Question

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

Answer 1

Let ABCD be a parallelogram

To prove ABCD be a rectangle provided the diagonals are equal.

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

`bar"BD" = bar"BC" + bar"CD"`

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

But `(bar"AC")^2 = (bar"BD")^2`

`(bar"AB" + bar"BC")^2 = (bar"BC" - bar"AB")^2`

`(bar"AB")^2 + (bar"BC")^2 + 2bar"AB" * bar"BC" = (bar"BC")^2 + (bar"AB")^2 - 2bar"AB" * bar"BC"`

`4bar"AB" * bar"BC"` = 0

`bar"AB" * bar"BC"` = 0

`vec"AB"` ⊥^r to `vec"BC"`

⇒ ABCD is a rectangle.