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Question
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.
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`
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