If the vertices $P, Q, R$ of a triangle $PQR$ are rational points (points with rational coordinates),which of the following points is/are not necessarily a rational point?

What is the distance of the orthocenter from the vertex of a triangle with vertices $(2, 3), (4, 5),$ and $(-1, 10)$?

Let the area of a $\triangle PQR$ with vertices $P(5, 4)$,$Q(-2, 4)$,and $R(a, b)$ be $35$ square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively,then $c+2d$ is equal to:

The incentre of a triangle with vertices $(7, 1)$,$(-1, 5)$,and $(3 + 2\sqrt{3}, 3 + 4\sqrt{3})$ is

The orthocentre of the triangle formed by the lines $xy = 0$ and $x + y = 1$ is

Let $A(6,8)$,$B(10 \cos \alpha, -10 \sin \alpha)$ and $C(-10 \sin \alpha, 10 \cos \alpha)$ be the vertices of a triangle. If $L(a, 9)$ and $G(h, k)$ are its orthocenter and centroid respectively,then $(5a - 3h + 6k + 100 \sin 2\alpha)$ is equal to . . . . . .