Question

Find the coordinates of circumcentre and radius of cirumcircle of $\Delta ABC$ if $A\left(7,1\right),B\left(3,5\right)$ and $C\left(2,0\right)$ are given .

Answer 1

Consider the given points of a triangle

$A(x_1,y_1)=(7,1)$

$B(x_2,y_2)=(3,5)$

$C(x_3,y_3)=(2,0)$

Let $O$ be the circumcentre

$O(x,y)=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

$O(x,y)=(\frac{7+3+2}{3},\frac{1+5+0}{3})$

$O(x,y)=(\frac{12}{3},\frac{6}{3})$

$O(x,y)=(4,2)$

Now, radius of circumcircle

Let $D(x_4,y_4)$ be the middle point of AB.

The

$D(x_4,y_4)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$D(x_4,y_4)=(\frac{7+3}{2},\frac{1+5}{2})$

$D(x_4,y_4)=(5,3)$

radius of circumcircle is

$=OD=\sqrt{{(x-x_4)}^2+{(y-y_4)}^2}$

$OD=\sqrt{{(4-5)}^2+{(2-3)}^2}$

$OD=\sqrt{1+1}=\sqrt{2}$

Hence, this is the answer.