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Question

In the given figure, DEBC.(i) If DE =4cm , BC= 8 cm andA(ADE)=25 cm2, find A(ABC).(ii) If DE:BC= 3:5, then find A(ADE):A(DBCE).

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Solution

(i)
In ABC and ADE, BAC=DAE (Common angles ) ...(1)ABC=ADE (Corresponding angles, as DEBC) ...(2)From equations (1) and (2), we get:ABC~ADE [By the AA test of similarity]Thus, A(ABC)A(ADE)=AB2AD2=BC2DE2=AC2AE2A(ABC)A(ADE)=BC2DE2On substituting A(ADE)=25 cm2, DE=4 cm and BC =8 cm, we get:A(ABC)25=(8)2(4)2A(ABC)=25×6416=100 cm2
(ii)
In (i) we have proved that ABC~ADE.Thus, A(ABC)A(ADE)=AB2AD2=BC2DE2=AC2AE2A(ABC)A(ADE)=BC2DE2A(ADE)A(ABC)=DE2BC2On substituting DE:BC =3:5, we get:A(ADE)A(ABC)=(3)2(5)2=925On subtracting 1 from both the sides , we get:A(ABC)A(ADE)-1=259-1A(ABC)-A(ADE)A(ADE)=25-99A(DBCE)A(ADE)=169A(ADE)A(DBCE)=916Thus, A(ADE):A(DBCE)=9:16

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