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Unit 5: Relationships in triangles
Homework 4: Centroids, Orthocenters, Review

Unit 5: Relationships in triangles Homework 4: Centroids, Orthocenters, Review-example-1

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Applying the centroid theorem, the missing measures are:

FM = 22

A.F = 66

CM = 36

ME = 18

MB = 46

DB = 69

The centroid theorem states that the distance from the centroid of the triangle to the opposite vertex is equal to one-third of the length of the segment that connects the midpoint of a side of a triangle to the opposite vertex.

AM = 44, CE = 54, and DM = 23, where M is the centroid of the triangle, thus:

FM = 1/3(FA) = 1/3(AM + FM)

Substitute

FM = 1/3(44 + FM)

3FM = 44 + FM

3FM - FM = 44

2FM = 44

FM = 44/2

FM = 22

A.F = AM + FM = 44 + 22

A.F = 66

CM = 2/3(CE)

CM = 2/3(54)

CM = 36

ME = CE - CM = 54 - 36

ME = 18

MB = 2/3(DB) = 2/3(DM + MB)

MB = 2/3(23 + MB)

3MB = 2(23 + MB)

3MB = 46 + 2MB

3MB - 2MB = 46

MB = 46

DB = DM + MB = 23 + 46

DB = 69

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