First slide

Cartesian plane

Question

If the coordinates of the vertices of a triangle an: rational numbers, then which of the following points of the triangle may not have rational coordinates?

Moderate

A

centroid
B

incentre
C

circumcentre
D

orthocentre

Solution

Let $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$
vertices of $△ A B C$ such that $x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}$ are rational numbers. Then,
$\frac{x_{1} + x_{2} + x_{3}}{3}$ and $\frac{y_{1} + y_{2} + y_{3}}{3}$ are also rational numbers
So, the coordinates of the centroid are always rational numbers.
Let $P$ be the circumcentre of . $∆ A B C$ . Then
$\begin{aligned} P A = P B = P C \\ \Rightarrow P A^{2} = P B^{2} = P C^{2} \end{aligned}$
$\Rightarrow P A^{2} = P B^{2} and P B^{2} = P C^{2}$

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