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Question

# Question 7 P and Q are the mid – points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

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Solution

## Given In a parallelogram ABCD, P and Q are the mid – points of AB and CD, respectively. To show PRQS is a parallelogram. Proof Since, ABCD is a parallelogram. AB || CD ⇒ AP || QC Also, AB || DC 12AB=12DC [dividing both sides by 2 ] ⇒ AP = QC [since, P and Q are the mid – points of AB and DC ] Now, AP || QC and AP = QC Thus APCQ is a parallelogram. ∴ AQ||PC or SQ || PR . . . . . . . . ..(i) Again, AB||DC or BP||DQ Also, AB=DC⇒12AB=12DC [dividing both sides by 2] ⇒ BP = QD [since, P and Q are the mid – points of AB and DC] Now, BP||QD and BP = QD So, BPDQ is a parallelogram. ∴ PD||BQ or PS||QR From Eqs. (i) and (ii), SQ||RP and PS||QR So, PRQS is a parallelogram. Hence proved.

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