If ratio of the areas of two squares is 1:4, the ratio of their perimeters is

Options

A. a. 1:2
B. b. 1:4
C. c. 1:6
D. d. 1:8

Answer

To find the ratio of the perimeters of two squares given that the ratio of their areas is 1:4, we can follow these steps:

### Step 1: Understand the relationship between area and side length of a square.
The area \( A \) of a square is given by the formula:
\[ A = \text{side}^2 \]
If we denote the side lengths of the two squares as \( a \) and \( b \), then the areas of the squares can be expressed as \( a^2 \) and \( b^2 \).

### Step 2: Set up the equation for the ratio of the areas.
According to the problem, the ratio of the areas of the two squares is given as:
\[ \frac{a^2}{b^2} = \frac{1}{4} \]

### Step 3: Take the square root of both sides to find the ratio of the side lengths.
Taking the square root of both sides gives us:
\[ \frac{a}{b} = \frac{1}{2} \]

### Step 4: Determine the ratio of the perimeters.
The perimeter \( P \) of a square is given by the formula:
\[ P = 4 \times \text{side} \]
Thus, the perimeters of the two squares are \( 4a \) and \( 4b \). The ratio of the perimeters can be expressed as:
\[ \frac{P_1}{P_2} = \frac{4a}{4b} = \frac{a}{b} \]

### Step 5: Substitute the ratio of the side lengths into the perimeter ratio.
From Step 3, we found that \( \frac{a}{b} = \frac{1}{2} \). Therefore, the ratio of the perimeters is:
\[ \frac{P_1}{P_2} = \frac{1}{2} \]

### Final Answer:
The ratio of the perimeters of the two squares is \( 1:2 \).

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