Exercise

Examine whether the given points $A(10,20),B(20,20),C(20,10),\text{ and }D(10,10)$ forms a square.

Using the section formula, prove that the three points $\left(-10, 8\right)$, $\left(2, 4\right)$ and $\left(8, 2\right)$ are collinear.

Point $X$ has co-ordinates $(4,7)$ and is the midpoint of $AB$. Given that $A$ has co-ordinates $(-3,11)$, find the co-ordinates of $B$.

Find the midpoint of pair of points: $(4,2)\text{ and }(6,12)$.

If $\left(2, 0\right), \left(4, 4\right) \text{and} \left(6, 2\right)$ are the vertices of the triangle $ABC$, then find the lengths of its medians.

If the points $P\left(a, 1\right), Q\left(1, b\right), R\left(-2, 11\right)$ are collinear and $Q$ be the midpoint of $\overline{PR}$, then find the value of $a \text{and} b$.

The straight line $4x+3y-k=0$ intersects the $x-$axis and $y-$axis at $A \text{and} B$, respectively. Find the coordinates of the point at which the straight line $\overline{AB}$ is divided internally into the ratio $2:1$. 

The coordinates of points $A,B$ and $C$ are $(1,2),(-2,1)$ and $\left(0,6\right)$ respectively. Verify that the medians of the triangle $ABC$ are concurrent. Also find the coordinates of the point of concurrence (centroid).