The ratio of the corresponding altitudes of two similar triangles is `2/5`. Is it correct to say that ratio of their areas is also `2/5`? Why?

Answer

To determine whether the ratio of the areas of two similar triangles is also \( \frac{2}{5} \) when the ratio of their corresponding altitudes is \( \frac{2}{5} \), we can use the properties of similar triangles.

### Step-by-Step Solution:

1. **Understand the Properties of Similar Triangles**:
- For two similar triangles, the ratio of their corresponding sides (or altitudes) is equal.
- The ratio of their areas is equal to the square of the ratio of their corresponding sides (or altitudes).

2. **Identify the Given Ratio**:
- The ratio of the corresponding altitudes of the two similar triangles is given as \( \frac{2}{5} \).

3. **Apply the Area of Similar Triangle Theorem**:
- According to the theorem, if the ratio of the corresponding altitudes (or sides) is \( \frac{a}{b} \), then the ratio of the areas of the triangles is given by:
\[ \text{Ratio of Areas} = \left(\frac{a}{b}\right)^2 \]
- In our case, \( a = 2 \) and \( b = 5 \).

4. **Calculate the Ratio of the Areas**:
- Substitute the values into the formula:
\[ \text{Ratio of Areas} = \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} \]

5. **Conclusion**:
- The ratio of the areas of the two triangles is \( \frac{4}{25} \).
- Since \( \frac{4}{25} \) is not equal to \( \frac{2}{5} \), it is incorrect to say that the ratio of their areas is also \( \frac{2}{5} \).

### Final Answer:
No, it is not correct to say that the ratio of their areas is \( \frac{2}{5} \). The correct ratio of their areas is \( \frac{4}{25} \).

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