A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate:
the length of LM, if L' M' = 5-4 cm. 

Answer

To solve the problem, we need to find the length of side LM of triangle LMN, given that the triangle has been reduced by a scale factor of 0.8 to triangle L'M'N', and that the length of side L'M' is 5.4 cm.

### Step-by-Step Solution:

1. **Understand the Scale Factor**: The scale factor of reduction is given as 0.8. This means that each side of the original triangle (LMN) is 1/0.8 times the corresponding side of the reduced triangle (L'M'N').

2. **Set Up the Relationship**: The relationship between the lengths of the sides of the original triangle and the reduced triangle can be expressed as:
\[ \frac{L'M'}{LM} = \text{Scale Factor} \]
Here, \(L'M' = 5.4 \, \text{cm}\) and the scale factor is \(0.8\).

3. **Substitute the Known Values**: Plugging in the known values into the equation gives:
\[ \frac{5.4}{LM} = 0.8 \]

4. **Cross-Multiply to Solve for LM**: To find LM, we can cross-multiply:
\[ 5.4 = 0.8 \times LM \]

5. **Isolate LM**: To isolate LM, divide both sides by 0.8:
\[ LM = \frac{5.4}{0.8} \]

6. **Calculate LM**: Performing the division:
\[ LM = 6.75 \, \text{cm} \]

### Final Answer:
The length of LM is **6.75 cm**.