Question

In ΔABC, M and N are mid-points of AB and AC, and NP || AB.

1. Prove that BMNP is a parallelogram.
2. If AB = 11 cm, BC = 12 cm, find the perimeter of BMNP.

Answer 1

Step 1:

M and N are midpoints of AB and AC

By the Midpoint Theorem, MN || BC

Also, MN = `1/2` BC

Step 2:

Given NP || AB, so NP || BM

Since MN || BC (from Step 1) and P is on BC, then MN || BP

Step 3:

Since both pairs of opposite sides are parallel (BM || NP and MN || BP), BMNP is a parallelogram

Step 4:

`BM = 1/2 AB = 1/2 xx 11  cm = 5.5  cm`

`MN = 1/2 BC = 1/2 xx 12  cm = 6  cm`

In a parallelogram, opposite sides are equal: NP = BM = 5.5 cm

In a parallelogram, opposite sides are equal: BP = MN = 6 cm

Step 5:

Perimeter = BM + MN + NP + BP

Perimeter = 5.5 cm + 6 cm + 5.5 cm + 6 cm

Perimeter = 2 × (5.5 cm + 6 cm) = 2 × 11.5 cm = 23 cm

BMNP is a parallelogram and its perimeter is 23 cm.