Question

10.
if j is the centroid of triangle def, dh=51, gf=60, and ei=57, find each missing measure.
jf =
11.
if j is the centroid of triangle def, dh=51, gf=60, and ei=57, find each missing measure.
ei =

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. For median $EJ$ (where $J$ is centroid, $EJ$ is part of median $EI$), $EJ:JI = 2:1$. First, find total length of median $EI$:
$EI = EJ + JI$, and $EJ = 2JI$. Given $EI=57$, substitute:
$2JI + JI = 57$
$3JI = 57$
$JI = 19$, so $EJ = 2\times19=38$.

Step2: Analyze median for $JF$

$JF$ is part of median $DJF$ (since $J$ is centroid, $DJ$ is from vertex $D$ to centroid, $JF$ is the segment to side $EF$). First, find total length of median $DH$: $DH$ is a median, so $DH = DJ + JH$, with $DJ:JH=2:1$. Given $DH=51$:
$2JH + JH = 51$
$3JH=51$
$JH=17$, $DJ=34$. But for $JF$, we use the median from $F$ to side $DE$: $GF$ is half the median? No, $GF=60$ is half of side $DE$? No, $GF$ is part of median $FGJ$: centroid $J$ divides median $FJ$ (from $F$ to centroid) and $JG$ (centroid to side $DE$) into $2:1$. Wait, no: the median from $F$ to $DE$ has total length $FJ + JG$, with $FJ:JG=2:1$. But $GF$ is the length from $G$ (midpoint of $DE$) to $F$, so total median length is $GF=60$? No, $G$ is midpoint of $DE$, so median is $FG$, with centroid $J$ on $FG$, so $FJ:JG=2:1$. Thus total median length $FG=FJ + JG$, $FJ=2JG$, so $FG=3JG$. But wait, no: the problem says $GF=60$, which is $FG=60$. So:
$FJ + JG = 60$, $FJ=2JG$
Substitute: $2JG + JG=60$
$3JG=60$
$JG=20$, so $FJ=2\times20=40$.

Wait, correction: For question 10, $JF$ is the segment from centroid $J$ to side $DE$? No, no: centroid divides each median into $2:1$, with the longer segment between vertex and centroid. So for median from $F$ to midpoint $G$ of $DE$, the median is $FG$, length $GF=60$ (so $FG=60$). Centroid $J$ is on $FG$, so $FJ$ is the segment from $F$ (vertex) to $J$ (centroid), $JG$ is from $J$ to $G$ (midpoint). So $FJ:JG=2:1$, so $FJ = \frac{2}{3} \times FG$, $JG=\frac{1}{3}\times FG$.
$FJ = \frac{2}{3} \times 60 = 40$.

For question 11, $EJ$ is given as part of median $EI=57$: centroid $J$ divides median $EI$ (from $E$ to midpoint $I$ of $DF$) into $EJ:JI=2:1$, so $EJ = \frac{2}{3} \times EI = \frac{2}{3} \times 57 = 38$.

Answer:

For question 10: $JF = 40$
For question 11: $EJ = 38$