Prove that $QM = PN$ [Coordinate Geometry]

Question

If the points $(P,Q)$, $(M,N)$ and $(P-M,Q-N)$ are collinear. Then show that $QM=PN$.

  Basic Points and formulae:

• Distance between two points = $\sqrt{(x_2-x_1)^2+(y_2-y_2)^2} $

Where, $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two points.

• The collinear points lie on the same line.

Answer 1

Hint: If your points are collinear, then the line through $(P,Q)$ and $(M,N)$ has the same slope as the line through $(P,Q)$ and $(P-M,Q-N)$. (They are, after all, the same line.)

There is a small thing you have to be careful about, and deal with separately. If $M-Q=0$, that is, if $M=Q$, the line is vertical.

For the case when $M-Q\ne 0$, use the standard formula for the slope of a line, and some algebra, to reach the desired conclusion.