Question

The sides of a triangle are $y=x,y=2x$ and $y=3x+4$. Find the equation of the right bisectors of the sides and hence the coordinates of t circumcentre.

Answer 1

Any line which passes through the middle point of any side and is perpendicular to it called its right bisector
Any line $\perp$ to $BC$ $y-x=0$ is $x+y+k=0$
Since it passes through mid point $D(-1,-1)$;
$\therefore$ $-1-1+k=0$ or $k=2$
Right bisector of $BC$ is $x+y+2=\mathrm{0.....}\left(1\right)$
Similarly, the right bisectors of sides $CA$ and $AB$ are respectively
$2y+x+10=\mathrm{0....}\left(2\right)$
$3y+x+18=\mathrm{0.....}\left(3\right)$
Solving any two of (1), (2) and (3) we get the point $(6,-8)$ which satisfies the third also. Hence the right bisectors of the sides of a triangle are concurrent and the point of concurrency is called the circumcentre of the triangle which is the centre of a circle that passes through the vertices of the triangle.