First slide

Cartesian plane

Question

If the vertices P, Q, R of a $∆$ PQR are rational points, which of the following points of the $∆$ PQR is (are) always rational point(s)?

Moderate

A

centroid
B

incentre
C

circumcentre
D

orthocentre

Solution

Let $P \equiv (x_{1}, y_{1}), Q \equiv (x_{2}, y_{2}); R \equiv (x_{3}, y_{3})$ , where $x_{i}, y_{i}$ (i = 1, 2, 3) are rational numbers.
Now, the centroid of $∆$ PQR is
$(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$
which is rational point. Incentre, circumcentre and orthocentre depend on sides of the triangle which may not be rational even if vertices are so. For example, for $P (0, 1)$ and $Q (1, 0); P Q = \sqrt{2}$

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