The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
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Answers
QUESTION :
The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.
ANSWER :
Let ABCD be a quadrilateral where P, Q, R, S are the midpoint of AB, BC, CD, DA. Diagonal AC and BD intersect at a right angle at point O. We need to show that PQRS rectangle
Proof:
Here, ABCD is a quadrilateral.
AC and BD are diagonals, which are perpendicular to each other.
P,Q,R and S are the mid-point of AB,BC,CD and AD respectively.
In △ABC,
P and Q are mid points of AB and BC respectively.
∴ PQ ∥ AC and PQ= 1/2AC ----- ( 1 ) [ By mid-point theorem ]
Similarly, in △ACD,
R and S are mid-points of sides CD and AD respectively.
∴ SR ∥ AC and SR= 1/2AC ----- ( 2 ) [ By mid-point theorem ]
From ( 1 ) and ( 2 ), we get
PQ ∥ SR and PQ = SR
∴ PQRS is a parallelogram.
Now, RS ∥ AC and QR ∥ BD.
⇒ Also, AC ⊥ BD [ Given ]
∴ RS ⊥ QR
∴ PQRS is a rectangle.