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If A D is a median of a triangle A B ...

If `A D` is a median of a triangle `A B C ,\ ` then prove that triangles `A D B\ a n d\ A D C` are equal in area. If `G` is the mid-point of median `A D ,` prove that `a r\ ( B G C)=2a r\ (\ A G C)`

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Proof : Since, ` AD` is the median of `ΔABC`
Therefore, `BD = DC`
`BD xx AM = DC xx AM`
` 1 /2 (BD xx AM) = 1/ 2 (DC xx AM) `
`Area (Δ ABD) = Area (Δ ACD)` ...(i)
Now, In `Δ BGC`,
` GD` is the median .
Therefore, Area `(ΔBGD) = Area (ΔCGD)` ...(ii)
Also, In `Δ ACD`, `CG` is the median .
Therefore, Area`(ΔAGC) = Area (ΔCGD)` ...(iii)
From (i), (ii) and (iii) we have Area `(ΔBGD) = Area (ΔAGC)`
But, Area `(ΔBGC) = 2 Area (ΔBGD)`
Therefore, `Area (ΔBGC) = 2 Area (ΔAGC)`
. Hence, proved
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