Question

If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in
area. If G is the mid-point of median AD, prove that ar (Δ BGC) = 2 ar (Δ AGC).

Answer 1

Draw AM ⊥ BC 
Since, AD is the median of ΔABC

∴  BD = DC

⇒ BD = AM = DC × AM

⇒ ` 1/2 (BD xx AM ) = 1/2 ( DC xx AM)`

⇒ ar (Δ ABC)  = ar (Δ ACD)   ........ (1)

In ΔBGC , GDis the median

∴ ar (BGD)  = area (OGD)      ......... (2)

In ΔACD , CG is the median

∴ area (AGC)  = area (Δ CGD)  ......... (3)

 From (1) and  (2) , we have 

Area (ΔBGD) = ar (Δ AGC)

But, ar (ΔBGC) = 2ar (BGD)

∴  ar (BGC ) = 2ar (Δ AGC)