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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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