Suppose a triangle is formed by $x+y=10$ and the coordinate axes. Then the number of points $(x, y)$ where $x$ and $y$ are natural numbers,lying inside the triangle is

Suppose $\triangle ABC$ is an isosceles triangle with $\angle C=90^{\circ}$,$A=(2,3)$ and $B=(4,5)$. Then the centroid of the triangle is

Let $ABCD$ be a square and let $P$ be a point on segment $CD$ such that $DP:PC=1:2$. Let $Q$ be a point on segment $AP$ such that $\angle BQP=90^{\circ}$. Then,the ratio of the area of quadrilateral $PQBC$ to the area of the square $ABCD$ is

If the straight lines $2x + 3y - 1 = 0$,$x + 2y - 1 = 0$,and $ax + by - 1 = 0$ form a triangle with the origin as the orthocentre,then $(a, b)$ is equal to

In a $\triangle ABC$ with $\angle A < \angle B < \angle C$,points $D, E, F$ are on the interior of segments $BC, CA, AB$ respectively. Which of the following triangles cannot be similar to $\triangle ABC$?

The vertices of a triangle are $(at_1t_2, a(t_1 + t_2))$,$(at_2t_3, a(t_2 + t_3))$,and $(at_3t_1, a(t_3 + t_1))$. Find the coordinates of its orthocentre.