`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. We are also given the ratio of the perimeters of the two triangles and the length of side PQ.

### Step-by-Step Solution:

1. **Understand the Similarity of Triangles**:
Since triangle XYZ is similar to triangle PQR, the ratio of their corresponding sides is equal to the ratio of their perimeters.

2. **Set Up the Ratio**:
The ratio of the perimeters of triangle XYZ to triangle PQR is given as 16:9. This means:
\[ \frac{\text{Perimeter of } \Delta XYZ}{\text{Perimeter of } \Delta PQR} = \frac{16}{9} \]

3. **Identify the Corresponding Sides**:
Let’s denote the sides of triangle PQR as follows:
- PQ = 3.6 cm
- QR = x (unknown)
- PR = y (unknown)

For triangle XYZ, we denote the corresponding sides as:
- XY = a (unknown)
- YZ = b (unknown)
- XZ = c (unknown)

4. **Use the Ratio to Find XY**:
Since the triangles are similar, we can write:
\[ \frac{XY}{PQ} = \frac{16}{9} \]
Substituting the value of PQ:
\[ \frac{XY}{3.6} = \frac{16}{9} \]

5. **Cross Multiply to Solve for XY**:
Cross multiplying gives us:
\[ 9 \cdot XY = 16 \cdot 3.6 \]
Simplifying the right side:
\[ 9 \cdot XY = 57.6 \]

6. **Divide Both Sides by 9**:
To find XY, we divide both sides by 9:
\[ XY = \frac{57.6}{9} = 6.4 \text{ cm} \]

### Final Answer:
The length of XY is **6.4 cm**.