Q. 53.7( 324 Votes )

# D, E and F are re

Answer :

**(i)** In triangle ABC it is given that,

EF is parallel to BC

And,

The**Midpoint Theorem**states that the segment joining two sides of a triangle at the midpoints of

those sides is parallel to the third side and is half the length of the third side.

EF = BC ( By using Mid – point theorem)

Also,

BD = BC (As D is the mid-point)

So,

BD = EF

BF and DE also parallel to each other

Therefore, the pair of opposite sides are equal and parallel in length.

Similarly BF = DE

Therefore,

BDEF is a parallelogram

**(ii)** Diagonal of a parallelogram divides it into two equal area

Therefore,

In parallelogram BDEFDF is the diagonal

⇒ Area of triangle BFD = Area of triangle DEF ......(1)

In parallelogram DCEF

⇒ Area of triangle AFE = Area of triangle DEF .....(2)

In parallelogram AFDEEF is the diagonal.

⇒ Area of triangle CDE = Area of triangle DEF ......(3)

So from (1), (2) and (3)

Area of triangle BFD = Area of triangle AFE = Area of triangle CDE = Area of triangle DEF

Area of triangle ABC = Area of triangle BFD + Area of triangle AFE + Area of triangle CDE + Area of triangle DEF4 (Area of triangle DEF) = Area of triangle ABC

Area of triangle DEF = × Area of triangle

**(iii)** Area of parallelogram BDEF = Area of triangle DEF + Area of triangle BDE

Area of parallelogram BDEF = Area of triangle DEF + Area of triangle DEF

= 2 × Area of triangle DEF

= 2 × × Area of triangle ABC

= × Area of triangle ABC

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