Question

P and Q are mid-points of sides AB and CD of a parallelogram ABCD. Prove that APCQ is a parallelogram.

Answer 1

Given:

P and Q are mid-points of sides AB and CD respectively of parallelogram ABCD.

To Prove:

APCQ is a parallelogram.

Proof:

1. Since ABCD is a parallelogram, opposite sides AB and CD are parallel and equal in length.

So, AB || DC and AB = DC.

2. P and Q are mid-points:

P is midpoint of AB → AP = PB = `1/2` AB

Q is midpoint of CD → CQ = QD = `1/2` DC

3. Consider segments AP and QC:

Since P is midpoint of AB, AP is half of AB

Since Q is midpoint of CD, QC is half of DC

4. Since AB || DC (as ABCD is a parallelogram) and AP and QC are half-segments on these parallel sides, it follows that AP || QC and AP = QC.

5. A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.

Here, in quadrilateral APCQ, AP and QC are opposite sides that are equal and parallel.

Therefore, APCQ is a parallelogram.