in triangle DEF,M and N are midpoints of sides EF and DE respectively IF ar(EMN)=4 sq cm,find ar(DEF) - Maths - Quadrilaterals

Question

in triangle DEF,M and N are midpoints of sides EF and DE
respectively IF ar(EMN)=4 sq cm,find ar(DEF)

Anuradha Sharma · Accepted Answer

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In ∆DEF, as M and N are the mid points of the sides EF, ED so using mid point theorem, MN|| DF and MN = 12DFSince MN∥FD and EF is a transversal, then∠EMN = ∠EFD  Correponding anglesSince MN∥FD and ED is a transversal, then∠ENM= ∠EDF  Correponding anglesIn ∆ENM and ∆EDF,∠EMN = ∠EFD  Correponding angles∠ENM= ∠EDF  Correponding angles⇒∆ENM ~ ∆EDF  AA similarityWe know that ratio of areas of 2 similar ∆'s is equal to the square of the ratio of their corresponding sides
So, ar∆ENMar∆EDF = NMDF2 = DF2DF2 = 14⇒4ar∆EDF = 14⇒ar∆EDF = 16 cm2

in triangle DEF,M and N are midpoints of sides EF and DE respectively.IF ar(EMN)=4 sq.cm,find ar(DEF)