1. 8.6 Proportions & Similar Triangles

2. Objectives/Assignments
Use proportionality theorems to calculate segment lengths.
To solve real-life problems, such as determining the dimensions of a piece of land.

3. Use Proportionality Theorems
In this lesson, you will study four proportionality theorems. Similar triangles are used to prove each theorem.

4. Theorems 8.4 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.
If TU ║ QS, then RT RU = TQ US

5. Theorems 8.5 Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side. RT RU If
, then TU ║ QS. = TQ US

6. Ex. 1: Finding the length of a segment
In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

7. Triangle Proportionality Thm.
Step:
DC EC
BD AE
4 EC
4(12) 8
6 = EC
Reason
Triangle Proportionality Thm.
Substitute
Multiply each side by 12.
Simplify. = = EC =
So, the length of EC is 6.

8. Ex. 2: Determining Parallels
Given the diagram, determine whether MN ║ GH. LM 56 8 = = MG 21 3 LN 48 3 = = NH 16 1 8 3 ≠ 3 1
MN is not parallel to GH.

9. Theorem 8.6
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
If r ║ s and s║ t and l and m intersect, r, s, and t, then UW VX = WY XZ

10. Theorem 8.7
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.
If CD bisects ACB, then AD CA = DB CB

11. Ex. 3: Using Proportionality Theorems
In the diagram 1  2  3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

12. 9 ● TU = 15 ● 11 Cross Product property
SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6 PQ ST
Parallel lines divide transversals proportionally. = QR TU 9 11 =
Substitute 15 TU
9 ● TU = 15 ● 11 Cross Product property
15(11) 55 TU = =
Divide each side by 9 and simplify. 9 3
So, the length of TU is 55/3 or 18 1/3.

13. Ex. 4: Using the Proportionality Theorem
In the diagram, CAD  DAB. Use the given side lengths to find the length of DC.

14. Solution:
Since AD is an angle bisector of CAB, you can apply Theorem Let x = DC. Then BD = 14 – x. AB BD =
Apply Thm. 8.7 AC DC 9
14-X
Substitute. = 15 X

15. Ex. 4 Continued . . . 9 ● x = 15 (14 – x) Cross product property
Distributive Property
Add 15x to each side
Divide each side by 24.
So, the length of DC is 8.75 units.

16. Use proportionality Theorems in Real Life
Example 5: Finding the length of a segment
Building Construction: You are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same length.

17. Use proportionality Theorems in Real Life
Because the studs AD, BE and CF are each vertical, you know they are parallel to each other. Using Theorem 8.6, you can conclude that DE AB = EF BC
Because the studs are evenly spaced, you know that DE = EF. So you can conclude that AB = BC, which means that the diagonal cuts at the tops of the strips have the same lengths.

18. Ex. 6: Finding Segment Lengths
In the diagram KL ║ MN. Find the values of the variables.

19. Solution To find the value of x, you can set up a proportion.9 x =
Write the proportion
Cross product property
Distributive property
Add 13.5x to each side.
Divide each side by 22.5
13.5 x
13.5(37.5 – x) = 9x
– 13.5x = 9x
= 22.5 x
22.5 = x
Since KL ║MN, ∆JKL ~ ∆JMN and JK KL = JM MN

20. Solution To find the value of y, you can set up a proportion.9
7.5 =
Write the proportion
Cross product property
Divide each side by 9. y
9y = 7.5(22.5)
y = 18.75