1. **State the problem:** We have two similar triangles, LMN and L'M'N', where triangle L'M'N' is a dilation of triangle LMN by a scale factor of 4. 2. **Given:** Scale factor $k = 4$ Side $L'M' = 36$ 3. **Find:** The length of side $LM$ in the original triangle. 4. **Formula:** For similar figures, corresponding sides are proportional by the scale factor: $$ L'M' = k \times LM $$ 5. **Solve for $LM$:** $$ LM = \frac{L'M'}{k} $$ 6. **Substitute values:** $$ LM = \frac{36}{4} $$ 7. **Simplify:** $$ LM = \cancel{\frac{36}{4}} = 9 $$ **Answer:** The length of side $LM$ is 9.