Question

$\LARGE{\underline{\frak{\purple{Quєstíon}}}}$


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

Answer 1

$\LARGE{\underline{\frak{\purple{Answєr :}}}}$

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 = 1212AC (mid-point formula)

In ∆ABC, P and Q are the mid points AB and BC.

∴ PQ || AC PQ = ½AC.

g ∴ 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.

Hᴏᴘᴇ Tʜɪs Hᴇʟᴘs Yᴏᴜ

Answer 2

$\huge\tt{\bold{\purple{\underbrace{\red{\mathfrak{ɾҽզմíɾҽժ⠀ αղsաҽɾ♡༄}}}}}}$

$\huge{\fbox{\pink{Given:-}}}$

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

⠀

$\huge{\fbox{\pink{To prove:-}}}$

PQRS is a rectangle.

⠀

$\huge{\fbox{\pink{Construction:-}}}$

Join AC and BD.

⠀

$\huge{\fbox{\pink{Proof:-}}}$

In ΔDRS and ΔBPQ,

DS = BQ (Halves of the opposite sides of the rhombus)

∠SDR = ∠QBP (Opposite angles of the rhombus)

DR = BP (Halves of the opposite sides of the rhombus)

Thus, ΔDRS ≅ ΔBPQ by SAS congruence condition.

RS = PQ by CPCT --- (i)

In ΔQCR and ΔSAP,

RC = PA (Halves of the opposite sides of the rhombus)

∠RCQ = ∠PAS (Opposite angles of the rhombus)

CQ = AS (Halves of the opposite sides of the rhombus)

Thus, ΔQCR ≅ ΔSAP by SAS congruence condition.

RQ = SP by CPCT --- (ii)

Now,

In ΔCDB,

R and Q are the mid points of CD and BC respectively.

• QR || BD

also,

P and S are the mid points of AD and AB respectively.

• PS || BD
• QR || PS

Thus, PQRS is a parallelogram.

also, ∠PQR = 90°

Now,

In PQRS,

RS = PQ and RQ = SP from (i) and (ii)

∠Q = 90°

Thus, PQRS is a rectangle.