The ratio between the corresponding sides of two similar triangles is 2 is to 5. Find the ratio between the areas of these triangles. 

Answer

To find the ratio between the areas of two similar triangles when the ratio of their corresponding sides is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio of Sides**:
We are given that the ratio of the corresponding sides of two similar triangles is \(2:5\).

2. **Use the Formula for Area Ratios**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This can be expressed mathematically as:
\[ \text{Ratio of Areas} = \left(\frac{\text{Side 1}}{\text{Side 2}}\right)^2 \]

3. **Substitute the Given Ratio**:
Here, the ratio of the sides is \( \frac{2}{5} \). Therefore, we can substitute this into the formula:
\[ \text{Ratio of Areas} = \left(\frac{2}{5}\right)^2 \]

4. **Calculate the Square of the Ratio**:
Now, we calculate the square of \( \frac{2}{5} \):
\[ \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} \]

5. **Final Result**:
Thus, the ratio of the areas of the two triangles is \( \frac{4}{25} \).

### Conclusion:
The ratio between the areas of the two similar triangles is \(4:25\).

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