Question
In figure, ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O. Show that ar(AABC) = ar(ΔABD).
Solution
Given: ABC and ABD are two triangles on the same base AB. Line segment CD is bisected by AB at O.
To Prove: ar(ΔABC) = ar(ΔABD).
Proof: ∵ Line segment CD is bisected by AB
at O.
∴ OC = OD BO is a median of ΔBCD and AO is a median of ΔACD
∵ BO is a median of ΔBCD
∴ ar(ΔOBC) = ar(ΔOBD) ...(1)
∵ A median of a triangle divides it into two triangles of equal areas
∵ AO is a median of ΔACD
∴ ar(ΔOAC) = ar(ΔOAD) ...(2)
∵ A median of a triangle divides it into two triangles of equal areas
Adding (1) and (2), we get ar(ΔOBC) + ar(ΔOAC)
= ar(ΔOBD) + ar(ΔOAD)
⇒ ar(ΔABC) = ar(ΔABD).
