Question

In a parallelogram ABCD, the diagonals AC and BD intersect at point O. If P is the mid-point of BC, prove that

1. PO || AB
2. PO = `1/2` AB

Answer 1

Given:

ABCD is a parallelogram.

Diagonals AC and BD meet at O.

P is the midpoint of BC.

To Prove:

1. PO || AB 
2. PO = `1/2` AB

Proof [Step-wise]:

1. In a parallelogram, the diagonals bisect each other.

Hence, AO = OC.

So, O is the midpoint of AC.

2. P is given as the midpoint of BC.

3. Consider triangle BCA.

P and O are midpoints of BC and CA, respectively.

4. By the Midpoint Theorem, the segment joining midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

PO is parallel to BA and PO = `1/2` BA.