Proportional Lengths
Lesson 7.6
Geometry Honors
Page 269
Lesson Focus
The focus of this lesson is on how the sides of a triangle can
be divided into proportional segments.
Two theorems that establish the conditions are developed.
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other
two sides, then it divides those sides proportionally.
Triangle Proportionality Theorem
Example 1: Find the value of x.
The lines
QR and ST
are parallel.
PS PT
Therefore, by the Triangle Proportionality Theorem, QS  RT
Substitute the values and solve for x: 6  9
2
Cross multiply: 6x = 18
6 x 18

Divide both sides by 6:
6
6
The value of x is 3.
x
Corollary to the
Triangle Proportionality Theorem
If three parallel lines intersect two transversals, then they
divide the transversals proportionally.
Triangle Angle-Bisector Theorem
If a ray bisects an angle of a triangle, then it divides the
opposite side into segments proportional to the other two
sides.
Proportional Lengths
Example 2: Given:
A
B
D
E
CD
(a) DA  _____
C
(b) If CD = 3, DA = 6, and DE = 3.5, then AB = _____
(c) If CB = 12, EB = 8, and CD = 6, then DA = _____
Proportional Lengths
Example 3: (a) If a = 2, b = 3, and c = 5, then d = _____
(b) If a = 4, b = 8, c = 5, then c + d = _____
a
b
c
d
Proportional Lengths
Example 4: CD bisects ACB. Find AB.
A
10
D
12
C
B
24
Written Exercises
Problem Set 7.6A, p.272: # 1 – 11
Self-Test 2, p.274: # 1 - 11