D, E and F are respectively the mid-points of the sides BC, CA and AB of a Δ ABC. Show that: (i) BDEF is a parallelogram. (ii) ar (DEF) = ar (ABC) (iii) ar (BDEF) = ar (ABC)

Question

D, E and F are respectively the mid-points of the sides BC, CA and AB of a &#x394; ABC. Show that: (i) BDEF is a parallelogram. (ii) ar (DEF) = ar (ABC) (iii) ar (BDEF) = ar (ABC)

Philoid · Accepted Answer

(i) In triangle ABC it is given that,

EF is parallel to BC

And,

EF = BC (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 opposite sides are equal and parallel in length

Therefore,

BDEF is a parallelogram

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

Therefore,

Area of triangle BFD = Area of triangle DEF (For parallelogram BDEF) (1)

Area of triangle AFE = Area of triangle DEF (For parallelogram DCEF) (2)

Area of triangle CDE = Area of triangle DEF (For parallelogram AFDE) (3)

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

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

4 (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

Area of parallelogram BDEF = 2 * Area of triangle DEF

Area of parallelogram BDEF = 2 * * Area of triangle ABC

Area of parallelogram BDEF = * Area of triangle ABC

D, E and F are respectively the mid-points of the sides BC, CA and AB of a Δ ABC. Show that:(i) BDEF is a parallelogram.(ii) ar (DEF) =ar (ABC)(iii) ar (BDEF) =ar (ABC)

D, E and F are respectively the mid-points of the sides BC, CA and AB of a Δ ABC. Show that:
(i) BDEF is a parallelogram.

(ii) ar (DEF) =ar (ABC)

(iii) ar (BDEF) =ar (ABC)