Question

In the figure, $△ABC$ and $△ABD$ are two triangles on the same base $AB$. If line segment $CD$ is bisected by $\overline{AB}$ at $O$, show that $area\left(△ABC\right)=area\left(△ABD\right)$.

Answer 1

Given In the figure, triangle ABC and triangle ABD are two triangles on the same base AB. If line segment CD is bisected by AB at O

Then $O$ is the midpoint of $CD$

In triangle $ACD$

$OC=OD$

So OA is the median

$\therefore area\left(\Delta AOC\right)=area\left(\Delta AOD\right)$...........................(1)

Then O is the midpoint of CD

In triangle $BCD$

$OC=OD$

So $OB$ is the median

$\therefore area\left(\Delta BOC\right)=area\left(\Delta BOD\right)$...........................(2)

Adding (1) and (2) we get

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

$\Rightarrow area\left(\Delta ABC\right)=area\left(\Delta ABD\right)$ $\left[henceproved\right]$