Question

Any point D is taken on the base BC of a triangle ABC and AD is produced to E, such that DE = AD.
Show that ar(triangle BCE) = ar(triangle ABC).​

Answer 1

ANSWER

Toprove:

AreaofΔBCE=areaofΔABC

proof:

InΔABE,

AD=DE(given)

∴BDisamedianofΔABE

⇒ar(ΔABD)=ar(ΔBED).......(1)

mediandividesatriangleintotwotrianglesofequalarea.

Similarly,

InΔACE

CDismedianofΔCED......(2)

Adding(1)and(2),

ar(ΔABD)ar(ΔACD)=ar(ΔBED)+ar(ΔCED)

orar(ΔABC)=ar(ΔBCE)

Hence,areaofΔBCE=areaofΔABC.

solution

Answer 2

ANSWER

Toprove:

AreaofΔBCE=areaofΔABC

proof:

InΔABE,

AD=DE(given)

∴BDisamedianofΔABE

⇒ar(ΔABD)=ar(ΔBED).......(1)

mediandividesatriangleintotwotrianglesofequalarea.

Similarly,

InΔACE

CDismedianofΔCED......(2)

Adding(1)and(2),

ar(ΔABD)ar(ΔACD)=ar(ΔBED)+ar(ΔCED)

orar(ΔABC)=ar(ΔBCE)

Hence,areaofΔBCE=areaofΔABC.

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