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In this isosceles triangle \( \triangle LMN \), we are asked to find the length of side \( LM \), given the following:

- \( \angle L = 62^\circ \)
- \( MN = 6.8 \, \text{cm} \)
- \( LM = LN \) (since the triangle is isosceles)

### Solution:

1. First, we need to find the other two angles, \( \angle M \) and \( \angle N \), in the isosceles triangle. In any triangle, the sum of the interior angles is \( 180^\circ \). For this triangle, the two base angles \( \angle M \) and \( \angle N \) are equal because it's an isosceles triangle. Let's denote these angles as \( \theta \).

\[
\theta + \theta + 62^\circ = 180^\circ
\]

\[
2\theta = 180^\circ - 62^\circ
\]

\[
2\theta = 118^\circ
\]

\[
\theta = 59^\circ
\]

So, \( \angle M = \angle N = 59^\circ \).

2. Now that we have all the angles, we can use the Law of Sines to find the length of \( LM \). The Law of Sines states:

\[
\frac{LM}{\sin(59^\circ)} = \frac{MN}{\sin(62^\circ)}
\]

Substitute the known values:

\[
\frac{LM}{\sin(59^\circ)} = \frac{6.8}{\sin(62^\circ)}
\]

Now, solve for \( LM \):

\[
LM = \frac{6.8 \times \sin(59^\circ)}{\sin(62^\circ)}
\]

Using a calculator to compute the sine values:

\[
LM = \frac{6.8 \times 0.8572}{0.8829}
\]

\[
LM \approx \frac{5.82896}{0.8829} \approx 6.6 \, \text{cm}
\]

Thus, the length of side \( LM \) is approximately \( 6.6 \, \text{cm} \).

Would you like more details on any of the steps?

### Related questions:
1. How would the solution change if the angle at \( L \) was different?
2. Can you use the Law of Cosines to solve this problem instead?
3. What if the triangle were equilateral instead of isosceles?
4. How does the Law of Sines apply to non-isosceles triangles?
5. How can we check the accuracy of the sine values used?

### Tip:
Always verify that your angle measurements are correct when using trigonometric formulas like the Law of Sines.

Find LM in this isosceles triangle with angle L = 62°, MN = 6.8 cm, and LM = LN.

Learn how to find the length of side LM in an isosceles triangle where angle L is 62°, MN is 6.8 cm, and LM = LN using the Law of Sines. This step-by-step solution covers key trigonometric concepts and geometry. Ideal for students in Grades 8-10.

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