Question

In ABC, D,E. Are the midpoints of sides AB and Ac Respectively show that ar ADE=1/4arABC

Answer 1

Answer:

Given: AD=DB;AE=EC

To Prove:Ar(ADE)=Ar(ABC)

Proof:We know that,

AD/DB=1.......(i)

AE/EC=1........(ii)

AD/DB=AE/EC

DE parallel to BC              (BY CONVERSE OF BPT)

NOW, BY MIDPOINT THEOREM

DE=1/2*BC......................(iii)

Construction=Draw perpendicular to BC such that ANC=<90

Now as  ME parallel to NC

<AME=<ANC=90           (CORRESPONDING ANGLES)

NOW IN AME & ANC,

<AME=<ANC=90

<MAE=<NAC                  (COMMON)

BY AA SIMILARITY CRITERION,

AME~ANC

AM/AN=AE/AC

AM/AN=1/2

AN=2AM...................(iv)

NOW,

AR(ADE)=1/2(DE*AM)

             =1/2(1/2BC*1/2AN)                    (BY EQUATIONS (iii) and (iv))

             =1/2(1/4(BC*AN))

             =1/2(1/2AR(ABC))

             =1/4(AR(ABC)

                                           HENCE PROVED