#### 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 P(x,y) of two points `A(x_1,y_1)` and `B(x_2, y_2)` we use section formula as,

`P(x,y) = ((x_1 + x_2)/2, (y_1 +y_2)/2)`

Therefore mid-point P of line segment AB can be written as,

`P(x,y) = ((5 + 3)/2, (7 + 9)/2)`

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) = `((8 + 0)/2 "," (6 + 10)/2)`

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.