Question

D,Eand F are the mid-points of sides AB,BC and CA triangleABC .Find the ratio of the area of triangle DEF and triangle ABC

Answer 1

ΔABC and ΔDEF are similar by (AA)
=> area of ΔDEF/area of ΔABC => (DE/AB)²=(EF/BC)²=(DF/AC)²

Answer 2

Solution: D, E and F are mid points hence FE || BC, DE || AB and DF || AC.

Therefore, quad. DEFB, DFEC & AFDE are parallelogram.

Hence angle E = B & F = C [opposite angles of || gm are equal]

By AA - criteria ∆DEF ~ ∆ABC

Also, DE = BF and DE = AF [opp. sides of || gm are equal]

→ DE = ½ AB

→ ar(∆DEF)/ar(∆ABC) = DE²/AB²

→ ar(∆DEF)/ar(∆ABC) = (½AB)²/AB²

→ ar(∆DEF)/ar(∆ABC) = ¼

Hence answer is ¼.