Page 118 - Math Course 2 (Book 2)

Perpendicular and Angle Bisectors

THEOREM 10.7 Centroid Theorem

The centroid of a triangle is located two thirds of the distance from a
vertex to the midpoint of the side opposite the vertex on a median. B
If L is the centroid of △ABC,
Example D L E
2
2
2
AL = AE, BL = BF, and CL = CD. Centroid
3 3 3

A F C

Find b.
Let’s Begin

XU = 3b + 2 + 8.7 Segment Addition
Postulate
2
Segment Measures 3b + 2 = XU Centroid Theorem
3
Example 3b + 2 = (3b + 2 + 8.7) Substitution
2
3

9b + 6 = 2(3b + 2 + 8.7) Multiply each side by 3
ALGEBRA
Points U,V, and W are the midpoints of YZ, ZX, XY,
respectively. Find a,b, and c. 9b + 6 = 6b + 21.4 Distribute and simplify.
Subtract 6b from each
Y 3b + 6 = 21.4 side.

Subtract 6 from each
W 7.4 U 3b = 15.4 side
5c 8.7
3b + 2 15.2
2a b = 5.13 Divide each side by 3

X V Z
Find c.
Find a. WZ = 5c + 15.2 Segment Addition
Postulate
2
VY = 2a + 7.4 Segment Addition Postulate 15.2 = WZ Centroid Theorem
3
2
7.4 = VY Centroid Theorem 2
3 15.2 = (5c + 15.2) Substitution
3
2
7.4 = (2a + 7.4) Substitution
3 45.6 = 2(5c + 15.2) Multiply each side by 3

22.2 = 2(2a + 7.4) Multiply each side by 3 45.6 = 10c + 30.4 Distribute

22.2 = 4a + 14.8 Distribute 15.2 = 10c Subtract 30.4 from
each side
Subtract 14.8 from each
7.4 = 4a
side 1.52 = c Divide each side by 10

1.85 = a Divide each side by 4 Answer a = 1.85, b = 5.13, c = 1.52

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