Question

the medians of $\triangle abc$ are $\overline{ae}$, $\overline{bf}$, and $\overline{cd}$. they meet at a single point $g$. (in other words, $g$ is the centroid of $\triangle abc$.) suppose $bf = 27$, $ag = 14$, and $cg = 20$. find the following lengths. note that the figure is not drawn to scale.

Explanation:

Step1: Recall centroid ratio rule

The centroid divides each median into a ratio of $2:1$, where the segment from vertex to centroid is twice the segment from centroid to the midpoint of the side.

Step2: Calculate $GD$

Since $CG:GD = 2:1$, $GD=\frac{1}{2}CG$.
$\displaystyle GD=\frac{1}{2} \times 20 = 10$

Step3: Calculate $GF$

Since $BG:GF = 2:1$, $GF=\frac{1}{3}BF$.
$\displaystyle GF=\frac{1}{3} \times 27 = 9$

Step4: Calculate $AE$

Since $AG:GE = 2:1$, $AG=\frac{2}{3}AE$, so $AE=\frac{3}{2}AG$.
$\displaystyle AE=\frac{3}{2} \times 14 = 21$

Answer:

$GD = 10$
$GF = 9$
$AE = 21$