**Question:**

Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).

**Solution:**

We have two points A (5, 7) and B (3, 9) which form a line segment and similarly

C (8, 6) and D (0, 10) form another line segment.

We have to prove that mid-point of AB is also the mid-point of CD.

In general to find the mid-point $\mathrm{P}(x, y)$ of 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 P of line segment AB can be written as,

$P(x, y)=\left(\frac{5+3}{2}, \frac{7+9}{2}\right)$

Now equate the individual terms to get,

$x=4$

$y=8$

So co-ordinates of P is (4, 8)

Similarly mid-point Q of side CD can be written as,

$Q(x, y)=\left(\frac{8+0}{2}, \frac{6+10}{2}\right)$

Now equate the individual terms to get,

$x=4$

$y=8$

So co-ordinates of Q is (4, 8)

Hence the point P and Q coincides.

Thus mid-point of AB is also the mid-point of CD.