Question

In the figure, ABC and ABD are two triangles on the same base AB. If line segment CD bisects AB at O show that ar (ABC)=ar (ABD).

Answer 1

Answer:

The median divides a triangle into two Triangles of equal areas.

Given:

∆ABC and ∆ABD are two triangles on the same base AB.

To show:

ar (ABC) = ar (ABD).

Proof:

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

OC= OD

In ∆ACD , We have OC=OD

So, AO is the median of ∆ACD

Also we know that the median divides a triangle into two Triangles of equal areas.

∴ ar(∆AOC) = ar(∆AOD) — (i)

Similarly,In ΔBCD,

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

∴ ar(∆BOC) = ar(∆BOD) — (ii)

On Adding eq (i) and (ii) we get,

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

 ar(∆ABC) = ar(∆ABD)

Hope this will help 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.