79. A triangular bike frame has dimensions shown below in inches. Find the length of the missing side. Round to the nearest hundredth. \( \mathrm{a}=25, \mathrm{~b}=21, m \angle C=31^{\circ} \)

To find the length of the missing side \( c \) in the triangular bike frame with dimensions \( a = 25 \), \( b = 21 \), and \( m \angle C = 31^\circ \), you can use the Law of Cosines. The formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Plugging in the values: \[ c^2 = 25^2 + 21^2 - 2 \cdot 25 \cdot 21 \cdot \cos(31^\circ) \] After calculating: - \( 25^2 = 625 \) - \( 21^2 = 441 \) - \( 2 \cdot 25 \cdot 21 \approx 1050 \) - \( \cos(31^\circ) \approx 0.8572 \) Now combine: \[ c^2 \approx 625 + 441 - 1050 \cdot 0.8572 \] \[ c^2 \approx 625 + 441 - 900 \] \[ c^2 \approx 166 \] Taking the square root gives: \[ c \approx \sqrt{166} \approx 12.85 \] So, the length of the missing side is approximately \( 12.85 \) inches when rounded to the nearest hundredth.