The radius of a circle is 20 cm. The radii (in cm) of three concentric circles drawn in such a manner that the whole area is divided into four equal parts, are :

Options

A. `20sqrt2, 20sqrt3,20`
B. `(10sqrt3)/(3),(10sqrt2)/(3),(10)/(3)`
C. `10sqrt3, 10sqrt2, 10`
D. `17, 14, 9`

Answer

To solve the problem of finding the radii of three concentric circles that divide the area of a larger circle with a radius of 20 cm into four equal parts, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a large circle with a radius of 20 cm. We need to find the radii of three smaller concentric circles such that the area of the larger circle is divided into four equal parts.

2. **Calculate the Area of the Large Circle**:
\[ \text{Area} = \pi r^2 = \pi (20)^2 = 400\pi \text{ cm}^2 \]

3. **Divide the Area into Four Equal Parts**:
Each part will have an area of:
\[ \frac{400\pi}{4} = 100\pi \text{ cm}^2 \]

4. **Determine the Area of Each Circle**:
- The area of the first smaller circle (let's call it Circle 1) should be \(100\pi\) cm².
- The area of the second smaller circle (Circle 2) should be \(200\pi\) cm² (which is the area of Circle 1 plus the area of the second part).
- The area of the third smaller circle (Circle 3) should be \(300\pi\) cm² (which is the area of Circle 1 plus the area of Circle 2 plus the area of the third part).

5. **Set Up the Equations**:
- For Circle 1:
\[ \pi r_1^2 = 100\pi \implies r_1^2 = 100 \implies r_1 = \sqrt{100} = 10 \text{ cm} \]
- For Circle 2:
\[ \pi r_2^2 = 200\pi \implies r_2^2 = 200 \implies r_2 = \sqrt{200} = 10\sqrt{2} \text{ cm} \]
- For Circle 3:
\[ \pi r_3^2 = 300\pi \implies r_3^2 = 300 \implies r_3 = \sqrt{300} = 10\sqrt{3} \text{ cm} \]

6. **Final Result**:
The radii of the three concentric circles are:
- \(r_1 = 10 \text{ cm}\)
- \(r_2 = 10\sqrt{2} \text{ cm}\)
- \(r_3 = 10\sqrt{3} \text{ cm}\)

### Summary of the Radii:
- First Circle: \(10 \text{ cm}\)
- Second Circle: \(10\sqrt{2} \text{ cm}\)
- Third Circle: \(10\sqrt{3} \text{ cm}\)