The ratio between the areas of two similar triangles is 16 : 25. Find the ratio between their :
perimeters. 

Answer

To find the ratio between the perimeters of two similar triangles given the ratio of their areas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that the ratio of the areas of two similar triangles is 16:25.

2. **Use the Property of Similar Triangles**:
For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This means:
\[ \text{Ratio of Areas} = \left(\text{Ratio of corresponding sides}\right)^2 \]

3. **Set Up the Equation**:
Let the ratio of the corresponding sides be \( k \). Then, we can express the ratio of the areas as:
\[ k^2 = \frac{16}{25} \]

4. **Find the Ratio of Corresponding Sides**:
To find \( k \), we take the square root of both sides:
\[ k = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \]

5. **Conclude the Ratio of Perimeters**:
Since the ratio of the perimeters of similar triangles is the same as the ratio of their corresponding sides, the ratio of the perimeters is also:
\[ \text{Ratio of Perimeters} = \frac{4}{5} \]

### Final Answer:
The ratio of the perimeters of the two similar triangles is \( 4:5 \).

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