Question

in the adjacent figure ABCD is a parallelogram and E and F are the midpoints of sides AB and CD respectively show that quadrilateral AECF is a parallelogram

Answer 1

Step-by-step explanation:

Given: A parallelogram ABCD, in which E and F are mid-points of AB and DC respectively.

Proof: Since E and F are mid-points of AB,DC respectively.

∴ AE = (1/2) AB and CF = (1/2) DC  ------- (i)

But, AB = DC and AB ║ DC   ------ (ii)

From (i) & (ii), we get

∴ AE = CF and AE ║ CF.

{Quadrilateral is a parallelogram if a pair of opposite sides is parallel and of equal length}.

∴ AECF is a parallelogram.

Hope it helps!

Answer 2

Now since E and F are mid points of AB and DC

⇒ AE = EB = AB

and DF= FC = DC

but AB = DC

⇒ AEFD and BCFE are parallelogram

⇒ AD EF BC ........ (1)

Now In ∆ ABG

AE = EB and EX AG (from (1))

⇒ X is the mid point of GB (mid point theorem)

and in ∆ GBH

GX = XB (∵ X is the mid point of GB)

and XP BH (from (1))

⇒ P is the mid point of GH

⇒ GP = PH