Question

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ABCD is a rectangle and P, Q, R, and S are mid - points of the sides AB, BC, CD and DA respectively. Show that quadrilateral ABCD is a rhombus.


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Answer 1

☆Let us join AC and BD.

In ΔABC,

P and Q are the mid-points of AB and BC respectively.

∴ PQ || AC and PQ = AC (Mid-point theorem) ... (1)

Similarly in ΔADC,

SR || AC and SR = AC (Mid-point theorem) ... (2)

Clearly, PQ || SR and PQ = SR

Since in quadrilateral PQRS, one pair of opposite sides is equal and parallel to 

each other, it 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)

However, 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

☆Therefore, PQRS is a rhombus

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Answer 2

Question :-

ABCD is a rectangle and P, Q, R, and S are mid - points of the sides AB, BC, CD and DA respectively. Show that quadrilateral ABCD is a rhombus.

Given :-

• ABCD is a rectangle and P, Q, R, and S are mid - points of the sides AB, BC, CD and DA respectively..

To prove :-

• PQRS is a rhombus

proof :-

a rhombus is a parallelogram with all sides equal Frist we will prove the PQRS is a parallelogram and then we will prove all sides are equal

In Triangle ABC :-

• p is mid point of AB
• Q is mid point of BC

therefore PQ ll AC and PQ = 1/2 AC ----(1)

In triangle ADC :-

• r is mid point of CD
• s is mid point of AD

therefore RS ll AC and RS = 1/2 AC -----(2)

from (1) and (2)

PQ ll RS and PQ = RS

now in PQRS

• one pair of opposite side is parallel and equal hence PQRS is a parallelogram.

now we will prove all sides are equal

in triangle APS and triangle BPQ

↬ AP = BP (p is the mid point)

↬ triangle PAS = triangle PBQ (each 90°)

↬ AS = BQ

↬ triangle APS = BPQ (sas rule)

↬ therefore PS = PQ (cpct)

but ps = rq and PQ = rs (opposite side of parallelogram)

therefore PQ = rs = ps = rq

therefore all sides are equal

thus, PQRS is a parallelogram with all sides equal

so, PQRS is a rhombus

hence proved

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