Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.Also, AC⊥BDTo prove PQRS is a rectangle.Proof Since, AC⊥BD∴ ∠COD=∠AOD=∠AOB=∠COB=90∘ ltBrgt In ΔADC, S and R are the mid-points of AD and DC respectively, then by mid-point theorem SR∣∣ACandSR=12AC ...(i)In ΔABC, P and Q are the mid-points of AB and BC respectively, then by mid-point theorem PQ∣∣ACandPQ=12AC ...(ii)From Eqs. (i) and (ii), PQ∣∣SRandPQ=SR=12AC ...(iii)Similarly, SP∣∣RQandSP=RQ=12BD ...(iv)Now, in quadrilateral EOFR, OE||FR,OF||ER∴ ∠EOF=∠ERF=90∘ [∵∠COD=90∘⇒∠EOF=90∘]...(v)So, PQRS is a rectangle. Hence proved.
Answers & Comments
Answer:
Given- 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,
AC and BD are joined.
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.
Verified answer
Ste
Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.Also, AC⊥BDTo prove PQRS is a rectangle.Proof Since, AC⊥BD∴ ∠COD=∠AOD=∠AOB=∠COB=90∘ ltBrgt In ΔADC, S and R are the mid-points of AD and DC respectively, then by mid-point theorem SR∣∣ACandSR=12AC ...(i)In ΔABC, P and Q are the mid-points of AB and BC respectively, then by mid-point theorem PQ∣∣ACandPQ=12AC ...(ii)From Eqs. (i) and (ii), PQ∣∣SRandPQ=SR=12AC ...(iii)Similarly, SP∣∣RQandSP=RQ=12BD ...(iv)Now, in quadrilateral EOFR, OE||FR,OF||ER∴ ∠EOF=∠ERF=90∘ [∵∠COD=90∘⇒∠EOF=90∘]...(v)So, PQRS is a rectangle. Hence proved.