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. `(9)/(2)`
B. `(11)/(2)`
C. `(5)/(2)`
D. `(7)/(2)`

Answer

To solve the problem step by step, we will follow these steps:

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

### Step 2: Add the equations
Now, we will add all three equations together:
\[ (AB + BC) + (BC + CA) + (CA + AB) = 12 + 14 + 18 \]
This simplifies to:
\[ 2(AB + BC + CA) = 44 \]

### Step 3: Solve for the sum of the sides
Now, divide both sides by 2 to find \( AB + BC + CA \):
\[ AB + BC + CA = \frac{44}{2} = 22 \text{ cm} \]
This value represents the perimeter of triangle ABC.

### Step 4: Relate the perimeter to the circle
The perimeter of the triangle is equal to the circumference of the circle. The formula for the circumference \( C \) of a circle is:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle.

### Step 5: Set the perimeter equal to the circumference
We can set the perimeter of the triangle equal to the circumference of the circle:
\[ 2\pi r = 22 \]

### Step 6: Solve for the radius \( r \)
Now, we will solve for \( r \):
\[ r = \frac{22}{2\pi} = \frac{11}{\pi} \]

### Step 7: Approximate the value of \( r \)
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{11}{\frac{22}{7}} = \frac{11 \times 7}{22} = \frac{77}{22} = \frac{7}{2} \text{ cm} \]

### Conclusion
Thus, the radius of the circle that has the same perimeter as the triangle is:
\[ \boxed{\frac{7}{2} \text{ cm}} \]