If the ratio of diagonals of two squares is 3:2 then the ratio of the areas of two squares is :

Options

A. `4:5`
B. `6:5`
C. `9:4`
D. `sqrt(3): sqrt(2)`

Answer

To find the ratio of the areas of two squares given the ratio of their diagonals, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side of a square
The diagonal \( d \) of a square with side length \( s \) can be calculated using the formula:
\[ d = s\sqrt{2} \]

### Step 2: Express the sides of the squares in terms of their diagonals
Let the side lengths of the two squares be \( s_1 \) and \( s_2 \). From the diagonal formula, we can express the sides in terms of the diagonals \( d_1 \) and \( d_2 \):
\[ s_1 = \frac{d_1}{\sqrt{2}} \quad \text{and} \quad s_2 = \frac{d_2}{\sqrt{2}} \]

### Step 3: Use the given ratio of the diagonals
According to the problem, the ratio of the diagonals is given as:
\[ \frac{d_1}{d_2} = \frac{3}{2} \]

### Step 4: Find the ratio of the sides of the squares
Using the relationship established in Step 2, we can find the ratio of the sides:
\[ \frac{s_1}{s_2} = \frac{\frac{d_1}{\sqrt{2}}}{\frac{d_2}{\sqrt{2}}} = \frac{d_1}{d_2} = \frac{3}{2} \]

### Step 5: Calculate the ratio of the areas of the squares
The area \( A \) of a square is given by \( A = s^2 \). Therefore, the ratio of the areas of the two squares is:
\[ \frac{A_1}{A_2} = \frac{s_1^2}{s_2^2} = \left(\frac{s_1}{s_2}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Conclusion
Thus, the ratio of the areas of the two squares is:
\[ \frac{9}{4} \]