With the vertices of a `Delta ABC` as centre three circles are described each touching the other two circles externally. If the sides of the triangles are 9 cm, 7 cm and 6 cm. Then, the radius of the circles (in cm) are

Options

A. 4 cm, 5 cm, 2 cm
B. 6 cm, 3 cm, 2cm
C. 4 cm, 3 cm, 2 cm
D. None of these

Answer

To find the radius of the circles described with the vertices of triangle ABC as their centers, we can follow these steps:

### Step 1: Define the Variables
Let the radius of the circle centered at vertex A be \( x \), at vertex B be \( y \), and at vertex C be \( z \). The sides of the triangle opposite to these vertices are given as follows:
- Side opposite A (BC) = 9 cm
- Side opposite B (AC) = 7 cm
- Side opposite C (AB) = 6 cm

### Step 2: Set Up the Equations
Since the circles touch each other externally, we can set up the following equations based on the distances between the centers of the circles:
1. \( x + y = 9 \) (Equation 1)
2. \( x + z = 6 \) (Equation 2)
3. \( y + z = 7 \) (Equation 3)

### Step 3: Add the Equations
Now, we will add all three equations:
\[ (x + y) + (x + z) + (y + z) = 9 + 6 + 7 \]
This simplifies to:
\[ 2x + 2y + 2z = 22 \]
Dividing the entire equation by 2 gives:
\[ x + y + z = 11 \quad \text{(Equation 4)} \]

### Step 4: Solve for Each Variable
Now we can use Equation 4 to find the values of \( x \), \( y \), and \( z \).

1. From Equation 1 and Equation 4:
\[ z = 11 - (x + y) = 11 - 9 = 2 \quad \text{(Radius at C)} \]

2. From Equation 2 and Equation 4:
\[ y = 11 - (x + z) = 11 - 6 = 5 \quad \text{(Radius at B)} \]

3. From Equation 3 and Equation 4:
\[ x = 11 - (y + z) = 11 - 7 = 4 \quad \text{(Radius at A)} \]

### Final Result
Thus, the radii of the circles are:
- \( x = 4 \) cm (Radius at A)
- \( y = 5 \) cm (Radius at B)
- \( z = 2 \) cm (Radius at C)

### Summary of the Radii
The radii of the circles are:
- Circle at A: 4 cm
- Circle at B: 5 cm
- Circle at C: 2 cm