Question

ABCD is a parallelogram. P and Q are midpoints. Show that DP and BW intersects AC

Answer 1

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Question



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 ;

i) APCQ is a parallelogram

ii) DPBQ is a parallelogram 

iii) PSQR is a parallelogram





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Solution



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Given:ABCD is a parallelogram

P,Q are the midpoints of AB and CD respectively.

(i)Since AB∥DC since opposite sides of parallelogram are parallel.

⇒AP∥QC since parts of parallel lines are parallel

Also,AB=CD since opposite sides of parallelogram are equal.

⇒21AB=21CD

⇒AP=QC since P is mid-point of AB and Q is the midpoint of DC

Since AP∥QC and AP=AC

One pair of opposite sides of APCQ are equal and parallel

∴APCQ is a parallelogram.

(ii)Since AN∥DC since opposite sides of parallelogram are parallel.

⇒PB∥DQ since parts of parallel lines are parallel

Also,AB=CD since opposite sides of parallelogram are equal.

⇒21AB=21CD

⇒PB=DQ since P is mid-point of AB and Q is the midpoint of DC

Since PB∥DQ and PB=DQ

One pair of opposite sides of DPBQ are equal and parallel

∴DPBQ is a parallelogram.