Question

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

Answer 1

Suppose that AD is a median of ∆ABC and D is the mid point of BC. AD divides ∆ABC in two triangles, i.e., ∆ABD and ∆ADC.
To prove: ar(∆ABD) = ar(∆ADC)
Construction: Draw AL ⊥ BC.
Proof: ar(∆ABD) = $\frac{1}{2}$ × base × height
⇒ ar(∆ABD) =$\frac{1}{2}$ × BD × AL
⇒ ar(∆ABD) = $\frac{1}{2}$ ×DC × AL ...(i) (BD = DC)
Also, ar(∆ADC) = $\frac{1}{2}$ × DC × AL ...(ii)
From equation (i) and (ii), we have:
ar(∆ABD) = ar(∆ADC)
Hence, the median of the triangle divides it into two triangles of equal areas.