Question

The distance between the circumcentre and centroid of a triangle whose vertices are (6, 0), (0, 6) and (6, 6), is _______.

Answer 1

In ∆ABC
A = (6, 0)
B = (0, 6)
C = (6, 6)

Clearly ∆ABC is a right-angled triangle at C and in right-angled triangle,
Circum-centre is mid point of hypotenuse
Since AB is hypotenuse and mid point of AB is (3, 3) $\left[\left(\frac{6+0}{2},\frac{0+6}{2}\right)\right]$
Hence circumcentre is (3, 3) and centroid is $\left(\frac{\left(6+0+6\right)}{3},\frac{\left(0+6+6\right)}{3}\right)$
i.e (4, 4)
∴ Distance between circumcentre (3, 3) and centroid (4, 4) is
$\begin{array}{rcl}\sqrt{{\left(4-3\right)}^2+{\left(4-3\right)}^2}& =& \sqrt{1+1}\\ & =& \sqrt{2}\end{array}$