In the following figure, G is centroid of the triangle ABC.

Prove that :

(i) Area (△ AGB) = $\dfrac{2}{3}$ x Area (△ ADB)

(ii) Area (△ AGB) = $\dfrac{1}{3}$ x Area (△ ABC)

(i) To Prove: Area (△ AGB) = $\dfrac{2}{3}$ x Area (△ ADB)

Proof: The centroid G of a triangle is the point of intersection of its three medians.

It divides each median into two segments, where the segment closer to the vertex is twice the length of the other.

If AD is a median, it divides △ ABC into two equal areas:

Area (△ ADB) = $\dfrac{1}{2}$ x Area (△ ABC)

Since G is the centroid, it divides △ ABC into three smaller triangles of equal area.

Area (△ AGB) = $\dfrac{1}{3}$ x Area (△ ABC)

$\dfrac{\text{(△AGB)}}{\text{(△ADB)}} = \dfrac{\dfrac{1}{3} \times \text{(△ABC)}}{\dfrac{1}{2} \times \text{(△ABC)}}$

= $\dfrac{1\times2}{3\times1} = \dfrac{2}{3}$

Hence, Area (△ AGB) = $\dfrac{2}{3}$ x Area (△ ADB).

(ii) As proved above,

Area (△ AGB) = $\dfrac{1}{3}$ x Area (△ ABC)

Hence, Area (△ AGB) = $\dfrac{1}{3}$ x Area (△ ABC).