8 6 Proportions Similar Triangles ObjectivesAssignments Use proportionality

8. 6 Proportions & Similar Triangles

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

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

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 TQ = RU US

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. If RT TQ = RU US , then TU ║ QS.

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?

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 ØSo, the length of EC is 6.

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

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 WY = VX XZ

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. AD CA = If CD bisects ACB, then DB CB

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?

SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8. 6 PQ QR 9 15 = = ST TU 11 TU Parallel lines divide transversals proportionally. Substitute 9 ● TU = 15 ● 11 Cross Product property TU 15(11) 55 = = 9 3 Divide each side by 9 and simplify. ØSo, the length of TU is 55/3 or 18 1/3.

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

Solution: Since AD is an angle bisector of CAB, you can apply Theorem 8. 7. Let x = DC. Then BD = 14 – x. AB AC 9 15 = = BD DC 14 -X X Apply Thm. 8. 7 Substitute.

Ex. 4 Continued. . . 9 ● x = 15 (14 – x) 9 x = 210 – 15 x 24 x= 210 x= 8. 75 Cross product property Distributive Property Add 15 x to each side Divide each side by 24. ØSo, the length of DC is 8. 75 units.

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.

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 EF n = AB 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.

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

Solution To find the value of x, you can set up a proportion. 9 13. 5 = 37. 5 - x x Write the proportion 13. 5(37. 5 – x) = 9 x Cross product property 506. 25 – 13. 5 x = 9 x Distributive property 506. 25 = 22. 5 x Add 13. 5 x to each side. 22. 5 = x Divide each side by 22. 5 ØSince KL ║MN, ∆JKL ~ ∆JMN and JK KL JM = MN

Solution To find the value of y, you can set up a proportion. 9 13. 5 + 9 = 7. 5 y 9 y = 7. 5(22. 5) y = 18. 75 Write the proportion Cross product property Divide each side by 9.