Question

In ΔABC, M and N are mid-points of sides AB and BC. P is a point on AC such that PN || AB. Prove that PMBN is a parallelogram.

Answer 1

Step 1:

M is the midpoint of AB

N is the midpoint of BC

By the Midpoint Theorem in ΔABC, MN || AC

Also, `MN = 1/2 AC`

Step 2:

In ΔABC, N is the midpoint of BC

PN || AB is given

By the converse of the Midpoint Theorem, if a line through the midpoint of one side of a triangle is parallel to another side, it bisects the third side.

Therefore, P is the midpoint of AC.

Step 3:

M is the midpoint of AB

P is the midpoint of AC

By the Midpoint Theorem in ΔABC, PM || BC

Also, `PM = 1/2 BC`

Step 4:

From Step 3, PM || BC

Since N is on BC, PM || BN

From Step 2, P is the midpoint of AC

From Step 1, MN || AC

Since PN || AB is given and M is on AB, PN || MB

Since both pairs of opposite sides are parallel (PM || BN and PN || MB), PMBN is a parallelogram.