Three circles touch each other externally. The distance between their centre is 5 cm, 6 cm and 7 cm. Find the radii of the circles :

Options

A. 2 cm, 3 cm, 4 cm
B. 3 cm, 4 cm, 1 cm
C. 1 cm, 2.5 cm, 3.5 cm
D. 1 cm, 2 cm, 4 cm

Answer

To find the radii of three circles that touch each other externally, given the distances between their centers as 5 cm, 6 cm, and 7 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables for Radii**:
Let the radii of the three circles be:
- Circle 1 (radius \( x \))
- Circle 2 (radius \( y \))
- Circle 3 (radius \( z \))

2. **Set Up Equations Based on Distances**:
Since the circles touch each other externally, we can write the following equations based on the distances between their centers:
- The distance between Circle 1 and Circle 2 is 5 cm:
\[ x + y = 5 \quad \text{(1)} \]
- The distance between Circle 2 and Circle 3 is 6 cm:
\[ y + z = 6 \quad \text{(2)} \]
- The distance between Circle 1 and Circle 3 is 7 cm:
\[ x + z = 7 \quad \text{(3)} \]

3. **Add the Equations**:
Now, we add all three equations:
\[ (x + y) + (y + z) + (x + z) = 5 + 6 + 7 \]
Simplifying the left side gives:
\[ 2x + 2y + 2z = 18 \]
Dividing the entire equation by 2:
\[ x + y + z = 9 \quad \text{(4)} \]

4. **Substitute to Find Individual Radii**:
Now, we can use equation (4) to find the individual radii:
- From equation (1):
\[ y = 5 - x \quad \text{(5)} \]
- Substitute equation (5) into equation (2):
\[ (5 - x) + z = 6 \implies z = 6 - (5 - x) \implies z = x + 1 \quad \text{(6)} \]
- Substitute equation (6) into equation (3):
\[ x + (x + 1) = 7 \implies 2x + 1 = 7 \implies 2x = 6 \implies x = 3 \]

5. **Find \( y \) and \( z \)**:
- Using \( x = 3 \) in equation (5):
\[ y = 5 - 3 = 2 \]
- Using \( x = 3 \) in equation (6):
\[ z = 3 + 1 = 4 \]

6. **Final Radii**:
The radii of the circles are:
- Circle 1: \( x = 3 \) cm
- Circle 2: \( y = 2 \) cm
- Circle 3: \( z = 4 \) cm

### Conclusion:
The radii of the circles are \( 2 \) cm, \( 3 \) cm, and \( 4 \) cm.