6.6 proportions & similar triangles

6.6 Proportions & Similar Triangles

Objectives/Assignments ,[object Object],[object Object]

Use Proportionality Theorems ,[object Object]

Theorems 6.4 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 6.5 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 TQ RU US = If , then TU ║ QS.

Ex. 1: Finding the length of a segment ,[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],= = = EC ,[object Object]

Ex. 2: Determining Parallels ,[object Object],LM MG 56 21 = 8 3 = LN NH 48 16 = 3 1 = 8 3 3 1 ≠ MN is not parallel to GH.

Theorem 6.6 ,[object Object],[object Object],UW WY VX XZ =

Theorem 6.7 ,[object Object],[object Object],AD DB CA CB =

Ex. 3: Using Proportionality Theorems ,[object Object]

SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6 PQ QR ST TU = 9 15 11 TU = 9 ● TU = 15 ● 11 Cross Product property 15(11) 9 55 3 = TU = Parallel lines divide transversals proportionally. Substitute Divide each side by 9 and simplify. ,[object Object]

Ex. 4: Using the Proportionality Theorem ,[object Object]

Solution: ,[object Object],AB AC BD DC = 9 15 14-X X = Apply Thm. 8.7 Substitute.

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. ,[object Object]

Use proportionality Theorems in Real Life ,[object Object],[object Object]

Use proportionality Theorems in Real Life ,[object Object],DE EF AB BC = ,[object Object]

Ex. 6: Finding Segment Lengths ,[object Object]

Solution ,[object Object],9 13.5 37.5 - x x = 13.5(37.5 – x) = 9x 506.25 – 13.5x = 9x 506.25 = 22.5 x 22.5 = x Write the proportion Cross product property Distributive property Add 13.5x to each side. Divide each side by 22.5 ,[object Object],JK JM KL MN =

Solution ,[object Object],9 13.5 + 9 7.5 y = 9y = 7.5(22.5) y = 18.75 Write the proportion Cross product property Divide each side by 9.

6.6 proportions & similar triangles

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6.6 proportions & similar triangles