Question

1. 1. if j is the centroid of $\triangle cde$, $de = 52$, $fc = 15$, and $he = 14$, find each missing measure.

a) $dg = \underline{\quad\quad}$
b) $ge = \underline{\quad\quad}$
c) $df = \underline{\quad\quad}$
d) $ch = \underline{\quad\quad}$
e) $ce = \underline{\quad\quad}$
start by drawing the diagram on your own paper and then fill in what you know and what you can figure out.
find the measure of $dg = \square$

Explanation:

Step1: Recall centroid segment ratio

The centroid divides a median into a $2:1$ ratio, with the longer segment between the vertex and centroid.

Step2: Solve for DG and GE

$DE = DG + GE$, and $\frac{DG}{GE} = \frac{2}{1}$, so $DG = \frac{2}{3}DE$, $GE = \frac{1}{3}DE$.
$DG = \frac{2}{3} \times 52 = \frac{104}{3} \approx 34.67$
$GE = \frac{1}{3} \times 52 = \frac{52}{3} \approx 17.33$

Step3: Solve for DF

$F$ is the midpoint of $CD$ (since $CF$ is part of a median), so $DF = FC = 15$.

Step4: Solve for CH and CE

$H$ is the midpoint of $CE$ (since $EH$ is part of a median), so $CH = HE = 14$, $CE = CH + HE = 14 + 14 = 28$.

Answer:

a) $DG = \boldsymbol{\frac{104}{3}}$
b) $GE = \boldsymbol{\frac{52}{3}}$
c) $DF = \boldsymbol{15}$
d) $CH = \boldsymbol{14}$
e) $CE = \boldsymbol{28}$