The diagonals of two squares are in the ratio of `3 : 2`. Find the ratio of their areas.

Options

A. `9 : 4`
B. `9 : 2`
C. `9 : 5`
D. `9 : 7`

Answer

To find the ratio of the areas of two squares given that their diagonals are in the ratio of 3:2, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the relationship between the diagonal and the area of a square:**
The area \( A \) of a square can be expressed in terms of its diagonal \( d \) using the formula:
\[ A = \frac{1}{2} d^2 \]

2. **Assigning values to the diagonals:**
Let the diagonal of the first square be \( 3x \) and the diagonal of the second square be \( 2x \), based on the given ratio of their diagonals (3:2).

3. **Calculating the area of the first square:**
Using the formula for the area of a square:
\[ A_1 = \frac{1}{2} (3x)^2 = \frac{1}{2} \cdot 9x^2 = \frac{9}{2} x^2 \]

4. **Calculating the area of the second square:**
Similarly, for the second square:
\[ A_2 = \frac{1}{2} (2x)^2 = \frac{1}{2} \cdot 4x^2 = 2x^2 \]

5. **Finding the ratio of the areas:**
Now, we find the ratio of the areas \( A_1 \) to \( A_2 \):
\[ \text{Ratio of areas} = \frac{A_1}{A_2} = \frac{\frac{9}{2} x^2}{2x^2} \]

6. **Simplifying the ratio:**
The \( x^2 \) terms cancel out:
\[ \frac{A_1}{A_2} = \frac{\frac{9}{2}}{2} = \frac{9}{4} \]

7. **Final ratio of the areas:**
Therefore, the ratio of the areas of the two squares is:
\[ \text{Ratio of areas} = 9:4 \]

### Summary:
The ratio of the areas of the two squares is \( 9:4 \).

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