Solve the triangle, if possible. \[ \mathrm{a}=200 \mathrm{~m}, \mathrm{~A}=39.02^{\circ}, \mathrm{C}=28.75^{\circ} \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round side lengths to the nearest whole number and angle measures to the nearest hundredth degree as needed.) A. There is only one possible solution for the triangle. The measurements for the remaining angle \( B \) and sides \( b \) and \( c \) are as follows. \( B \approx \) \( \square \) \( { }^{\circ} \) \( b= \) \( \square \) m \[ c= \] \( \square \) m B. There are two possible sốutions for the triangle. The measurements for the solution with the smaller angle \( B \) are as follows. \( \mathrm{B}_{1} \approx \) \( \square \) \( { }^{\circ} \) \[ b_{1}= \] \( \square \) m \[ c_{1}= \] \( \square \) m The measurements for the solution with the larger angle \( B \) are as follows. \( \mathrm{B}_{2} \approx \) \( \square \) - \[ \mathrm{b}_{2}= \] \( \square \) m \[ c_{2}=\square \] m C. There are no possible solutions for this triangle.

To solve the triangle given \( a = 200 \, \text{m} \), \( A = 39.02^\circ \), and \( C = 28.75^\circ \), we first find angle \( B \) using the fact that the angles in a triangle sum to 180 degrees. \[ B = 180^\circ - A - C = 180^\circ - 39.02^\circ - 28.75^\circ \approx 112.23^\circ \] Now we apply the Law of Sines to find sides \( b \) and \( c \). \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] 1. To find side \( b \): \[ \frac{200}{\sin(39.02^\circ)} = \frac{b}{\sin(112.23^\circ)} \] \[ b \approx 200 \cdot \frac{\sin(112.23^\circ)}{\sin(39.02^\circ)} \approx 200 \cdot \frac{0.9272}{0.6293} \approx 294 \] 2. To find side \( c \): \[ \frac{200}{\sin(39.02^\circ)} = \frac{c}{\sin(28.75^\circ)} \] \[ c \approx 200 \cdot \frac{\sin(28.75^\circ)}{\sin(39.02^\circ)} \approx 200 \cdot \frac{0.4848}{0.6293} \approx 154 \] So, the first choice is valid: A. There is only one possible solution for the triangle. The measurements for the remaining angle \( B \) and sides \( b \) and \( c \) are as follows. \( B \approx 112.23 \, { }^{\circ} \) \( b= 294 \, \text{m} \) \[ c= 154 \, \text{m} \]