The ratio of the areas of two squares one having double than the other .find the ratio of their diagonal?

Options

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

Answer

To solve the problem, we need to find the ratio of the diagonals of two squares given that the area of one square is double that of the other. Let's break this down step by step.

### Step 1: Define the Areas of the Squares
Let the area of the first square be \( A_1 \) and the area of the second square be \( A_2 \). According to the problem, we have:
\[ A_2 = 2 \times A_1 \]

### Step 2: Relate Area to Side Length
The area of a square is given by the formula:
\[ A = \text{side}^2 \]
Let the side length of the first square be \( s_1 \) and the side length of the second square be \( s_2 \). Therefore, we can express the areas as:
\[ A_1 = s_1^2 \]
\[ A_2 = s_2^2 \]

### Step 3: Substitute the Areas
Substituting the expressions for the areas into the equation from Step 1, we get:
\[ s_2^2 = 2 \times s_1^2 \]

### Step 4: Solve for Side Lengths
Taking the square root of both sides gives us:
\[ s_2 = \sqrt{2} \times s_1 \]

### Step 5: Find the Diagonal of Each Square
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = s \sqrt{2} \]
Thus, the diagonals of the squares are:
\[ d_1 = s_1 \sqrt{2} \]
\[ d_2 = s_2 \sqrt{2} = (\sqrt{2} \times s_1) \sqrt{2} = 2s_1 \]

### Step 6: Find the Ratio of the Diagonals
Now we can find the ratio of the diagonals \( d_2 \) to \( d_1 \):
\[ \text{Ratio} = \frac{d_2}{d_1} = \frac{2s_1}{s_1 \sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Conclusion
Thus, the ratio of the diagonals of the two squares is:
\[ \text{Ratio of diagonals} = \sqrt{2} : 1 \]