How can i solve this tough question of class 9th 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)
AnilSinghBoraGuru
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)