First slide

Cartesian plane

Question

Statement 1: If in a triangle, orthocentre, circumcentre and centroid are rational points, then its vertices must also be rational points.
Statement 2: If the vertices of a triangle are rational points, then the centroid, circumcentre are also rational points.

Moderate

A

Statement 1 is true, statement 2 is true and statement 2 is correct explanation for statement 1
B

Statement 1 is true, statement 2 is true and statement 2 is NOT the correct explanation for statement 1
C

Statement 1 is true, statement 2 is false.
D

Statement 1 is false, statement 2 is true.

Solution

Statement 2 is obviously correct.
For statement 1, let thecircumcentre be at (0,0) and the vertices of the triangle be $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ . Then centroid is $(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$ , and orthocentre of the triangles becomes $(x_{1} + x_{2} + x_{3}, y_{1} + y_{2} + y_{3})$ .
This implies that if the centroid is rational then orthocentre is also rational but $(x_{1} + x_{2} + x_{3})$ can be rational even if $x_{1}, x_{2}, x_{3}$ are not all rational.
For example, $A (0, 1), B (- 1 / 2, \sqrt{3} / 2) and C (- 1 / 2, - \sqrt{3} / 2)$ where G, H and C are at (0,0) i.e., rational points

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