In triangle $△\mathrm{klm}$, $l=210$ inches, $\angle k=116°$, and $\angle l=11°$. Find the length of $m$ to the nearest inch.

• A. $102$
• B. $1051$
• C. $41$
• D. $1056$

Answer: (B)

Verified step by step guidance

1. Identify the given elements in triangle \( \triangle klm \): side \( l = 210 \) inches, angle \( \angle k = 116^\circ \), and angle \( \angle l = 11^\circ \). We need to find the length of side \( m \).
2. Calculate the measure of the third angle \( \angle m \) using the triangle angle sum property: \( \angle m = 180^\circ - \angle k - \angle l \).
3. Use the Law of Sines, which states that \( \frac{l}{\sin(\angle m)} = \frac{m}{\sin(\angle l)} = \frac{k}{\sin(\angle k)} \), to set up a proportion involving side \( m \) and the known side \( l \).
4. Express side \( m \) in terms of the known quantities: \[ m = \frac{l \cdot \sin(\angle l)}{\sin(\angle m)} \].
5. Substitute the known values of \( l \), \( \angle l \), and \( \angle m \) into the formula and solve for \( m \). Round the result to the nearest inch.