Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

#### Solution

Let ABCD be a quadrilateral in which

`|bar"AB"| = |bar"BC"| = |bar"CD"| = |bar"DA"|` ....(1)

and AB || DC and AD || BC

∴ `bar"AB" = bar"DC"` and `bar"AD" = bar"BC"` ...(2)

Now, `bar"AC" = bar"AB" + bar"BC"`

and `bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"BC"` ...[By(2)]

`= bar"BC" - bar"AB"`

∴ `bar"AC".bar"BD" = (bar"AB" + bar"BC").(bar"BC" - bar"AB")`

`= bar"AB".(bar"BC" - bar"AB") + bar"BC" . (bar"BC" - bar"AB")`

`= bar"AB".bar"BC" - bar"AB".bar"AB" + bar"BC".bar"BC" - bar"BC".bar"AB"`

`= |bar"BC"|^2 - |bar"AB"|^2` ....`[∵ bar"AB".bar"BC" = bar"BC".bar"AB"]`

= 0 ...[By(1)]

∵ `bar"AC", bar"BD"` are non-zero vectors

∴ `bar"AC"` is perpendicular to `bar"BD"`

Hence, the diagonals are perpendicular.