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D, E and F respectively are the mid-points of the sides BC, CA and AB of ABC. Show that ar(DEF) =

Asked by Topperlearning User 4th June 2014, 1:23 PM
Answered by Expert
Answer:

D and E are mid points of BC and AC.

DE||AB and DE = (Mid-point Theorem)

Similarly, E and F are mid points of AC and AB.

EF||BC and EF =

In quadrilateral BDEF, DE||BF and FE||BD

BDEF is a parallelogram

Similarly, DCEF and AFDE are parallelograms.

We know that diagonal of a parallelogram divides it into two triangles of equal area.

Area (BFD) = Area (DEF) (For parallelogram BDEF)

Area (CDE) = Area (DEF) (For parallelogram DCEF)

Area (AFE) = Area (DEF) (For parallelogram AFDE)

Area (AFE) = Area (BFD) = Area (CDE) = Area (DEF)

Also,

Area (AFE) + Area (BDF) + Area (CDE) + Area (DEF) = Area (ABC)

Area (DEF) + Area (DEF) + Area (DEF) + Area (DEF) = Area (ABC)

4 Area (DEF) = Area (ABC)

Area (DEF) = Area (ABC)

Answered by Expert 4th June 2014, 3:23 PM
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