How can i solve this tough question of class 9^{th} ncert of Areas of Parallelograms and Triangles of math of exercise 9.3 of question no.5(2). Give me the best and simple way for solving this question. D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC. Show that(ii) ar(DEF) = ¼ ar(ABC)

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# D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC. Show that(ii) ar(DEF) = ¼ ar(ABC) Q.5(2)

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(ii) Proceeding from the result of (i),

BDEF, DCEF, AFDE are parallelograms.

Diagonal of a parallelogram divides it into two triangles of equal area.

∴ar(ΔBFD) = ar(ΔDEF) (For parallelogram BDEF) — (i)

also,

ar(ΔAFE) = ar(ΔDEF) (For parallelogram DCEF) — (ii)

ar(ΔCDE) = ar(ΔDEF) (For parallelogram AFDE) — (iii)

From (i), (ii) and (iii)

ar(ΔBFD) = ar(ΔAFE) = ar(ΔCDE) = ar(ΔDEF)

⇒ ar(ΔBFD) +ar(ΔAFE) +ar(ΔCDE) +ar(ΔDEF) = ar(ΔABC)

⇒ 4 ar(ΔDEF) = ar(ΔABC)

⇒ ar(DEF) = ¼ ar(ABC)