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We are given a triangle \( \triangle KLM \) with the following information:

- \( LM = 12 \)
- \( \angle K = 60^\circ \)
- \( \angle M = 45^\circ \)

To find the perimeter of the triangle, we need to find the lengths of the sides \( KL \) and \( KM \).

### Step 1: Calculate \( \angle L \)
Using the angle sum property of a triangle:
\[
\angle K + \angle M + \angle L = 180^\circ
\]
\[
60^\circ + 45^\circ + \angle L = 180^\circ
\]
\[
\angle L = 180^\circ - 105^\circ = 75^\circ
\]

### Step 2: Use the Law of Sines
The Law of Sines relates the sides and angles of a triangle:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a = KL \), opposite \( \angle M \)
- \( b = KM \), opposite \( \angle L \)
- \( c = LM = 12 \), opposite \( \angle K \)

Using the ratio \( \frac{c}{\sin K} \), we can find the other sides.

#### Find \( KL \):
\[
\frac{KL}{\sin 45^\circ} = \frac{12}{\sin 60^\circ}
\]
\[
KL = \frac{12 \cdot \sin 45^\circ}{\sin 60^\circ}
\]

#### Find \( KM \):
\[
\frac{KM}{\sin 75^\circ} = \frac{12}{\sin 60^\circ}
\]
\[
KM = \frac{12 \cdot \sin 75^\circ}{\sin 60^\circ}
\]

I'll calculate these values now.Using the Law of Sines, we have found the following:

- \( KL \approx 9.80 \)
- \( KM \approx 13.38 \)

Now, the perimeter of \( \triangle KLM \) is the sum of all the sides:
\[
\text{Perimeter} = LM + KL + KM \approx 12 + 9.80 + 13.38 = 35.18
\]

Thus, the perimeter of \( \triangle KLM \) is approximately **35.18 units**.

Would you like more details or clarifications?

Here are some related questions to expand on this:

1. How does the Law of Cosines differ from the Law of Sines?
2. Can the Law of Sines be applied to any type of triangle?
3. What is the significance of knowing all angles in a triangle?
4. How can we find the area of \( \triangle KLM \) using trigonometry?
5. How would the solution change if we had a different side length instead of \( LM \)?

**Tip:** When solving problems involving angles and sides in triangles, converting degrees to radians is often necessary for accurate calculations using trigonometric functions.

Given a triangle KLM with LM = 12, angle K = 60°, and angle M = 45°, find the perimeter of triangle KLM.

Solve for the perimeter of triangle KLM where LM = 12, angle K = 60°, and angle M = 45°. This problem involves using the Law of Sines and the angle sum property of triangles. Suitable for students in Grades 9-12.

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