Question

Show that a median of a triangle divides it into two triangle of equal area.

Answer 1

R.E.F image

To Prove :- The median of triangle divided it into triangles of

equal area.

Given :- $\Delta ABC$ with AD as the median

$BD=CD=\frac{1}{2}BC$

To Prove :- $ar\left(\Delta ABD\right)=ar\left(\Delta ACD\right)$

Construction :- Draw line $AN\perp BC$

Proof :- To find area, we use formula

area of triangle $=\frac{1}{2}\times Base\times Altitude$

In $\Delta ABD,$

BD is Base and AN is altitude $\therefore ar\left(ABD\right)=\frac{1}{2}\times Base\times Altitude$

$ar\left(ABD\right)=\frac{1}{2}\times BD\times AN...\left(1\right)$

In $\Delta ACD,$

CD is Base and AN is altitude $\therefore ar\left(ACD\right)=\frac{1}{2}\times Base\times Altitude$

$=\frac{1}{2}\times CD\times AN$

But $BD=CD$

$\therefore ar\left(ACD\right)=\frac{1}{2}\times BD\times AN...\left(2\right)$

from (1) and (2)

$ar\left(\Delta ABD\right)=ar\left(\Delta ACD\right)$

Hence proved