the formula to get \(B = \sin^{-1}\left(\frac{9.9 \cdot \sin(77.4)}{11.7}\right)\), which gives \(B = 55.7^{\circ}\).
Step 4 ：Find angle C using the formula: \(C = 180 - A - B\).
Step 5 ：Substitute the values of A and B into the formula to get \(C = 180 - 77.4 - 55.7\), which gives \(C = 46.9^{\circ}\).
Step 6 ：Find side c using the formula: \(c = a \cdot \sin(C) / \sin(A)\).
Step 7 ：Substitute the values of a, C, and A into the formula to get \(c = 11.7 \cdot \sin(46.9) / \sin(77.4)\), which gives \(c = 8.8\).
Step 8 ：Final Answer: There is only 1 possible solution for the triangle. The measurements for the remaining angles B and C and side C are as follows: \(\boxed{\mathrm{m} \angle \mathrm{B}= 55.7^{\circ}}\), \(\boxed{\mathrm{m} \angle \mathrm{C}= 46.9^{\circ}}\), and the length of side \(c=\boxed{8.8}\).
