Similar figures are two-dimensional shapes that have the same shape but can differ in size. To be similar, the corresponding sides of the figures must be proportional, meaning they differ by a constant scale factor. Similar figures have the same angle measures but different side lengths. Problems involving similar figures can be solved by identifying the scale factor between corresponding sides and using proportional reasoning.

1. Similar Figures (Not exactly the same, but pretty close!)

2. Let’s do a little review work before discussing similar figures.

11. Determine the length of the unknown side. 12 9 15 4 3 ?

12. These triangles differ by a factor of 3. 12 9 15 4 3 ? 15 3= 5

13. Determine the length of the unknown side. 4 2 24 ?

14. These dodecagons differ by a factor of 6. 4 2 24 ? 2 x 6 = 12

15. Sometimes the factor between 2 figures is not obvious and some calculations are necessary. 18 12 15 12 10 8 ? =

16. To find this missing factor, divide 18 by 12. 18 12 15 12 10 8 ? =

17. 18 divided by 12 = 1.5

18. The value of the missing factor is 1.5. 18 12 15 12 10 8 1.5 =

19. When changing the size of a figure, will the angles of the figure also change? ? ? ? 70 70 40

20. Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees. If the size of the angles were increased, the sum would exceed 180 degrees. 70 70 40 70 70 40

21. We can verify this fact by placing the smaller triangle inside the larger triangle. 70 70 40 70 70 40

22. 70 70 70 70 40 The 40 degree angles are congruent.

23. 70 70 70 70 70 40 40 The 70 degree angles are congruent.

24. 70 70 70 70 70 40 4 The other 70 degree angles are congruent.

25. Find the length of the missing side. 30 40 50 6 8 ?

26. This looks messy. Let’s translate the two triangles. 30 40 50 6 8 ?

27. Now “things” are easier to see. 30 40 50 8 ? 6

28. The common factor between these triangles is 5. 30 40 50 8 ? 6

29. So the length of the missing side is…?

30. That’s right! It’s ten! 30 40 50 8 10 6

32. Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

33. You can measure the length of the tree’s shadow. 10 feet

34. Then, measure the length of your shadow. 10 feet 2 feet

35. If you know how tall you are, then you can determine how tall the tree is. 10 feet 2 feet 6 ft

36. The tree must be 30 ft tall. Boy, that’s a tall tree! 10 feet 2 feet 6 ft

37. Similar figures “work” just like equivalent fractions. 5 30 66 11

38. These numerators and denominators differ by a factor of 6. 5 30 66 11 6 6

39. Two equivalent fractions are called a proportion. 5 30 66 11

41. Practice Time!

42. 1) Determine the missing side of the triangle. 3 4 5 12 9 ? 

43. 1) Determine the missing side of the triangle. 3 4 5 12 9 15 

44. 2) Determine the missing side of the triangle. 6 4 6 36 36  ?

45. 2) Determine the missing side of the triangle. 6 4 6 36 36  24

46. 3) Determine the missing sides of the triangle. 39 24 33 ? 8  ?

47. 3) Determine the missing sides of the triangle. 39 24 33 13 8  11

48. 4) Determine the height of the lighthouse. 2.5 8 10  ?

49. 4) Determine the height of the lighthouse. 2.5 8 10  32

50. 5) Determine the height of the car. 5 3 12  ?

51. 5) Determine the height of the car. 5 3 12  7.2

52. THE END!