plz give me answer in step by step 3 question class 9th
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Given,
P, Q, R, S are the midpoints of AB, BC, CD and DA respectively.
ABCD is a rectangle
To Prove
PQRS is a rhombus.
Construction
Join AC.In ΔABC,
∴ PQ || AC and PQ = AC (Mid-point theorem) ... (1)
Similarly in ΔADC,
SR || AC and SR = AC (Mid-point theorem) ... (2)equ
PQ || SR and PQ = SR
in PQRS one pair of opposite side is parallel and equal
hence,PQRS is a parallelogram
∴ PS || QR and PS = QR (Opposite sides of parallelogram)... (3)
In ΔBCD, Q and R are the mid-points of side BC and CD respectively.
∴ QR || BD and QR =BD (Mid-point theorem) ... (4)
the diagonals of a rectangle are equal.
∴ AC = BD …(5)
By using equation (1), (2), (3), (4), and (5), we obtain
PQ = QR = SR = PS
, PQRS is a rhombus
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