Question

1. if m is the centroid of $\triangle abc$, $am = 44$, $ce = 54$, and $dm = 23$, find each measure.

a) $fm = \underline{\quad\quad}$
b) $af = \underline{\quad\quad}$
c) $cm = \underline{\quad\quad}$
d) $me = \underline{\quad\quad}$
e) $mb = \underline{\quad\quad}$
f) $db = \underline{\quad\quad}$

Explanation:

Step1: Find FM using centroid ratio

The centroid divides a median into a 2:1 ratio, so $AM:FM = 2:1$. Rearranged, $FM = \frac{AM}{2}$
$FM = \frac{44}{2} = 22$

Step2: Calculate AF (total median length)

$AF = AM + FM$
$AF = 44 + 22 = 66$

Step3: Find CM from median CE

Centroid divides CE into $CM:ME = 2:1$, so $CM = \frac{2}{3} \times CE$
$CM = \frac{2}{3} \times 54 = 36$

Step4: Calculate ME from median CE

$ME = \frac{1}{3} \times CE$
$ME = \frac{1}{3} \times 54 = 18$

Step5: Find MB using centroid ratio

Centroid divides DB into $MB:DM = 2:1$, so $MB = 2 \times DM$
$MB = 2 \times 23 = 46$

Step6: Calculate DB (total median length)

$DB = DM + MB$
$DB = 23 + 46 = 69$

Answer:

a) $FM = 22$
b) $AF = 66$
c) $CM = 36$
d) $ME = 18$
e) $MB = 46$
f) $DB = 69$