The diagonals of two squares are in the ratio of 3:7. What is the ratio of their areas ?

Options

A. `3:7`
B. `7:49`
C. `4:7`
D. `7:3`

Answer

To solve the problem, we need to find the ratio of the areas of two squares given that the diagonals of the squares are in the ratio of 3:7.

### Step-by-Step Solution:

1. **Understanding the Diagonal of a Square**:
The diagonal \( d \) of a square with side length \( x \) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2} \]

2. **Expressing Side Length in Terms of Diagonal**:
From the diagonal formula, we can express the side length \( x \) in terms of the diagonal \( d \):
\[ x = \frac{d}{\sqrt{2}} \]

3. **Finding the Area of the Square**:
The area \( A \) of a square is given by:
\[ A = x^2 \]
Substituting \( x \) from the previous step, we get:
\[ A = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2} \]

4. **Calculating the Areas for Both Squares**:
Let the diagonals of the two squares be \( d_1 \) and \( d_2 \) such that:
\[ \frac{d_1}{d_2} = \frac{3}{7} \]
We can express \( d_1 \) and \( d_2 \) as:
\[ d_1 = 3k \quad \text{and} \quad d_2 = 7k \quad \text{for some } k \]

5. **Finding the Areas**:
Now, we can find the areas \( A_1 \) and \( A_2 \) of the squares:
\[ A_1 = \frac{(d_1)^2}{2} = \frac{(3k)^2}{2} = \frac{9k^2}{2} \]
\[ A_2 = \frac{(d_2)^2}{2} = \frac{(7k)^2}{2} = \frac{49k^2}{2} \]

6. **Finding the Ratio of the Areas**:
Now, we can find the ratio of the areas \( A_1 \) to \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{\frac{9k^2}{2}}{\frac{49k^2}{2}} = \frac{9}{49} \]

### Final Answer:
The ratio of the areas of the two squares is:
\[ \boxed{9:49} \]