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We are tasked with finding the perimeter of \( \triangle KLM \) using the given side lengths and the angles from the diagram.

### Known values:
- \( LM = 12 \)
- \( KM = 15.6 \)
- Angle \( \angle LKM = 50^\circ \)
- Angle \( \angle KLM = 55^\circ \)

We need to find the missing side \( KL \).

### Step 1: Apply the Law of Cosines

Using the Law of Cosines to find \( KL \), we have:
\[
KL^2 = KM^2 + LM^2 - 2 \cdot KM \cdot LM \cdot \cos(\angle KML)
\]
The angle \( \angle KML \) is found using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[
\angle KML = 180^\circ - 55^\circ - 50^\circ = 75^\circ
\]
Now applying the Law of Cosines:
\[
KL^2 = 15.6^2 + 12^2 - 2 \cdot 15.6 \cdot 12 \cdot \cos(75^\circ)
\]
\[
KL^2 = 243.36 + 144 - 2 \cdot 15.6 \cdot 12 \cdot \cos(75^\circ)
\]
Using \( \cos(75^\circ) \approx 0.2588 \):
\[
KL^2 = 243.36 + 144 - 2 \cdot 15.6 \cdot 12 \cdot 0.2588
\]
\[
KL^2 = 387.36 - 96.694
\]
\[
KL^2 \approx 290.666
\]
\[
KL \approx \sqrt{290.666} \approx 17.05
\]

### Step 2: Find the Perimeter

Now, the perimeter of \( \triangle KLM \) is:
\[
\text{Perimeter} = KL + LM + KM = 17.05 + 12 + 15.6 = 44.65
\]

### Final Answer:
The perimeter of \( \triangle KLM \) is approximately **44.7** units, rounded to the nearest tenth.

Would you like a further breakdown of any part of this process, or have any additional questions?

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Here are 5 related questions to explore:
1. How is the Law of Cosines applied in non-right triangles?
2. What is the relationship between angles and sides in any triangle?
3. Can the Law of Sines also be used to solve this problem? Why or why not?
4. What are the real-world applications of the Law of Cosines?
5. How do we calculate angles and sides in obtuse triangles differently?

**Tip:** Always ensure the angle sum in any triangle is exactly \(180^\circ\) when calculating unknown angles.

If HI = 11, IJ = 10, HJ = 13, LM = 12, and KM = 15.6, find the perimeter of triangle KLM.

Find the perimeter of triangle KLM using the Law of Cosines. Given side lengths KM = 15.6, LM = 12, and angle KML = 75°, calculate the missing side and the perimeter. Suitable for students in Grades 9-12.

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