If a circle of radius 12 cm is divided into two equal parts by one concentric circle, then radius of inner circle is:

Options

A. 6 cm
B. 4 cm
C. `6sqrt(2) cm`
D. `4sqrt(2) cm`

Answer

To find the radius of the inner circle when a circle of radius 12 cm is divided into two equal parts by a concentric circle, we can follow these steps:

### Step 1: Calculate the area of the outer circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

For the outer circle with a radius of 12 cm:
\[ A_{\text{outer}} = \pi (12)^2 = \pi \times 144 = 144\pi \, \text{cm}^2 \]

### Step 2: Find the area of each part when divided into two equal parts
Since the outer circle is divided into two equal parts, the area of each part will be half of the total area of the outer circle:
\[ A_{\text{each part}} = \frac{A_{\text{outer}}}{2} = \frac{144\pi}{2} = 72\pi \, \text{cm}^2 \]

### Step 3: Set up the equation for the area of the inner circle
Let the radius of the inner circle be \( r \). The area of the inner circle can also be expressed using the area formula:
\[ A_{\text{inner}} = \pi r^2 \]

### Step 4: Equate the area of the inner circle to the area of one part of the outer circle
Since the area of the inner circle is equal to the area of one part of the outer circle:
\[ \pi r^2 = 72\pi \]

### Step 5: Simplify and solve for \( r \)
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 = 72 \]
Now, take the square root of both sides:
\[ r = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \, \text{cm} \]

### Conclusion
The radius of the inner circle is \( 6\sqrt{2} \, \text{cm} \).

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