Question

ΔABC and ΔABD are two triangles on the same base
AB. If line segment CD is bisected by AB at O. Show that
ar(ABC) = ar(ABD)

Answer 1

Consider ΔACD.
Given that line segment CD is bisected by AB at O.
Therefore, OA is the median of ΔACD
∴Area (ΔACO) = Area (ΔADO) ... (1)
In ΔBCD, OB is the median.
∴ ar(ΔBCO) = ar(ΔBDO) ... (2)
Adding equations (1) and (2), we obtain
ar(ΔACO) + ar(ΔBCO) = ar(ΔADO) + ar(ΔBDO)
⇒ ar(ΔABC) = ar(ΔABD)

Answer 2

In triangle ABC, AO is the median (CD is bisected by AB at O)

So, ar(AOC)=ar(AOD)..........(i)

Also,

triangle BCD,BO is the median. (CD is bisected by AB at O)

So, ar(BOC) = ar(BOD)..........(ii)

Adding (i) and (ii),

We get,

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

⇒ ar(ABC) = ar(ABD)

Hence showed.