Medians and Centroid of a Triangle

A median connects a vertex to the midpoint of the opposite side, so $BD=CD$.

The centroid is where all three medians of a triangle meet. It always lies inside the triangle, no matter the triangle's shape.

The centroid divides the median in a $2:1$ ratio. Since $AE$ is the longer segment, $ED$ is half of $AE$, so $ED=6\text{ cm}÷2=3\text{ cm}$.

Since $CG$ is a median, it divides triangle $ABC$ into two triangles of equal area. Triangle $AGC$ is one of those halves, so the total area is $2\times 10{\text{ cm}}^2=20{\text{ cm}}^2$.

The centroid divides the median in a $2:1$ ratio. The longer segment $BE$ is two-thirds of the total length $BF$: $BE=\frac{2}{3}\times 12=8\text{ cm}$.

The centroid divides the median in a $2:1$ ratio, so $AE$ is two-thirds of $AD$. Therefore, $AD=\frac{3}{2}\times AE=\frac{3}{2}\times 10=15\text{ cm}$.