Question

If $K(5,4)$ is the mid-point of the line segment $PQ$ and co-ordinates of $Q$ are $(2,3),$ then find the co-ordinates of point $P$.

Answer 1

Given that, PQ is a line segment with K its mid-point

Let, P$\equiv (x_1,y_1)$ and Q$\equiv (x_2,y_2)$

Given, Co-ordinates of of Q are $(2,3)$ and that of mid-point K are $(5,4)$

By defination of mid-point,

$x$ co-ordinates of mid-point $=\frac{x_1+x_2}{2}$

$y$ co-ordinates of mid-point $=\frac{y_1+y_2}{2}$

$\therefore$ $5=\frac{x_1+2}{2}$ & $4=\frac{y_1+3}{2}$

$\therefore$ $10=x_1+2$ & $8=y_1+3$

$\therefore$ $x_1=8$ & $y_1=5$

hence, co-ordinates of point P are $(8,5)$.