In a triangle ABC, the lengths of in-radius and circum-radius are 2 units and 6 units respectively. Find the distance (in units) between the in-centre and circum centre.

Options

A. `sqrt(3)`
B. `2sqrt(3)`
C. `3sqrt(3)`
D. `4sqrt(3)`

Answer

To find the distance between the in-center and circum-center of triangle ABC, we can use the formula:

\[ d^2 = R \cdot r - 2r^2 \]

where:
- \(d\) is the distance between the in-center and circum-center,
- \(R\) is the circum-radius,
- \(r\) is the in-radius.

Given:
- \(r = 2\) units (in-radius),
- \(R = 6\) units (circum-radius).

### Step 1: Substitute the values into the formula

We start by substituting the values of \(R\) and \(r\) into the formula:

\[ d^2 = R \cdot r - 2r^2 \]

Substituting \(R = 6\) and \(r = 2\):

\[ d^2 = 6 \cdot 2 - 2 \cdot (2^2) \]

### Step 2: Calculate the terms

Now we calculate each term:

1. Calculate \(6 \cdot 2\):
\[ 6 \cdot 2 = 12 \]

2. Calculate \(2 \cdot (2^2)\):
\[ 2^2 = 4 \quad \text{so} \quad 2 \cdot 4 = 8 \]

### Step 3: Substitute back into the equation

Now we substitute these results back into the equation:

\[ d^2 = 12 - 8 \]

### Step 4: Simplify the equation

Now simplify the right-hand side:

\[ d^2 = 4 \]

### Step 5: Take the square root

To find \(d\), we take the square root of both sides:

\[ d = \sqrt{4} = 2 \]

### Conclusion

Thus, the distance between the in-center and circum-center is:

\[ \boxed{2} \text{ units} \]