Question

In Fig. $ ABC$ and $ ABD$ are two triangles on the same base $ AB$. If line- segment $ CD$ is bisected by $ AB$ at $ O$, show that: $ ar\left(ABC\right)=ar\left(ABD\right)$

Answer 1

Prove the areas of given triangle equal based on given information

We need to prove, $ar\left(ABC\right)=ar\left(ABD\right)$

In $∆ABC$, $AO$ is the median. ($CD$ is bisected by $AB$ at $O$)

Hence,

$ar\left(AOC\right)=ar\left(AOD\right)..................\left(1\right)$

Also, $∆BCD$, $BO$ is the median. ($CD$ is bisected by $AB$ at $O$)

Therefore,

$ar\left(BOC\right)=ar\left(BOD\right)..................\left(2\right)$

On adding equation $\left(1\right)$ and $\left(2\right)$ we get

$$\begin{gathered} \Rightarrow ar\left(AOC\right)+ar\left(BOC\right)=ar\left(AOD\right)+ar\left(BOD\right) \\ \Rightarrow ar\left(ABC\right)=ar\left(ABD\right) \end{gathered}$$

Hence proved, $ar\left(ABC\right)=ar\left(ABD\right)$.