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 and orthocentre are also rational points.

Options

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 exlpanation for Statement 1.
C. Statement 1 is true, Statement 2 is false.
D. Statement 1 is false, Statement 2 is true.

Answer

D

Explanation

Statement 2 is obvisouly correct. For statement 1. let the circumcentre be at (0,0) and the vertices of the triangle be `(x_1,y_1),(x_1,y_1),(x_2,y_2)` and `(x_3,y_3)`. Then centroid is `((x_1+x_2+x_3)/(3),(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 then othocentre 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(1,0), B(-1//2,sqrt3//2)` and `C(-1//2,-sqrt3//2)`, where G,H and C are at (0,0) i,e., rational points.