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Preeti Dabral 2 years, 1 month ago
Given: △ABC ∼ △DEF
AM is a median in △ABC and DN is the corresponding median in △DEF
To prove: area△ABCarea△DEF=AM2DN2
Proof: △DBC ∼ △DEF
⇒ ∠ A = ∠ D, ∠ B = ∠ E, ∠ C = ∠ F and
ABDE=BCEF=ACDF
Also, area△ABCarea△DEF=AB2DE2=BC2EF2=AC2DF2........(i)[area theorem]
Now,ABDE=BCEF=12BC12EF=BMEN..............(ii)
In △ABM and DEN
∠ B= ∠ E and ABDE=BMEN [From (ii)]
⇒ △ABM ∼△DEN [SAS similarity]
⇒area△ABMarea△DEN=AB2DE2=AM2DN2=BM2EN2............(iii)
From (i) and (iii), we get
area△ABCarea△DEF=AM2DN2
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