If a square ABCD is inscribed in a circle of radius 'r' and AB = 4 cm, then the value of r is :

Options

A. 2 cm
B. `2 sqrt(2)` cm
C. 4 cm
D. `4 sqrt(2)` cm

Answer

To find the radius \( r \) of the circle in which the square ABCD is inscribed, follow these steps:

### Step 1: Understand the relationship between the square and the circle
A square inscribed in a circle means that all four corners of the square touch the circle. The distance from the center of the circle to any vertex of the square is the radius \( r \).

### Step 2: Identify the side length of the square
Given that the side length \( AB \) of the square is \( 4 \) cm, we can denote this side length as \( s \):
\[ s = 4 \text{ cm} \]

### Step 3: Relate the side length of the square to the radius of the circle
The radius \( r \) of the circle can be calculated using the formula that relates the side length of the square to the radius of the circumscribed circle:
\[ r = \frac{s \sqrt{2}}{2} \]
This formula arises because the diagonal of the square is equal to the diameter of the circle, and the diagonal \( d \) of a square can be calculated as:
\[ d = s \sqrt{2} \]
Since the diameter \( D \) of the circle is twice the radius, we have:
\[ D = 2r = s \sqrt{2} \]
Thus, rearranging gives:
\[ r = \frac{s \sqrt{2}}{2} \]

### Step 4: Substitute the value of \( s \)
Now, substitute \( s = 4 \) cm into the formula:
\[ r = \frac{4 \sqrt{2}}{2} \]

### Step 5: Simplify the expression
Simplifying the expression gives:
\[ r = 2 \sqrt{2} \text{ cm} \]

### Final Answer
Thus, the value of \( r \) is:
\[ r = 2\sqrt{2} \text{ cm} \]

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