Three circles touch each other externally. A triangle is formed when the centres of these circles are joined together. Find the radii of the circles, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm.

Options

A. 4.2cm,2.5cm and 5.8 cm
B. 4.5cm,2.5cm and 7.5 cm
C. 3.5cm,2.5cm and 5.5 cm
D. 3.5cm,12.5cm and 7.5 cm

Answer

To find the radii of the three circles that touch each other externally and form a triangle with sides 6 cm, 8 cm, and 9 cm, we can follow these steps:

### Step 1: Define Variables
Let the radii of the three circles be \( R_1 \), \( R_2 \), and \( R_3 \). The sides of the triangle formed by the centers of the circles are given as:
- \( AB = 6 \) cm
- \( AC = 8 \) cm
- \( BC = 9 \) cm

### Step 2: Set Up Equations
From the properties of the circles touching each other externally, we can set up the following equations based on the sides of the triangle:
1. \( R_1 + R_2 = 6 \) (Equation 1)
2. \( R_2 + R_3 = 9 \) (Equation 2)
3. \( R_3 + R_1 = 8 \) (Equation 3)

### Step 3: Add the Equations
Now, we can add all three equations together:
\[ (R_1 + R_2) + (R_2 + R_3) + (R_3 + R_1) = 6 + 9 + 8 \]
This simplifies to:
\[ 2R_1 + 2R_2 + 2R_3 = 23 \]
Dividing both sides by 2 gives:
\[ R_1 + R_2 + R_3 = 11.5 \quad (Equation 4) \]

### Step 4: Solve for Each Radius
Now we can use Equation 4 along with the individual equations to find the values of \( R_1 \), \( R_2 \), and \( R_3 \).

#### Finding \( R_1 \):
From Equation 1:
\[ R_1 + R_2 = 6 \implies R_2 = 6 - R_1 \]
Substituting \( R_2 \) in Equation 2:
\[ (6 - R_1) + R_3 = 9 \implies R_3 = 9 - (6 - R_1) = 3 + R_1 \]
Now substitute \( R_3 \) in Equation 3:
\[ (3 + R_1) + R_1 = 8 \implies 2R_1 + 3 = 8 \implies 2R_1 = 5 \implies R_1 = 2.5 \text{ cm} \]

#### Finding \( R_2 \):
Now substitute \( R_1 \) back into Equation 1:
\[ R_1 + R_2 = 6 \implies 2.5 + R_2 = 6 \implies R_2 = 6 - 2.5 = 3.5 \text{ cm} \]

#### Finding \( R_3 \):
Now substitute \( R_1 \) into the expression for \( R_3 \):
\[ R_3 = 3 + R_1 = 3 + 2.5 = 5.5 \text{ cm} \]

### Final Values
Thus, the radii of the circles are:
- \( R_1 = 2.5 \) cm
- \( R_2 = 3.5 \) cm
- \( R_3 = 5.5 \) cm

### Summary
The radii of the circles are:
- \( R_1 = 2.5 \) cm
- \( R_2 = 3.5 \) cm
- \( R_3 = 5.5 \) cm