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8.6 Proportions and Similar Triangles

8.6 Proportions and Similar Triangles. Triangle Proportionality Theorem. If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If TU ll QS, then . R. U. T. >. S. >. Q. Given AB ll ED, find EA. C.

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8.6 Proportions and Similar Triangles

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  1. 8.6 Proportions and Similar Triangles

  2. Triangle Proportionality Theorem • If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If TU ll QS, then R U T > S > Q

  3. Given AB ll ED, find EA C 8 12 D E > 4 A > B 8 EA = 4(12) 8 EA = 48 EA = 6

  4. Converse of the Triangle Proportionality Theorem

  5. Determining Parallels • Given the diagram, determine whether MN || GH. LM = LNNOTICE:WE ARE COMPARING MG NH SEGMENTS TO SEGMENTS! 56 = 48 21 16 G 21 M 56 56(16) ≠ 48(21) 16 L 48 N H Since the cross-products are not equal, the ratios are not the same, and the segment MN is not parallel to segment GH.

  6. Theorems • If three parallel lines intersect two transversals, then they divide the transversals proportionally. UW=VX WY XZ U W Y Z X V

  7. Theorems • 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=DB AC CB A D B C

  8. Given CD=14, find DB. A 9 15 15 X = 9(14) 15 X = 126 X = 8.4 B C D 14 X

  9. Solve 8 = 10 f 20 160 = 10f 16 = f 20 f 10 8

  10. Solve. 10 = p 32 24p = 320 p ≈ 13.3 10 p 14 32

  11. Solve. 18 = 32 s 18s = 1024 s = 56 8/9 18 32 14 s

  12. x = 12 25 23 23x = 300 x ≈ 13 x 12 23 25 END OF 8.6

  13. 8.7 Dilations

  14. Warmup Find the value of the variable. 8 6(t+3) = 5(8) 6t + 18 = 40 6t = 22 t ≈ 3.67 6 t+3 5

  15. Vocabulary • Dilation: type of transformation which contains reduction and enlargement. • Reduction: 0< scale factor <1 • Enlargement: scale factor >1

  16. Identify the dilation and find its scale factor. Examples on Identifying Dilations P 5 preimage P’ 2 image Since k<1, it’s a reduction And scale factor is 2:5 C

  17. Dilation in a coordinate plane. • Draw a dilation of rectangle ABCD with A(2,2), B(6,2), C(6,4), D(2,4). Name the dilation EFGH. Use the origin as the center and use a scale factor of ½. • How does the perimeter of the preimage compare to the perimeter of the image? • The perimeter of EFGH is ½ the perimeter of ABCD.

  18. Dilation • http://www.geogebra.org/webstart/geogebra.html

  19. Dilation in a coordinate plane • Draw a dilation of rectangle ABCD with A(2,2), B(6,2), C(6,4), D(2,4). Use the origin as the center and use a scale factor of 2.

  20. dilation • http://www.geogebra.org/webstart/geogebra.html

  21. Ch 8 Review Day 4 Part 2

  22. Review 1 • Find the geometric mean of 4 and 6 • Find the geometric mean of 32 and 96

  23. Review 2 • What is the perimeter of ΔABC? B 20 30 8 10 A C 7 D

  24. Review 3 • Find the value of JN. J N 18 L 16 M 3 K

  25. Review 4 • Simplify the followings.

  26. Review 5 • Identify the dilation and find its scale factor. P 14 P’ 6 C

  27. Review 6 • The ratio of AB : BC is 3 : 8. Solve for x. D C X A B 6

  28. Review 7 • The perimeter of a rectangle is 54. The ratio of the lengths of the sides is 2:7. What are the lengths of the sides?

  29. Review 8 • Draw the given triangles. Then decide whether they are similar or not similar. • ΔPQR with 16, 8, 18 and ΔXYZ with 9,8,4 Q 8 18 Y ΔPQR is not similar to ΔXYZ R 16 P 4 8 Z X 9

  30. Review 9 Find the value of the variable. 33 q 21 17.5

  31. Review 10 • If ΔABC ~ ΔXYZ, AB = 6, and XY = 4, what is the scale factor of the triangle? C Scale Factor 6 : 4 3 : 2 Z A B 6 Y X 4

  32. Review 11

  33. Review 12 Define scale factor, geometric mean and ratio. Provide an example for each term.

  34. Review 13 T S 20° V 65° U X W What is the measure of <X? What is the measure of <W? What is the measure of <T? What is the measure of <U? Which side is congruent to TU?

  35. Review 14 • Solve the proportion 9x = 13(37 – x) 9x = 481 – 13x +13x +13x 22x = 481 x = 21.9 9x = 15 ( 14 – x ) 9x = 210 – 15x 24x = 210 x = 8.75

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