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.

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

Answer 1

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)`

Similarly,

`ar(ΔCFE) = 1/4ar(CBA) `

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

`ar(ΔDEF) = ar(ΔABC) - 3/4 ar(ΔABC)`

`ar(ΔDEF) = `1/4 ar(ΔABC)`

`(ar(ΔDEF))/(ar(ΔABC)) = 1/4`

Hence, area of (ΔDEF) : area of (ΔABC) = 1 : 4 or area of (△ABC) : area of (△DEF) = 4 : 1.