Similar Triangles Perimeter Calculation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Given Information

• The two triangles, $\Delta\text{XYZ}$ and $\Delta\text{LMN}$, are similar.
• Perimeter of $\Delta\text{XYZ}$ ($P_{\text{XYZ}}$) = $48\text{ cm}$.
• Perimeter of $\Delta\text{LMN}$ ($P_{\text{LMN}}$) = $32\text{ cm}$.
• Length of side $\text{LM}$ in $\Delta\text{LMN}$ = $12\text{ cm}$.
• We need to find the length of the corresponding side $\text{XY}$ in $\Delta\text{XYZ}$.

Applying Perimeter-Side Ratio Theorem

The theorem states that for similar triangles:

$ \frac{\text{Perimeter of } \Delta\text{XYZ}}{\text{Perimeter of } \Delta\text{LMN}} = \frac{\text{XY}}{\text{LM}} $

Calculating XY Length

Substitute the given values into the formula:

$ \frac{48\text{ cm}}{32\text{ cm}} = \frac{\text{XY}}{12\text{ cm}} $

Simplify the ratio of the perimeters:

$ \frac{48}{32} = \frac{3 \times 16}{2 \times 16} = \frac{3}{2} $

Now, the equation becomes:

$ \frac{3}{2} = \frac{\text{XY}}{12\text{ cm}} $

To find $\text{XY}$, multiply both sides by $12\text{ cm}$:

$ \text{XY} = \frac{3}{2} \times 12\text{ cm} $

$ \text{XY} = 3 \times \frac{12}{2}\text{ cm} $

$ \text{XY} = 3 \times 6\text{ cm} $

$ \text{XY} = 18\text{ cm} $

Final Answer

The length of side $\text{XY}$ is $18\text{ cm}$.