ABCD is a parellelogram in which P and Q are midpoints of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R. show that
(1) APCQ is a parellogram
(2) DPBQ is a parellogram
Answers
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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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