If sides of a triangle are respectively 5 cm, 6 cm and 7 cm then radius of the circumcircle of the triangle is

Options

A. 9 cm
B. `35/sqrt(6)` cm
C. `35/(4sqrt(6))` cm
D. `17/2` cm

Answer

To find the radius of the circumcircle of a triangle with sides 5 cm, 6 cm, and 7 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the sides of the triangle**: Let the sides of the triangle be \( a = 5 \) cm, \( b = 6 \) cm, and \( c = 7 \) cm.

2. **Calculate the semi-perimeter (s)**: The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{5 + 6 + 7}{2} = \frac{18}{2} = 9 \text{ cm} \]

3. **Use Heron's formula to find the area (Δ)**: Heron's formula states that the area of the triangle can be calculated as:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ \Delta = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} \]
Simplifying further:
\[ \Delta = \sqrt{9 \times 24} = \sqrt{216} = 6\sqrt{6} \text{ cm}^2 \]

4. **Apply the circumradius formula (R)**: The radius \( R \) of the circumcircle of a triangle can be calculated using the formula:
\[ R = \frac{abc}{4\Delta} \]
Substituting the values:
\[ R = \frac{5 \times 6 \times 7}{4 \times 6\sqrt{6}} = \frac{210}{24\sqrt{6}} = \frac{35}{4\sqrt{6}} \text{ cm} \]

### Final Answer:
The radius of the circumcircle of the triangle is:
\[ R = \frac{35}{4\sqrt{6}} \text{ cm} \]