Abcd is a paralleogram in which P and Q are the mid points of opposite Sides AB and CD. If AQ intersect DP at S and BQ intersect CP at R shoe that
(A) abcd is a paralleogram
B. DPBQ is a paralleogram
C. PQRS is a paralleogram
Answers
a) abcd is a parallelogram (given)
b)<D=<b- opposite angles of a parallelogram
therefore half <d=half <b
so, <dpb=<bqd
similarly <pbq=<qdp
therefore dpbq is a parallelogram with 2 pair of opposite anles equal
c) prove that s is the mid point of aq and r is the mid point of cp
t
so sq=qr=rp=ps also aq is a bisector so , sq//pr//qr//ps
therefore pqrs is a parallelogram
The Proof is given below:
If P is the mid point of AB and Q is the mid point of CD then
AP = PB = AB/2
CQ = QD = CD/2
But ABCD is a parallelogram
∴ AB = CD
∴ AP = PB = CQ = QD
Also
∵ AB ║ CD
∴ AP ║ CQ ║ QD
And PB ║ CQ ║ QD
In quadrilateral APCQ
∵ AP ║ CQ
And, AP = CQ
Therefore, APCQ is a parallelogram (By theorem: If in a quadrilateral the opposite sides are equal and parallel then the quadrilateral is a parallelogram)
In quadrilateral DPBQ
∵ PB ║ QD
And, PB = QD
Therefore, DPBQ is a parallelogram
It can be easliy proved that APQD and PBCQ are parallelograms
AQ, DP and PC, QB are diagonals of the parallelograms APQD and PBCQ
∴ AS = SQ and DS = SP
And PR = RC and BR = RQ
∴ APCQ is a parallelogram
∴ AQ = PC
Therefore, (1/2)AQ = (1/2)PC
or, SQ = PR
And AQ ║ PC
Therefore, (1/2)AQ ║ (1/2)PC
or, SQ¦ ║ PR
? SQ = PR
And SQ ║ PR
Therefore, quadrilateral PSQR is a parallelogram (Hence Proved)
Hope this answer is helpful.
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