In triangleABC, D and E are the midpoints of sides AB and AC respectively, and F is any point on BC
Prove that area(quadrilateral ADFE) =1/2area(ABC).
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ABC, D, E and F are midpoints of sides AB, BC and CA respectively. BC = EC Recall that the line joining the midpoints of two sides of a triangle is parallel to third side and half of it. Hence DF = (1/2) BC ⇒ (DF/BC) = (1/2) → (1) Similarly, (DE/AC) = (1/2) → (2) (EF/AB) = (1/2) → (3) From (1), (2) and (3) we have But if in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar Hence ΔABC ~ ΔEDF [By SSS similarity theorem] Hence area of ΔDEF : area of ΔABC = 1 :
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