Question

If the coordinates of the vertices of a triangle are rational then which if the following points is not always rational ?

• A. Orthocenter
• B. Centroid
• C. Incenter
• D. Circumcenter

Answer 1

Answer: (C)

Incenter

Given the coordinates of vertices of a triangle are rational .

To get the centroid,we need to compute the mean of x and y coordinates of the vertex respectively which are rational

The radius of incircle is given by $r=\frac{2s}{a+b+c}$

Consider a right triangle , $a=1,b=1$

$\therefore c=\sqrt{2.}$ So its area is $\frac{1}{2}$ which is rational and its perimeter$P=2+\sqrt{2}$ , which is irrational

$\therefore x$ is irrational too.