If the ratio of the areas of two squares is 9:1, then the ratio of their perimeters is

Options

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

Answer

To solve the problem, we need to find the ratio of the perimeters of two squares given that the ratio of their areas is 9:1.

### Step-by-Step Solution:

1. **Understanding the Ratio of Areas**:
The ratio of the areas of the two squares is given as 9:1. This means if we denote the area of the first square as \( A_1 \) and the area of the second square as \( A_2 \), we can write:
\[ \frac{A_1}{A_2} = \frac{9}{1} \]

2. **Assigning Values to Areas**:
For simplicity, we can assign the area of the first square \( A_1 = 9 \) and the area of the second square \( A_2 = 1 \).

3. **Finding the Side Lengths of the Squares**:
The area of a square is given by the formula \( A = s^2 \), where \( s \) is the side length of the square.
- For the first square:
\[ A_1 = s_1^2 \implies 9 = s_1^2 \implies s_1 = \sqrt{9} = 3 \]
- For the second square:
\[ A_2 = s_2^2 \implies 1 = s_2^2 \implies s_2 = \sqrt{1} = 1 \]

4. **Calculating the Perimeters**:
The perimeter \( P \) of a square is given by the formula \( P = 4s \).
- For the first square:
\[ P_1 = 4 \times s_1 = 4 \times 3 = 12 \]
- For the second square:
\[ P_2 = 4 \times s_2 = 4 \times 1 = 4 \]

5. **Finding the Ratio of Perimeters**:
Now, we can find the ratio of the perimeters of the two squares:
\[ \frac{P_1}{P_2} = \frac{12}{4} = 3 \]
Thus, the ratio of the perimeters of the two squares is:
\[ \frac{P_1}{P_2} = 3:1 \]

### Final Answer:
The ratio of the perimeters of the two squares is **3:1**.