Question

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

a) fm =
b) af =
c) cm =
d) me =
e) mb =
f) db =

Explanation:

Step1: Recall centroid property

The centroid divides each median in a 2:1 ratio. That is, the distance from the vertex to the centroid is twice the distance from the centroid to the mid - point of the opposite side.

Step2: Find FM

Since the centroid divides the median from vertex A to side BC (with median AF) in a 2:1 ratio and AM = 44, then FM=$\frac{AM}{2}=\frac{44}{2} = 22$.

Step3: Find AF

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

Step4: Find CM

Since the centroid divides the median from vertex C to side AB (with median CE) in a 2:1 ratio and CE = 54, and CM is the part from the vertex to the centroid, CM = $\frac{2}{3}\times CE=\frac{2}{3}\times54 = 36$.

Step5: Find ME

ME=$\frac{1}{3}\times CE=\frac{1}{3}\times54 = 18$.

Step6: Find MB

Since the centroid divides the median from vertex B to side AC (with median BD) in a 2:1 ratio and DM = 23, then MB = 2DM=2×23 = 46.

Step7: Find DB

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

Answer:

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