If the ratio of the sides of two similar triangle is `3:5`, then ratio of the areas is :

Options

A. `sqrt3 :sqrt5`
B. `9:25`
C. `5:3`
D. None of these.

Answer

To find the ratio of the areas of two similar triangles when the ratio of their 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 sides of two similar triangles is \(3:5\).

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. This can be expressed mathematically as:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \left(\frac{\text{Side of Triangle 1}}{\text{Side of Triangle 2}}\right)^2 \]

3. **Substitute the Given Ratio**:
Substitute the given ratio of the sides into the formula:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \left(\frac{3}{5}\right)^2 \]

4. **Calculate the Square of the Ratio**:
Now, calculate the square of \(\frac{3}{5}\):
\[ \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \]

5. **State the Final Ratio of Areas**:
Thus, the ratio of the areas of the two triangles is:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \frac{9}{25} \]

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

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