Ex .8.2, 3
ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Given: ABCD is rectangle where
P, Q, R and S are mid-points of the
sides AB, BC, CD and DA respectively
To prove: PQRS is a rhombus
Construction: Join A & C
Proof: A rhombus is a parallelogram with all sides equal
First we will prove PQRS is a parallelogram,
and then prove all sides equal
From (1) & (2)
PQ ∥ RS and PQ = RS
In PQRS,
one pair of opposite side is parallel and equal.
Hence, PQRS is a parallelogram.
Now we prove all sides equal
In ∆ APS & ∆ BPQ
AP = BP
∠ PAS = ∠ PBQ
AS = BQ
∴ ∆ APS ≅ ∆ BPQ
∴ PS = PQ
But PS = RQ & PQ = RS
∴ PQ = RS = PS = RQ
Hence, all sides are equal
Thus, PQRS is a parallelogram with all sides equal
So, PQRS is a rhombus
Hence proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.