In triangle $△\mathrm{lmn}$, side $l$ is $170$ inches, angle $\angle m$ is $121°$, and angle $\angle n$ is $40°$. Find the length of side $n$ to the nearest inch.

• A. $74$ inches
• B. $112$ inches
• C. $99$ inches
• D. $130$ inches

Answer: (B)

Verified step by step guidance

1. Identify the given elements in triangle \( \triangle lmn \): side \( l = 170 \) inches, angle \( \angle m = 121^\circ \), and angle \( \angle n = 40^\circ \).
2. Calculate the measure of the third angle \( \angle l \) using the triangle angle sum property: \( \angle l = 180^\circ - \angle m - \angle n \).
3. Use the Law of Sines, which states that \( \frac{\text{side } l}{\sin(\angle l)} = \frac{\text{side } n}{\sin(\angle n)} \), to set up an equation to find side \( n \).
4. Rearrange the Law of Sines equation to solve for side \( n \): \[ n = \frac{l \cdot \sin(\angle n)}{\sin(\angle l)} \].
5. Substitute the known values of \( l \), \( \angle n \), and \( \angle l \) into the equation and compute the sine values to find the length of side \( n \).