Example 6
ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD. If AQ intersects DP at S and BQ intersects CP at R, show that:
APCQ is a parallelogram.
Given: ABCD is a parallelogram
P , Q are mid-points of AB & CD respectively
To prove: APCQ is a parallelogram
Proof:
Since AB ∥ DC
⇒ AP ∥ QC
Also,
AB = CD
1/2 AB = 1/2 CD
AP = QC
Since AP ∥ QC & AP = QC
One pair of opposite sides of APCQ are equal and parallel
∴ APCQ is a parallelogram
Example 6
ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD.If AQ intersects DP at S and BQ intersects CP at R, show that:
(ii) DPBQ is a parallelogram.
Since
AB ∥ DC
⇒ PB ∥ DQ
Also, AB = CD
∴ 1/2 AB = 1/2 CD
PB = DQ
Now, PB ∥ DQ & PB = DQ
Since one pair of opposite sides of DPBQ are equal and parallel
∴ DPBQ is a parallelogram
Example 6
ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD.If AQ intersects DP at S and BQ intersects CP at R, show that:
(iii) PSQR is a parallelogram.
In part(ii) we proved
DPBQ is a parallelogram
So, DP ∥ QB
∴ SP ∥ QR
Also, in part(i) we proved
APCQ is a parallelogram
So, AQ ∥ PC
∴ SQ ∥ PR
From (1) & (2)
SP ∥ QR & SQ ∥ PR
Now, in PSQR
both pairs of opposite sides of PSQR are parallel
∴ PSQR is a parallelogram

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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