Question: The image contains a triangle labeled \( \triangle MNP \) with the following information: -…

The image contains a triangle labeled \( \triangle MNP \) with the following information:

• Angle \( M = 35^\circ \)
• Angle \( P = 65^\circ \)
• Side \( p = 20 \)

The question asks for the length of side \( m \) in \( \triangle MNP \). The options provided are:

• 12.7
• 38.5
• 10.4
• 31.6

The triangle is not drawn to scale.

Solution

To find the length of side \( m \) in \(\triangle MNP\), we can use the Law of Sines. Given \( \angle M = 35^\circ \), \( \angle P = 65^\circ \), and \( p = 20 \). First, find \( \angle N \): \[ \angle N = 180^\circ - \angle M - \angle P \] \[ \angle N = 180^\circ - 35^\circ - 65^\circ = 80^\circ \] Next, apply the Law of Sines: \[ \frac{m}{\sin 65^\circ} = \frac{p}{\sin 80^\circ} \] Substitute \( p = 20 \): \[ \frac{m}{\sin 65^\circ} = \frac{20}{\sin 80^\circ} \] Now, solve for \( m \): \[ m = \frac{20 \cdot \sin 65^\circ}{\sin 80^\circ} \] Calculate \( m \) using a calculator: \[ m \approx \frac{20 \cdot 0.9063}{0.9848} \] \[ m \approx \frac{18.126}{0.9848} \] \[ m \approx 18.4 \] Rounding to the nearest tenth, the length of side \( m \) is 18.4.