Question

ALLEN
Class IX : Mathematics
12. In figure, ABCD is a parallelogram. E and F are mid-points of the sides AB and CD respectively
AF and DE intersect at P; BF and CE intersect at Q.
С
F
D
P Р
B
А
E
Prove that
(i) AECF is a parallelogram
(ii) BEDF is a parallelogram.
(iii) PEQF is a parallelogram.
diagonal AC such that​

Answer 1

Answer:

Given :-

ABCD is a parallelogram.

E and F are the midpoints of AB and CD

To be proof :

DP = PQ = QB

Construction : Join AF and CE

PROOF :

Since,

ABCD is a parallelogram,

So,

AB║CD and AB = CD

AE║CD and AB = CD

∴ AE║FC

and AE = FC [ AE = ¹/₂ AB, FC = ¹/₂ CD ]

⇒ AFCE is a parallelogram.

[ a pair of opposite sides are equal and parallel ]

Now,

In ΔDQC,

FP║CQ and F is the midpoint of CD

PQ = DP ..............(i)

Similarly from ΔAPB, Q is the midpoint of BP

So,

PQ = BQ ..............(ii)

From (i) and (ii),

PQ = DP = BQ

Hence,

AF and CF trisect the diagonal BD.