Question

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

Answer 1

heya,
as we all know,
it is given that : ∆ABC and ∆ABD are two triangles on the same base AB.

we have to show: ar(ABC) = ar(ABD)

the line segment CD is the bisected by AB at O.

in ΔACD, OC=OD.
hence, AO is the median of ΔACD.
∴ ar(AOC) = ar(AOD) ------------ (i)
in ΔBCD, BO is the median of AB.
∴ ar(BOC)=ar(BOD) -------------- (ii)

now, adding (i) and (ii), we get -
ar(AOC) + ar(BOC) = ar(AOD)+ ar(BOD)
ar(ABC) = ar(ABD).

hence, proved.
thank you. i hope this will helps you.

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)+(BOD)

⇒ ar(ABC) = ar(ABD)

Hence showed.