If ratio of sides of a triangle are 4 : 5 : 6 then what is the ratio of circumradius and inradius ?

Options

A. `2:1`
B. `16:7`
C. `12:7`
D. `16:5`

Answer

To find the ratio of the circumradius (R) and inradius (r) of a triangle with sides in the ratio 4:5:6, we can follow these steps:

### Step 1: Define the sides of the triangle
Let the sides of the triangle be represented as:
- \( A = 4k \)
- \( B = 5k \)
- \( C = 6k \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is given by:
\[ s = \frac{A + B + C}{2} = \frac{4k + 5k + 6k}{2} = \frac{15k}{2} \]

### Step 3: Calculate the area (A) using Heron's formula
Heron's formula states that the area of the triangle is:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values:
\[ s - A = \frac{15k}{2} - 4k = \frac{15k - 8k}{2} = \frac{7k}{2} \]
\[ s - B = \frac{15k}{2} - 5k = \frac{15k - 10k}{2} = \frac{5k}{2} \]
\[ s - C = \frac{15k}{2} - 6k = \frac{15k - 12k}{2} = \frac{3k}{2} \]
Now substituting these into Heron's formula:
\[ \text{Area} = \sqrt{\frac{15k}{2} \cdot \frac{7k}{2} \cdot \frac{5k}{2} \cdot \frac{3k}{2}} = \sqrt{\frac{15 \cdot 7 \cdot 5 \cdot 3 \cdot k^4}{16}} = \frac{\sqrt{1575} k^2}{4} \]

### Step 4: Calculate the circumradius (R)
The circumradius \( R \) is given by the formula:
\[ R = \frac{abc}{4 \times \text{Area}} \]
Substituting \( a = 4k, b = 5k, c = 6k \):
\[ R = \frac{(4k)(5k)(6k)}{4 \times \frac{\sqrt{1575} k^2}{4}} = \frac{120k^3}{\sqrt{1575} k^2} = \frac{120k}{\sqrt{1575}} \]

### Step 5: Calculate the inradius (r)
The inradius \( r \) is given by the formula:
\[ r = \frac{\text{Area}}{s} \]
Substituting the area and semi-perimeter:
\[ r = \frac{\frac{\sqrt{1575} k^2}{4}}{\frac{15k}{2}} = \frac{\sqrt{1575} k^2}{4} \cdot \frac{2}{15k} = \frac{\sqrt{1575} k}{30} \]

### Step 6: Find the ratio \( \frac{R}{r} \)
Now we can find the ratio of circumradius to inradius:
\[ \frac{R}{r} = \frac{\frac{120k}{\sqrt{1575}}}{\frac{\sqrt{1575} k}{30}} = \frac{120k \cdot 30}{\sqrt{1575} \cdot \sqrt{1575} k} = \frac{3600}{1575} \]
Simplifying this gives:
\[ \frac{3600 \div 225}{1575 \div 225} = \frac{16}{7} \]

### Final Answer
Thus, the ratio of the circumradius to the inradius is:
\[ \frac{R}{r} = 16 : 7 \]