`DeltaXYZ` is similar to `DeltaPQR`.If ratio of perimeter of `DeltaXYZ` and perimeter of `DeltaPQR` is 16 : 9 and PQ = 3.6 cm, then what is the length (in cm) of XY?

Options

A. `4.8`
B. `3.2`
C. `6.4`
D. `8.6`

Answer

To solve the problem, we need to find the length of segment XY in triangle XYZ, given that triangle XYZ is similar to triangle PQR. The ratio of the perimeters of these triangles is given as 16:9, and the length of side PQ is 3.6 cm.

### Step-by-Step Solution:

1. **Understand the Ratio of Similar Triangles:**
Since triangle XYZ is similar to triangle PQR, the ratio of corresponding sides will be the same as the ratio of their perimeters. Therefore, if the ratio of the perimeters is 16:9, the ratio of corresponding sides will also be 16:9.

2. **Set Up the Ratio:**
Let the length of side XY be represented as \( XY \). According to the similarity ratio:
\[ \frac{XY}{PQ} = \frac{16}{9} \]

3. **Substitute the Known Length:**
We know that \( PQ = 3.6 \) cm. Substituting this value into the ratio gives:
\[ \frac{XY}{3.6} = \frac{16}{9} \]

4. **Cross-Multiply to Solve for XY:**
To find \( XY \), we can cross-multiply:
\[ XY \cdot 9 = 16 \cdot 3.6 \]

5. **Calculate the Right Side:**
Now, calculate \( 16 \cdot 3.6 \):
\[ 16 \cdot 3.6 = 57.6 \]

6. **Solve for XY:**
Now, we have:
\[ 9XY = 57.6 \]
To find \( XY \), divide both sides by 9:
\[ XY = \frac{57.6}{9} = 6.4 \]

7. **Final Answer:**
Therefore, the length of \( XY \) is \( 6.4 \) cm.

### Summary:
The length of segment XY in triangle XYZ is \( 6.4 \) cm.