Question

"Question 4 In the given 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 (ABC) = ar (ABD).

Class 9 - Math - Areas of Parallelograms and Triangles Page 162"

Answer 1

1.CO=OD    (Given)
2. Area of triangle ABC= 1/2xbasexheight
                                        =1/2x AB x CO
3. Area Of triangle ABD= 1/2 x base x height
                                        = 1/2 x AB x OD
 From 1 & 3, we know that,
 CO= OD
therefore
4.Area of triangle ABD = 1/2 x AB xCO
Therefore,from 2 & 4,
Ar(ABC)= Ar(ABD)

Answer 2

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

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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)

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Hope this will help you...