Question

The figure formed by joining the mid points of the sides of a quadrilateral is a

• A. trapezium
• B. rhombus
• C. parallelogram
• D. rectangle

Answer 1

Answer: (A), (C)

The correct options are
A trapezium
C parallelogram

Parallelogram:

1. A parallelogram is a simple quadrilateral with two pairs of parallel sides.

2. The opposite or facing sides of a parallelogram are of equal length.

Proof:

In quadrilateral ABCD points P, Q, R, S are midpoints of side AB, BC, CD and AD respectively.

To prove :

PS || QR and SR || PQ. i.e. Quadrilateral PQRS is a parallelogram

Proof:

1. Draw diagonal BD.

2. As PS is the midsegment of ▲ ABD, we can say that PS || BD.

3. As QR is the midsegment of ▲ BCD, we can say that QR || BD.

4. ∵ PS || BD and QR || BD by transitivity, we can say that PS || QR.

5. Now draw diagonal AC.

6. As SR is the midsegment of ▲ ACD, we can say that SR || AC.

7. As PQ is the midsegment of ▲ ABC, we can say that PQ || AC.

8. ∵ SR || AC and PQ || AC by transitivity, we can say that SR || PQ.

9. ∵ PS || QR and SR || PQ, ∴ quadrilateral PQRS is a parallelogram (by definition)

All parallelogram are trapezium.

Ans- A and C