Question

In the parallelogram ABCD, E and F are the mid-points of AB and DC respectively. Prove that AEFD is a parallelogram.

Answer 1

Given: ABCD is a parallelogram.

E is the midpoint of AB.

F is the midpoint of DC.

To Prove: AEFD is a parallelogram.

Proof (Step-wise):

1. From ABCD being a parallelogram.

AB || CD and AD || BC

2. E is the midpoint of AB and F is the midpoint of DC.

So, AE is a segment of AB and DF is a segment of DC.

Because AB || CD, we have AE || DF.

3. Place vectors:

Let AB = 2u and AD = v.

Then A = 0, B = 2u, D = v, C = 2u + v.

Hence, E = midpoint of AB = u

And F = midpoint of DC

= `d + (c - d)/2`

= v + u

4. Compute EF = F – E

= (v + u) – u

= v

= AD

Thus, EF is parallel to AD in fact EF = AD.

5. From steps 2 and 4,

Both pairs of opposite sides of the quadrilateral AEFD are parallel: 

AE || DF and EF || AD 

Therefore, AEFD is a parallelogram.