Question

Assertion :If in a triangle orthocenter, circumcenter & centroid are rational points, then its vertices must be rational. Reason: If the vertices of a triangle are rational points then the centroid, circumcenter & orthocenter are also rational points.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Let $A\left(a_1{b}_1\right),B(a_2,b_2),C(a_3,b_3)$ are the vertices of $\Delta ABC$ where $a_1,b_1,c_1$. $a_2,b_2,c_2$ are rational points
Now centroid of ABC is $\frac{a_1+b_2+c_3}{3},\frac{b_1+b_2+b_3}{3}$ which is rational point.
Similarly, the circumcentre & orthocentre are rational points.

Hence, Reason (R) is correct.

Now for Assertion (A) let circumcentre be at (0, 0) & the vertices of triangle be $A(a_1,b_1),B(a_2,b_2),C(a_3,b_3)$, then orthocentre of the triangle becomes $(\sum a_1,\sum b_1)$.

This implies that if centroid is rational, then orthocentre is also rational & $a_1+a_2+a_3$ can be rational even if $a_1,a_2,a_3$ are not all rational, so the Assertion (A) is not correct.