Use the given information to solve right triangle \( A B C \) for all missing parts. \( A=31.5^{\circ}, a=7 \frac{3}{4} \mathrm{in} \). \( B=58.5^{\circ} \) (Type an integer or a decimal.) \( \mathrm{b} \approx 12.6 \) in. (Type an integer or a decimal. Round to the nearest tenth as needed.) \( \mathrm{c} \approx \square \) in. (Type an integer or a decimal. Round to the nearest tenth as needed.)

To solve the right triangle \( ABC \) with the given information, we'll follow these steps: ### Given: - Angle \( A = 31.5^\circ \) - Side \( a = 7 \frac{3}{4} \) inches \( = 7.75 \) inches - Angle \( B = 58.5^\circ \) - Side \( b \approx 12.6 \) inches Since this is a right triangle, the right angle is at \( C \), so \( C = 90^\circ \). ### Objective: Find the length of the hypotenuse \( c \). ### Steps to Find \( c \): 1. **Using the Sine Function:** \[ \sin A = \frac{a}{c} \] Rearranging to solve for \( c \): \[ c = \frac{a}{\sin A} \] 2. **Calculate \( \sin(31.5^\circ) \):** \[ \sin(31.5^\circ) \approx 0.5225 \] 3. **Compute \( c \):** \[ c = \frac{7.75}{0.5225} \approx 14.8 \text{ inches} \] ### Conclusion: The length of the hypotenuse \( c \) is approximately **14.8 inches**. **Answer:** \( \mathrm{c} \approx 14.8 \) in.