8.6 Proportions & Similar Triangles

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. Step: DC EC BD AE 4 EC 8 12 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. 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 8.7. 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) 9x = 210 – 15x 24x= 210 x= 8.75 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 37.5 - 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 506.25 – 13.5x = 9x 506.25 = 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. 13.5 + 9 y 9y = 7.5(22.5) y = 18.75