D, E and F are respectively the mid-points of sides AB, BC and CA of ΔABC. Find the ratio of the area of ΔDEF and ΔABC.

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#### Solution

D and E are the mid-points of ΔABC

:.DE || AC and DE = `1/2AC`

In ΔBED and ΔBCA

∠BED = ∠BCA (Corresponding angle)

∠BDE = ∠BAC (Corresponding angle)

∠EBD = ∠CBA (Common angles)

∴ΔBED ~ ΔBCA (AAA similarity criterion)

`(ar(ΔBED))/(ar(ΔBCA)) = 1/4`

`=>ar(ΔBED) = 1/4 ar(ΔBCA)`

Similary

`ar(ΔCFE) = 1/4ar(CBA) `

Also ar(ΔDEF) = ar(ΔABC) - [ar(ΔBED) + ar(ΔCFE) + ar(ΔADF)]

`=>ar(ΔDEF) = ar(ΔABC) - 3/4 ar(ΔABC) = 1/4 ar(ΔABC)`

`=>ar(ΔDEF) = ar(ΔABC) - 3/4 ar(ΔABC) = 1/4 ar(ΔABC)`

`=>(ar(ΔDEF))/(ar(ΔABC)) = 1/4`

Concept: Areas of Similar Triangles

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