The ratio between the altitudes of two similar triangles is 3 : 5, write the ratio between their :
 areas. 

Answer

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

### Step-by-Step Solution:

1. **Understand the Relationship**: We know that the ratio of the altitudes of two similar triangles is given as \(3:5\). Let's denote the altitudes as \(AM\) for triangle \(ABC\) and \(PN\) for triangle \(PQR\). Thus, we have:
\[ \frac{AM}{PN} = \frac{3}{5} \]

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 (or altitudes). Therefore, if the ratio of the altitudes is \( \frac{3}{5} \), the ratio of the areas can be found using:
\[ \text{Area Ratio} = \left(\frac{AM}{PN}\right)^2 \]

3. **Calculate the Area Ratio**: Substitute the given ratio of the altitudes into the formula:
\[ \text{Area Ratio} = \left(\frac{3}{5}\right)^2 \]
\[ \text{Area Ratio} = \frac{3^2}{5^2} = \frac{9}{25} \]

4. **Conclusion**: Therefore, the ratio of the areas of triangle \(ABC\) to triangle \(PQR\) is:
\[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle PQR} = \frac{9}{25} \]

### Final Answer:
The ratio of the areas of the two triangles is \(9:25\).

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