Three circles of radii 3.5 cm, 4.5 cm and 5.5 cm touch each other externally. Then the perimeter of the triangle formed by joining the centres of the circles, in cm, is

Options

A. 27
B. `pi[(3.5)^(2) + (4.5)^(2) + (5.5)^(2)]`
C. `27pi`
D. `13.5`

Answer

To find the perimeter of the triangle formed by joining the centers of three circles with given radii, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
- Let the radius of the first circle (Circle A) be \( r_1 = 3.5 \, \text{cm} \)
- Let the radius of the second circle (Circle B) be \( r_2 = 4.5 \, \text{cm} \)
- Let the radius of the third circle (Circle C) be \( r_3 = 5.5 \, \text{cm} \)

2. **Determine the Distances Between the Centers:**
- The distance between the centers of Circle A and Circle B (denoted as AB) is the sum of their radii:
\[ AB = r_1 + r_2 = 3.5 \, \text{cm} + 4.5 \, \text{cm} = 8.0 \, \text{cm} \]
- The distance between the centers of Circle B and Circle C (denoted as BC) is:
\[ BC = r_2 + r_3 = 4.5 \, \text{cm} + 5.5 \, \text{cm} = 10.0 \, \text{cm} \]
- The distance between the centers of Circle C and Circle A (denoted as CA) is:
\[ CA = r_3 + r_1 = 5.5 \, \text{cm} + 3.5 \, \text{cm} = 9.0 \, \text{cm} \]

3. **Calculate the Perimeter of the Triangle:**
- The perimeter \( P \) of the triangle formed by the centers of the circles is the sum of the lengths of its sides:
\[ P = AB + BC + CA \]
- Substituting the values we found:
\[ P = 8.0 \, \text{cm} + 10.0 \, \text{cm} + 9.0 \, \text{cm} = 27.0 \, \text{cm} \]

### Final Answer:
The perimeter of the triangle formed by joining the centers of the circles is \( 27.0 \, \text{cm} \).

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