1. Ch. 11 - Similarity Class Notes

3. Monday 1/5 – Similarity Ch.11 Textbook p. 564 – one inside table Use patty paper to complete step 1. Use the ruler on your protractor to complete step 2. In five minutes, you will be quizzed for credit.

4. Quiz Question 1 Whiteboard, marker & rag in table. Hold up answer, facing forward, 1 min. Q: Are these shapes congruent? A.Yes, they have the same angles. B.Yes, they can be superimposed. C.No, they have the same angles but not the same sides. D.Cannot be determined. (CBD)

5. Polygons are similar if, Their corresponding angles are congruent (same) and, Their corresponding sides are proportional. All the corresponding pairs of sides have the same ratio.

6. Calculate the ratio of each pair of corresponding sides in these shapes. Record on patty paper. Example: RS/CD = 4.1/2.3 = 1.8 In five minutes, you will be quizzed for credit.

7. Quiz Question 2 (1 min) Whiteboard, marker & rag in table. Hold up answer, facing forward. Q: Are these shapes similar? A.Yes, they have the same angles and side ratios. B.Yes, they can be superimposed. C.No, they have the same angles but not the same sides. D.Cannot be determined. (CBD)

8. Notes Date… Title…Similarity & Dilations – Ch. 11

9. Polygons are similar if, Their corresponding angles are congruent (same) and, Their corresponding sides are proportional. All the corresponding pairs of sides have the same ratio.

10. Dilation – Process of growing Example: Eye pupils dilate in dark. Scale factor – How much the pre- image has grown to create the image. Example: 3:1 = scale factor 3 Calculate scale factor by dividing length of corresponding sides image/pre-image.

11. Pre-image 3 cm -> Image 9 cm = 9/3 = scale factor 3 Pre-image 8 cm -> Image 2 cm = 2/8 = scale factor ¼ = 0.25 Scale factor > 1 is growing Scale factor < 1 is shrinking

12. Quiz Question 3 (1 minute) Q: Are these shapes, A.Congruent B.Similar C.Congruent & Similar D.Neither

13. Are corresponding angles congruent? YES. (90 o ) Are corresponding sides proportional? NO. Conclusion – These polygons are not similar or congruent.

14. Quiz Question 4 Answer # 1, p. 568 on whiteboard.

15. Quiz Question 5 Answer # 8, p. 568 on whiteboard.

16. Quiz Question 6 Answer # 5, p. 568 on whiteboard. Also write the scale factor.

17. Cleanup Grade Start at 100% each quarter. Lose credit if don’t: – Push in chair to touch table. – Pick up trash near desk, even if not yours. – Remove trash inside desk. – Erase whiteboard. – Return textbook, whiteboard, marker, rag into desk. Also start with 100% for following rules…

18. HW p. 568 # 1-10 # 1 & 2 Find two shapes that have same angles and corresponding parts, but have different sizes. # 3 – 5 Draw original shape on graph paper. Then repeat but make all sides bigger by same amount. (2x?)

20. Have HW open in first minute of class to earn credit. Open textbook to p. 568. Tuesday 1/6 – Similarity Ch.11

21. 1.You will be quizzed for credit on this in 10 min. 2.Place a point on the left of a new page in your notebook. 3.Draw three rays ‘radiating’ out from this point, to the upper, middle & lower right. 4.Randomly place an additional point on each ray. Connect them to form a triangle. 5.Place a second point on each ray, at twice the distance as the first. Connect to make a second triangle. 6.What do we notice about the two triangles?

22. Quiz Question 1 (1 minute) Q: What is the relationship between these triangles? A.Congruent B.Similar C.Congruent & Similar D.Neither

23. Quiz Question 2 (1 minute) Q: What is the scale factor? A.0.5 B.1.0 C.2.0 D.None of the above

24. Quiz Question 3 (1 minute) Q: Corresponding sides in these two triangles are: A.congruent B.parallel C.perpendicular D.None of the above

25. Dilations Centered on Same Point

26. Dilations work for all polygons… Center of dilation can be inside shapes. Scale factor can be negative… What is the scale factor in this dilation?

27. Similar Shapes Centered on Same Point of Dilation... Corresponding angles are congruent (same.) Parallel corresponding sides. Ratios of every pair of corresponding sides is the same. Ratios of center-to-vertex distances are the same for corresponding vertices in each shape. Ratios between sides in the same shape, are the same in both shapes.

29. You will be quizzed on the following graph in 10 minutes. On your personal whiteboard graph: (3,2) (4, -2) (-2, -3) Dilate triangle with scale factor of 2, Draw rays centered on origin. Write a coordinate rule for this. Wednesday 1/7 – Similarity Ch.11

30. Dilation Centered on Origin

31. Coordinate rules for dilations 2x dilation:(x, y) → (2x, 2y) 3x dilation:(x, y) → (3x, 3y) Stretch - shapes not similar! (x, y) → (2x, y) Another stretch: (x, y) → (x, 2y)

32. Are corresponding angles congruent? YES. (90 o ) Are corresponding sides proportional? NO. Conclusion – These polygons are not similar or congruent. If all corresponding angles remain congruent, must sides be proportional?

33. If all sides are proportional, must corresponding angles remain congruent? Scale factor consistently 18/12=1.5 But…are corresponding angles congruent? NO!

34. How to Know if Shapes are Similar Corresponding angles congruent? YES. Corresponding sides proportional? YES. Conclusion – These polygons are similar. CORN ~ PEAS

35. AA Triangle Similarity Shortcut However, if two angles in a triangle are the same…

36. AA Triangle Similarity Shortcut However, if two corresponding angles in two triangles are the same…then both triangles must be similar! Do not need to measure third pair of angles (Triangle Sum Conjecture says they must be the same.) Do not need to measure sides or calculate scale factor… sides must all grow/shrink by same ratio to keep angles congruent.

37. HW p. 574 # 1-10

39. Have HW open in first minute of class to earn credit. Open textbook to p. 574 Thursday 1/8 – Similarity Ch.11

40. Measuring Height w Similar Triangles

42. Measuring Height or Distance with Similar Triangles

43. Measuring Height w Similar Triangles

45. Highest edge of U.S. flag in commons Gold ball on flagpole in front of school Water tower (extra credit) Measure two ways each: mirror & ruler Each partner records data & calculations You will be quizzed on your answers & methods next week. Friday 1/9 – Measure Heights of Tall Objects with Similar Triangles

46. Mirror Ruler Friday 1/9 – Measure Heights of Tall Objects with Similar Triangles

47. Place mirror, meter stick, little ruler on desk to be checked in.

49. Continue to measure: – Highest edge of U.S. flag in commons – Gold ball on flagpole in front of school – Water tower (extra credit) Measure two ways each: mirror & ruler Each partner records data & calculations You will be quizzed on your answers & methods next week. Monday 1/12 – Measure Heights of Tall Objects with Similar Triangles

50. Place mirror, meter stick, little ruler on desk to be checked in.

51. HW HW due tomorrow 11.3 #1-10. Quiz tomorrow, sections 1-3. May use your notes. May NOT use cell phone calculator for scale factors.

53. Have HW open in first minute of class to earn credit. Have textbook open to p. 582. Tuesday 1/13 – Similarity Ch.11

54. 14.0 cm on Google Maps 4.0 cm on Google Maps

56. Wednesday 1/14 QUIZ May use notes & calculator. No cell phones.

58. Thursday 1/15 Review for Test 1.In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5). Tape in graph paper if needed. 2.Draw rays from origin through each vertex. 3.Dilate above shape with scale factors of 0.5 and 2. 4.Write as complete sentences: “In a dilation, _________ angles ___________.” “We can control the scale factor by _____________________________________.”

59. Dilation using ‘ray method’

60. Thursday 1/15 Review for Test 1.In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5). Tape in graph paper if needed. 2.Draw rays from origin through each vertex. 3.Dilate above shape with scale factors of 0.5 and 2. 4.Write as complete sentences: “In a dilation, corresponding angles remain congruent.” “We can control the scale factor by increasing the distance between center-of-dilation & vertices by that factor.”

61. Dilations Centered on Same Point Dilations work for all polygons… Center of dilation can be inside shapes. Center of dilation can be outside shapes.

62. Dilations Centered on Same Point Scale factor can be negative… What is the scale factor in this dilation?

63. 1.Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (5, 0) (6, -3) (-1, 1) (0, -7). 2.Write as complete sentence: “These polygons are: congruent/similar/neither (may be more than one) because __________ _____________________. 3. “If similar, the scale factor is ____________.”

64. 1.Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (1.5, 2) (2, -1) (-1.5, 3) (-1, -5). 2.Write as complete sentence: “These polygons are: congruent/similar/neither (chose one) because _____________________. 3. “If similar, the scale factor is ____________.”

65. 1.Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (1.5,1) (2,-0.5) (-1.5,1.5) (-1,-2.5). 2.Write as complete sentence: “These polygons are: congruent/similar/neither (chose one) because _____________________. 3. “If similar, the scale factor is ____________.”

67. Friday 1/16 30 min No notes No cell phone Yes calculator 10/10 = A+ 9/10 = A- 7/10 = B 5/10 = C 3/10 = D TEST – Similarity & Dilations

69. A level topics 11.4 & 11.5 Verify geometric properties of dilations

70. WarmUp – 5 min Draw a square 2 cm on each side. Draw a square 6 cm on each side. Draw dashed lines to show how many little squares fit in the big square. Calculate the area of each and write in each square with units. Fill in this Conjecture: “If corresponding sides of two similar polygons compare in a ratio of m/n, then their areas compare in the ratio of ______________.”

71. Proportional Volume Conjecture Draw a cube 2 cm on each side. Draw a bigger cube 6 cm on each side. Draw dashed lines to show one little cube in the corner of the big cube. Calculate the volume of each cube. Fill in this Conjecture: “If corresponding edges (or radii or heights) of two similar solids compare in a ratio of m/n, then their volumes compare in the ratio of ______________.”

72. For shapes with a scale factor of 2, how do these ‘scale up’? Perimeter? Area? For solids with scale factor of 2, how do these ‘scale up’? Surface Area? Volume?

73. For shapes with a scale factor of 3, how do these ‘scale up’? Perimeter? Area? For solids with scale factor of 3, how do these ‘scale up’? Surface Area? Volume?

74. Worksheet to Complete Lesson 11.5 – Proportions with Area and Volume

75. 1) Which two HW Q’s would you most like to see? 2) Working with your partner, read pp. 599-602 ‘Why Do Elephants Have Big Ears?’ Discuss, then write your answers for Q’s 1-15. Show Mr. Sidman.

76. 1.You will need your compass. 2.Place a point in the center of the notebook page. 3.Draw three rays ‘radiating’ out from this point. 4.Randomly place one additional point on each ray. Place them at the edges of the paper. Connect them to form a BIG triangle. 5.Place a second point at half the distance along each ray as the first. This makes a similar triangle with a scale factor of 0.5 compared to the first. 6.Bisect one corresponding side of each triangle. 7.Construct one corresponding median for each triangle. Find the ratio of big-to-little medians. 8.Bisect a corresponding angle in each. Find the ratio of the big-to-little angle bisector segments.

77. Conclusion Corresponding (matching) dimensions of similar triangles all have the same scale factor (ratio): little side = little median = little angle bisector big side big median angle bisector little altitude = little midsegment = little perpendicular bisector big altitude big midsegment big perpendicular bisector

78. 1.You will need your compass. 2.Place a point in the center of the notebook page. 3.Draw three rays ‘radiating’ out from this point. 4.Randomly place one additional point on each ray. Place them at the edges of the paper. Connect them to form a BIG triangle. 5.Place a second point at half the distance along each ray as the first. This makes a similar triangle with a scale factor of 0.5 compared to the first. 6.Find the ratio of the small-to-big perimeters. 7.For one corresponding angle in each triangle, drop a perpendicular bisector to the opposite side. 8.Use this altitude (height) to find the ratio of areas.

79. WarmUp – 5 min 1.Write a step-by-step proof that ∆LMN ̴ ∆EMO. 2.What is length y?

80. 1.Draw two rays forming an acute angle. 2.On one ray, use a ruler to mark off lengths 8 cm and then an additional 10 cm from vertex. Label these segments. 3.On the other ray, mark off segments 12 cm and an additional 15cm. 4.Connect points to make a little triangle inside the big. 5.Are these triangles similar? 6.Calculate the ratio 8cm to 10cm. 12cm to 15cm. What do you notice? 7.What else do you notice about these triangles?