Questions: m = 3 n = 5 M, D = ?, N

Solution Steps

• Triangle \( \triangle OMN \) with \( \angle O = 15^\circ \), \( \angle M = 120^\circ \), and \( OM = 11 \).

• Use the fact that the sum of angles in a triangle is \( 180^\circ \). \[ \angle N = 180^\circ - \angle O - \angle M \] \[ \angle N = 180^\circ - 15^\circ - 120^\circ \] \[ \angle N = 45^\circ \]

• The Law of Sines states: \[ \frac{ON}{\sin(\angle M)} = \frac{OM}{\sin(\angle N)} \] \[ \frac{ON}{\sin(120^\circ)} = \frac{11}{\sin(45^\circ)} \] \[ \frac{ON}{\frac{\sqrt{3}}{2}} = \frac{11}{\frac{\sqrt{2}}{2}} \] \[ ON \cdot \frac{\sqrt{2}}{2} = 11 \cdot \frac{\sqrt{3}}{2} \] \[ ON \cdot \sqrt{2} = 11 \sqrt{3} \] \[ ON = \frac{11 \sqrt{3}}{\sqrt{2}} \] \[ ON = 11 \sqrt{\frac{3}{2}} \]

Final Answer