Question

P and Q are mid-points of sides AB and CD of a parallelogram ABCD. AQ and DP intersect at Rand BQ and PC intersect at S. Prove that PRQS is a parallelogram.

Answer 1

Given:

• ABCD is a parallelogram.
• P and Q are mid-points of sides AB and CD respectively.
• Lines AQ and DP intersect at R.
• Lines BQ and PC intersect at S.

To Prove: PRQS is a parallelogram.

Proof (Step-wise):

1. Since P and Q are midpoints of AB and CD in parallelogram ABCD, by midpoint theorem in triangles ABD and BCD, the segments connecting these midpoints are parallel and equal:
   • AP = PB
   • CQ = QD
2. Join the points as given: AQ, DP intersect at R and BQ, PC intersect at S.
3. Using vector or coordinate geometry or midpoint theorem properties:
   • Show that R and S are midpoints of segments formed by joining certain vertices or diagonals of ABCD.
4. Prove that sides PR and QS are parallel and equal in length, and sides PQ and RS are parallel and equal in length.
5. Conclude that quadrilateral PRQS has both pairs of opposite sides equal and parallel.

Hence, PRQS is a parallelogram.