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Question
If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle,
show that the diagonals AC and BD intersect at the right angle.
Solution
The figure is shown below
Let ABCD be a quadrilateral where P, Q, R, S are the midpoint of AB, BC, CD, DA.PQRS is a rectangle. Diagonal AC and BD intersect at point O. We need to show that AC and BD intersect at a right angle.
Proof:
PQ || AC, therefore ∠AOD = ∠PXO ...[ Corresponding angle ]...(1)
Again BD || RQ, therefore ∠PXO = ∠RQX = 90° ....[ Corresponding angle and angle of a rectangle ]...(2)
From (1) and (2) we get ,
∠AOD = 90°
Similarly, ∠AOB = ∠BOC = ∠DOC = 90°
Therefore diagonals AC and BD intersect at right angle.
Hence proved.
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