Find the distance of the point (1, 2) from the mid-point

We have to find the distance of a point A (1, 2) from the mid-point of the line segment joining P (6, 8) and Q (2, 4).

In general to find the mid-point $\mathrm{P}(x, y)$ of any two points $\mathrm{A}\left(x_{1}, y_{1}\right)$ and $\mathrm{B}\left(x_{2}, y_{2}\right)$ we use section formula as,

$\mathrm{P}(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

Therefore mid-point B of line segment PQ can be written as,

$B(x, y)=\left(\frac{6+2}{2}, \frac{4+8}{2}\right)$

Now equate the individual terms to get,

$x=4$

$y=6$

So co-ordinates of B is (4, 6)

Therefore distance between A and B,

$\mathrm{AB}=\sqrt{(4-1)^{2}+(6-2)^{2}}$

$=\sqrt{9+16}$

$=5$