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the base BC of triangle ABC is divided at D such that BD is half DC prove that the area of triangle ABC is equal to 1 by 3 of the area of triangle ABC ​

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student-name Aparna asked in Math

the base BC of triangle ABC is divided at D such that BD=1/2DC. Prove that ar(triangle ABD)=1/3. ar(triangleABC).

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student-name Ajanta answered this

36357 helpful votes in Math, Class XII-Science

given : BD = 1/2 DC

construction: draw a line perpendicular to BC . AE perpendicular to BC.

let AE = h (the height of the triangle ABC)

since BD + DC = BC

BD + 2BD = BC

3BD = BC

BD = 1/3 BC

area of the triangle ABD =frac{1}{2}*BD*AE=1/2*(1/3BC)*AE

=frac{1}{3}*frac{1}{2}BC*AE

=frac{1}{3}*area(ABC)