Question

Show that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer 1

Given:

Let ABCD be any quadrilateral.

Let P, Q, R, S be the mid-points of AB, BC, CD, DA respectively, so PR and QS are the lines joining mid-points of opposite sides.

Step-wise calculation:

1. Put position vectors or coordinates for the vertices:

Let A = a, B = b, C = c, D = d (vectors).

2. Then the mid-points are

`P = (a + b)/2`

`Q = (b + c)/2`

`R = (c + d)/2`

`S = (d + a)/2`

3. Compute the midpoint of PR:

`"Mid" (PR) = (P + R)/2` 

= `((a + b)/2 + (c + d)/2)/2` 

= `(a + b + c + d)/4`

4. Compute the midpoint of QS:

`"Mid" (QS) = (Q + S)/2` 

= `((b + c)/2 + (d + a)/2)/2`

= `(a + b + c + d)/4`

5. Since Mid (PR) = Mid (QS), the two segments PR and QS share the same midpoint.

Hence, they bisect each other.

Therefore, the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.