Objective: Students will use proportional parts of
triangles and divide a segment into parts.
S. Calahan
2008
Triangle Proportionality Theorem
 If a line is parallel to one side of a
triangle and intersects the other two
sides in two distinct points, then it
separates these sides into segments of
proportional lengths.
C
B
If BD || AE, BA ≈ DE
CB CD
A
D
E
Find the length of a side
 In
ABC, BD || AE, AB = 9, BC = 21,
and DE = 6. Find DC.
C
B
A
D
E
Find the length of a side
 From the Triangle Proportionality
Theorem BA ≈ DE
CB CD
 Substitute the know measures.
C
B
A
D
E
Find the length of a side
9 = 6
Let x = DC
21 x
9(x) =21(6)
9x = 126
x = 14
Therefore, DC = 14
B
A
C
D
E
Triangle Midsegment Theorem
 A midsegment of a triangle is parallel to one
side of the triangle, and its length is ½ the
length of that side.
If B and D are midpoints
of AC and EC
respectively, BD || AE
and BD = ½ AE.
C
B
A
D
E
Midsegment of a Triangle
 If AE = 12 then BD = ½(12) = 6.
C
B
A
D
E
Divide Segments Proportionally
 If three or more parallel lines intersect
two transversals, then they cut off the
transversal proportionally.
F
E
D
AB = DE, AC = BC, and AC = DF
BC EF DF EF
BC EF
A
B
C