Find the ratio of circum radius (R) to inradius (r) of a triangle whose sides are 4,6 and 8.

Options

A. `2:3`
B. `16:5`
C. `5:16`
D. `4:7`

Answer

To find the ratio of the circumradius (R) to the inradius (r) of a triangle with sides 4, 6, and 8, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 4 \)
- \( b = 6 \)
- \( c = 8 \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{4 + 6 + 8}{2} = \frac{18}{2} = 9 \]

### Step 3: Calculate the area (Δ) of the triangle using Heron's formula
Heron's formula states:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ \Delta = \sqrt{9(9-4)(9-6)(9-8)} = \sqrt{9 \times 5 \times 3 \times 1} = \sqrt{45} = 3\sqrt{5} \]

### Step 4: Calculate the circumradius (R)
The circumradius \( R \) is given by the formula:
\[ R = \frac{abc}{4\Delta} \]
Substituting the values:
\[ R = \frac{4 \times 6 \times 8}{4 \times 3\sqrt{5}} = \frac{192}{12\sqrt{5}} = \frac{16}{\sqrt{5}} = \frac{16\sqrt{5}}{5} \]

### Step 5: Calculate the inradius (r)
The inradius \( r \) is given by the formula:
\[ r = \frac{\Delta}{s} \]
Substituting the values:
\[ r = \frac{3\sqrt{5}}{9} = \frac{\sqrt{5}}{3} \]

### Step 6: Calculate the ratio \( \frac{R}{r} \)
Now, we can find the ratio of circumradius to inradius:
\[ \frac{R}{r} = \frac{\frac{16\sqrt{5}}{5}}{\frac{\sqrt{5}}{3}} = \frac{16\sqrt{5}}{5} \times \frac{3}{\sqrt{5}} = \frac{16 \times 3}{5} = \frac{48}{5} \]

### Final Answer
The ratio of circumradius to inradius of the triangle is:
\[ \frac{R}{r} = \frac{48}{5} \]

---