In a triangle `ABC, AB + BC = 12 cm, BC + CA = 14 cm `and `CA + AB = 18 cm`. Find the radius of the circle (in cm) which has the same perimeter as the triangle.

Options

A. `5/2`
B. `7/2`
C. `9/2`
D. `11/2`

Answer

To solve the problem, we need to find the lengths of the sides of triangle ABC using the given equations and then calculate the radius of the circle that has the same perimeter as the triangle.

### Step-by-Step Solution:

1. **Set up the equations**:
We are given three equations based on the sides of the triangle:
- \( AB + BC = 12 \) (Equation 1)
- \( BC + CA = 14 \) (Equation 2)
- \( CA + AB = 18 \) (Equation 3)

2. **Add all three equations**:
\[ (AB + BC) + (BC + CA) + (CA + AB) = 12 + 14 + 18 \]
This simplifies to:
\[ 2AB + 2BC + 2CA = 44 \]

3. **Divide by 2**:
\[ AB + BC + CA = 22 \]
This means the perimeter of triangle ABC is 22 cm.

4. **Express each side in terms of the others**:
From the equations, we can express each side:
- From Equation 1: \( AB = 12 - BC \)
- From Equation 2: \( CA = 14 - BC \)
- From Equation 3: \( CA = 18 - AB \)

5. **Substitute to find the lengths**:
Substitute \( AB \) from Equation 1 into Equation 3:
\[ CA = 18 - (12 - BC) = 6 + BC \]
Now we have two expressions for \( CA \):
\[ 14 - BC = 6 + BC \]
Solving for \( BC \):
\[ 14 - 6 = 2BC \implies 8 = 2BC \implies BC = 4 \]

6. **Find \( AB \) and \( CA \)**:
Substitute \( BC = 4 \) back into the equations:
- From Equation 1:
\[ AB + 4 = 12 \implies AB = 8 \]
- From Equation 2:
\[ 4 + CA = 14 \implies CA = 10 \]

Thus, the sides of the triangle are:
- \( AB = 8 \) cm
- \( BC = 4 \) cm
- \( CA = 10 \) cm

7. **Calculate the radius of the circle with the same perimeter**:
The perimeter of the triangle is 22 cm. The radius \( r \) of a circle with the same perimeter \( P \) is given by:
\[ r = \frac{P}{2\pi} \]
Substituting the perimeter:
\[ r = \frac{22}{2\pi} = \frac{11}{\pi} \text{ cm} \]

### Final Answer:
The radius of the circle which has the same perimeter as the triangle is \( \frac{11}{\pi} \) cm.