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ABCD is a parallelogram and P, Q are midpoints of sides AB and DC respectively. What can you conclude about quadrilateral APQD?

Class 9QuadrilateralsGeometry — Quadrilaterals
  • AAPQD is necessarily a parallelogram
  • BAPQD is necessarily a rectangle
  • CAPQD is a trapezium but not a parallelogram
  • DNo definite conclusion can be drawn about APQD

Step-by-Step Solution

AB ∥ DC and AB = DC. Their halves AP and DQ are therefore equal and parallel. One pair of opposite sides equal and parallel makes APQD a parallelogram.
Answer: A) APQD is necessarily a parallelogram

Key idea: Mid-point theorem applied to quadrilaterals

**Result**: The figure formed by joining the mid-points of the sides of any quadrilateral (in order) is a **parallelogram**.

**Proof sketch**: Let ABCD be any quadrilateral. Let P, Q, R, S be midpoints of AB, BC, CD, DA respectively. Draw diagonal AC.
- In ΔABC: PQ ∥ AC and PQ = (1/2)AC (mid-point theorem).
- In ΔACD: SR ∥ AC and SR = (1/2)AC (mid-point theorem).
- So PQ ∥ SR and PQ = SR. One pair equal AND parallel → PQRS is a parallelogram. ∎

If ABCD is a rectangle, PQRS becomes a rhombus. If ABCD is a rhombus, PQRS becomes a rectangle. Surprising and elegant.

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