$A$ quadrilateral has distinct integer side lengths. If the second-largest side has length $10$,then the maximum possible length of the largest side is

When are the points with coordinates $(2a, 3a)$,$(3b, 2b)$,and $(c, c)$ collinear?

If the circumcenter of the triangle formed by the points $A(a, 3)$,$B(b, 5)$,and $C(a, b)$ is $(1, 1)$,then out of all the possible coordinates of $C$,the sum of the absolute values of the distinct coordinates of $C$ is

Let $A(\alpha, -2)$,$B(\alpha, 6)$,and $C\left(\frac{\alpha}{4}, -2\right)$ be the vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$,then which of the following is $NOT$ correct about $\triangle ABC$?

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

The diagonals of a parallelogram $ABCD$ are along the lines $x+3y=4$ and $6x-2y=7$. Then $ABCD$ must be a