D,E and F are respectively the midpoints of sides AB,BCand CA of triangle ABC. find the ratios of the areas of DEF and ABC

Asked by chandershekhar sharma | 19th May, 2013, 09:39: PM

Expert Answer:

Given - A triangle ABC with D, E and F being the mid points of AB, BC and CA respectively. 
Join DE, EF and FD. 
Using mid point theorem, a line joining the mid points of 2 sides in a triangle is parallel to the third side and equal to half the length of third side. 
Hence, DE = 1/2 AC, EF = 1/2AB and FD = 1/2BC
Hence, by SSS similarity, 
triangle DEF is similar to triangle CAB
Now, the ratio of areas of similar triangles is equal to the square of the ratios of their sides
Hence, triangle(DEF)/triangle (CAB) = DE^2/CA^2 = 1/4

Answered by  | 20th May, 2013, 09:26: AM

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