Question

If the coordinates of the vertices of a triangle are rational numbers then which of the following points of the triangle will always have rational coordinates?

• A. Centroid
• B. Incentre
• C. Circumcentre
• D. Orthocentre

Answer 1

Answer: (A)

Centroid

Let $P(x_1,y_1),Q(x_2,y_2)$ and $R(x_3,y_3)$ be coordinates of triangle where $x_i,y_i(i=1,2,3)$ are rational numbers.
Now the centroid of triangle is,
$(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$ which is rational point.
Incentre, circumcentre and orthocenter depend on sides of the triangle which may not be rational even if the co-ordinates of vertices are so.