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Question

ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O, show that ar(ABC)=ar(ABD).
1300026_ceb85f5832e44768addca181ce5bc9d6.PNG

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Solution

Given:ABC and ABD on the same base AB and AB bisects CD
OC=OD .....(1)

To prove:ar(ABC)=ar(ABD)

Proof:
In ACD,

Since OC=OD from (1)

,OA is the median

ar(AOC)=ar(AOD) ....(2)

Since median divides the triangle into equal areas

Similarly, in BCD

since OC=OD from (1)

OB is the median

ar(BOC)=ar(BOD) ....(3)

Since median divides the triangle into equal areas

Adding eqns(2) and (3) we get

ar(AOC)+ar(BOC)=ar(AOD)+ar(BOD)

ar(ABC)=ar(ADB)

Hence proved.

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