In ΔKLM, l = 210 inches, mm∠K=116° and mm∠L=11°. Find the length of m, to the nearest inch.

To find the length of side \( m \) in triangle \( \Delta KLM \), we can use the Law of Sines, which states:

\[
\frac{m}{\sin(\angle K)} = \frac{l}{\sin(\angle M)}
\]

Given:
- \( l = 210 \) inches
- \( \angle K = 116^\circ \)
- \( \angle L = 11^\circ \)

First, we need to find \( \angle M \), the remaining angle in the triangle:

\[
\angle M = 180^\circ - \angle K - \angle L = 180^\circ - 116^\circ - 11^\circ = 53^\circ
\]

Now apply the Law of Sines:

\[
\frac{m}{\sin(116^\circ)} = \frac{210}{\sin(53^\circ)}
\]

To find \( m \):

\[
m = \frac{210 \times \sin(116^\circ)}{\sin(53^\circ)}
\]

Now calculate the sine values and solve for \( m \):

\[
m = \frac{210 \times \sin(116^\circ)}{\sin(53^\circ)} \approx \frac{210 \times 0.8829}{0.7986} \approx \frac{185.409}{0.7986} \approx 232.2 \text{ inches}
\]

To the nearest inch:

\[
m \approx 232 \text{ inches}
\]

The length of side \( m \) is approximately **232 inches**.

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How would the length of \( m \) change if \( \angle K \) were smaller?
2. What are some common applications of the Law of Sines in real life?
3. Can you use the Law of Cosines to find side \( m \) in this triangle? Why or why not?
4. How does the sum of the angles in a triangle help in solving for unknown sides?
5. What is the importance of accurate angle measurements in trigonometry?

**Tip:** When using trigonometric functions, remember that the accuracy of your result depends on the precision of your angle values and sine calculations.

Learn how to find the length of side m in triangle KLM using the Law of Sines. Given side l = 210 inches, angle K = 116°, and angle L = 11°, calculate the length of side m to the nearest inch. Detailed step-by-step solution included.

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