Data : ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.
To Prove: PQRS is a rectangle.
Construction: Diagonals AC and BD are drawn.
Proof: To prove PQRS is a rectnagle, one of its angle should be right angle.
In ∆ADC, S and R are the mid points of AD and DC.
∴ SR || AC
SR = \(\frac{1}{2}\)AC (mid-point formula)
In ∆ABC, P and Q are the mid points AB and BC.
∴ PQ || AC
PQ = \(\frac{1}{2}\)AC.
∴ SR || PQ and SR = PQ
∴ PQRS is a parallelogram.
But diagonals of a rhombus bisect at right angles. 90° angle is formed at ’O’.
∴ ∠P = 90°
∴ PQRS is a parallelogram, each of its angle is right angle.
This is the property of rectangle.
∴ PQRS is a rectangle.
