In triangle $△\mathrm{klm}$, side $m$ has length $63$ inches, angle $\angle k$ is $24°$, and angle $\angle l$ is $42°$. Find the length of side $l$ to the nearest inch.

• A. $29$ inches
• B. $38$ inches
• C. $54$ inches
• D. $45$ inches

Answer: (A)

Verified step by step guidance

1. Identify the given elements in triangle \( \triangle klm \): side \( m = 63 \) inches, angle \( \angle k = 24^\circ \), and angle \( \angle l = 42^\circ \).
2. Calculate the measure of the third angle \( \angle m \) using the triangle angle sum property: \( \angle m = 180^\circ - \angle k - \angle l = 180^\circ - 24^\circ - 42^\circ \).
3. Use the Law of Sines, which states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: \[ \frac{l}{\sin(\angle l)} = \frac{m}{\sin(\angle m)} \].
4. Substitute the known values into the Law of Sines formula: \[ \frac{l}{\sin(42^\circ)} = \frac{63}{\sin(\angle m)} \].
5. Solve for side \( l \) by isolating it: \[ l = \frac{63 \times \sin(42^\circ)}{\sin(\angle m)} \]. Then, calculate the numerical value and round to the nearest inch.