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 (ABC) = ar (ABD).

Answer 1

Given

$\Delta ABC$ and $\Delta ABD$ are two triangles on the same base AB.

To show :

$ar\left(ABC\right)=ar\left(ABD\right)$

Proof :

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

In $\Delta ACD$, We have OC=OD.

So, AO is the median of $\Delta ACD$

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

$\therefore ar\left(\Delta AOC\right)=ar\left(\Delta AOD\right)$ _______ (1)

Similarly , In $\Delta BCD$,

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

$\therefore ar\left(\Delta BOC\right)=ar\left(\Delta BOD\right)$ _______ (2)

On adding equation (1) and (2) we get,

$ar\left(\Delta AOC\right)+ar\left(\Delta BOC\right)=ar\left(\Delta AOD\right)+ar\left(\Delta BOD\right)$

$\therefore ar\left(\Delta ABC\right)=ar\left(\Delta ABD\right)$