Question

Show that the median of a triangle divides it into two triangles of equal areas.

Answer 1

Let ABC be a triangle and Let AD be one of its medians.

In $△ABD$ and $△ADC$ the vertex is common and these bases BD and DC are equal.

Draw $AE\perp BC.$

Now $area\left(△ABD\right)=\frac{1}{2}\times base\times altitudeof△ADB$

$=\frac{1}{2}\times BD\times AE$

$=\frac{1}{2}\times DC\times AE(\because BD=DC)$

but DC and AE is the base and altitude of $△ACD$

$=\frac{1}{2}\times$ base DC $\times$ altitude of $△ACD$

$=area△ACD$

$\Rightarrow area\left(△ABD\right)=area\left(△ACD\right)$

Hence the median of a triangle divides it into two triangles of equal areas.