Sides of two similar triangles are in the ratio `7 : 8` .Areas of these triangles are in the ratio

Options

A. `8 : 7`
B. `49 : 64`
C. `7 : 15`
D. `64 : 49`

Answer

To find the ratio of the areas of two similar triangles given that the ratio of their corresponding sides is \(7 : 8\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio of Sides**:
The sides of the two similar triangles are given in the ratio \(7 : 8\). Let's denote the sides of the first triangle as \(7x\) and the sides of the second triangle as \(8x\), where \(x\) is a common multiplier.

2. **Use the Area Ratio Theorem**:
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. **Calculate the Area Ratio**:
Substitute the ratio of the sides into the formula:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \left(\frac{7}{8}\right)^2 \]
Now, calculate \(\left(\frac{7}{8}\right)^2\):
\[ \left(\frac{7}{8}\right)^2 = \frac{7^2}{8^2} = \frac{49}{64} \]

4. **Conclusion**:
Therefore, the ratio of the areas of the two triangles is:
\[ \frac{\text{Area of Triangle 1}}{\text{Area of Triangle 2}} = \frac{49}{64} \]

### Final Answer:
The areas of the two triangles are in the ratio \(49 : 64\).