The ratio of the areas of two squares one having double its diagonal than the other is ?

Options

A. `3:2`
B. `2:1`
C. `4:1`
D. `3:1`

Answer

To find the ratio of the areas of two squares where one square has a diagonal that is double the diagonal of the other square, we can follow these steps:

### Step-by-Step Solution

1. **Define the Diagonals**:
Let the diagonal of the smaller square (Square S2) be \(d\) cm. Therefore, the diagonal of the larger square (Square S1) will be \(2d\) cm.

2. **Area of Square Formula**:
The area \(A\) of a square can be calculated using the formula:
\[ A = \frac{1}{2} \times (\text{diagonal})^2 \]

3. **Calculate the Area of S1**:
Using the diagonal of Square S1:
\[ A_{S1} = \frac{1}{2} \times (2d)^2 = \frac{1}{2} \times 4d^2 = 2d^2 \text{ cm}^2 \]

4. **Calculate the Area of S2**:
Using the diagonal of Square S2:
\[ A_{S2} = \frac{1}{2} \times d^2 = \frac{1}{2}d^2 \text{ cm}^2 \]

5. **Find the Ratio of Areas**:
Now, we need to find the ratio of the areas of S1 to S2:
\[ \text{Ratio} = \frac{A_{S1}}{A_{S2}} = \frac{2d^2}{\frac{1}{2}d^2} \]

6. **Simplify the Ratio**:
Simplifying the expression:
\[ \text{Ratio} = \frac{2d^2}{\frac{1}{2}d^2} = 2d^2 \times \frac{2}{d^2} = 4 \]

7. **Final Ratio**:
Therefore, the ratio of the areas of the two squares is:
\[ \text{Ratio} = 4:1 \]

### Conclusion
The ratio of the areas of the two squares is \(4:1\).