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