The ratio of the sides of two similar triangles is 3:7, ratio of areas of these two triangles is :

Options

A. `15:3.5`
B. `9:49`
C. `6:14`
D. `49:9`

Answer

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

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that the ratio of the sides of two similar triangles is \(3:7\).

2. **Set Up the Ratio of Sides**:
Let the sides of the two triangles be represented as:
\[ \text{Side of Triangle 1} = 3k \quad \text{and} \quad \text{Side of Triangle 2} = 7k \]
where \(k\) is a common multiplier.

3. **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:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \left(\frac{\text{Side of Triangle 1}}{\text{Side of Triangle 2}}\right)^2 \]

4. **Substitute the Ratio of Sides**:
Substitute the ratio of the sides into the formula:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \left(\frac{3}{7}\right)^2 \]

5. **Calculate the Square of the Ratio**:
Now, calculate \(\left(\frac{3}{7}\right)^2\):
\[ \left(\frac{3}{7}\right)^2 = \frac{3^2}{7^2} = \frac{9}{49} \]

6. **State the Final Ratio of Areas**:
Therefore, the ratio of the areas of the two triangles is:
\[ \text{Ratio of Areas} = 9:49 \]

### Final Answer:
The ratio of the areas of the two triangles is \(9:49\).

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