Prove analytically that the diagonals of a rectangle are equal.

$A (1,4)$ and $B (4,8)$ are two points. $P$ is a point on $AB$ such that $AP = AB + BP. If AP = 10$ units. Find the coordinates of $P$.

The line segment joining $A (2, 3)$ and $B (-3, 5)$ is extended through each end by a length equal to its original length. Find the coordinates of the new ends.

The line segment joining $A(6,3)$ to $B(-1,-4)$ is doubled in length by having half its length added to each end. Find the coordinates of the new ends.

The coordinates of two points $A$ and $B$ are $(-1,4)$ and $(5,1)$ respectively. Find the coordinates of the point $P$ which lies on extended line $AB$ such that it is three times as far from $B$ as from $A$.

Find the distances ( in units ) of that point from the origin which divides the line segment joining the points $(5,-4)$ and$(3,-2)$ in the ratio $4:3$.