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

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

3. Congruent Figures In order to be congruent, two figures must be the same size and same shape.

4. Similar Figures Similar figures must be the same shape, but their sizes may be different.

5. Similar Figures This is the symbol that means “similar.” These figures are the same shape but different sizes.

6. SIZES Although the size of the two shapes can be different, the sizes of the two shapes must differ by a factor. 33 2 1 6 6 2 4

7. SIZES In this case, the factor is x 2. 33 2 1 6 6 2 4

8. SIZES Or you can think of the factor as 2. 33 2 1 6 6 2 4

9. Enlargements When you have a photograph enlarged, you make a similar photograph. X 3

10. Reductions A photograph can also be shrunk to produce a slide. 4

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

12. Similar Figures So, similar figures are two figures that are the same shape and whose sides are proportional.

13. Proportions We can use our knowledge of proportions to solve problems with similar figures. If two figures are similar, each part of the smaller shape will be proportional to the corresponding part of the larger.

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

15. In this case we know the measurement of both the bases and the heights; we’re looking for the measurement of the smaller triangle’s hypotenuse. We’ll want our proportion to use either the base or height (a known ratio) and the hypotenuse (our unknown) 12 9 15 4 3 ?

16. We can set up the proportion using either of the known quantities for the first ratio:: 12 9 15 4 3 ?

17. We can set up the proportion using either of the known quantities for the first ratio: 12 9 15 4 3 ?

18. We can set up the proportion using the known quantities for the first ratio: Then, we solve for the unknown side in the usual way: (15 x 3) ÷ 9The hypotenuse of the 45÷9smaller triangle measures 5five.

19. Sometimes it won’t be obvious what to name the parts of the shape. In that case, you can label them. Make sure the right labels go on the right parts! 18 12 ? 108

20. Sometimes it won’t be obvious what to name the parts of the shape. In that case, you can label them with letters. Make sure the right labels go on the right parts! 18 c 12 a ? b 12 c 10 b 8a8a

21. We know the “a”s and the “c”s, we’re missing one of the “b”s. Our ratio could use either a or c, and it must use b.

22. We know the “a”s and the “c”s, we’re missing one of the “b”s. Our ratio could use either a or c, and it must use b. We have several options: Let’s use

23. Starting with our known ratio: 18 c 12 a ? b 12 c 10 b 8a8a

24. Starting with our known ratio: (12 x 10) ÷8 120 ÷ 8 15Our missing side measures 15 18 c 12 a ? b 12 c 10 b 8a8a

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

26. 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 40 70 40

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

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

29. Now things are easier to see. 30 40 50 ? 6

30. The only side that has measurements listed on both the small and large triangle is the height, so we need to use that as our reference. We need to find the missing hypotenuse, so that will be our other side. 40 50 ? 30 6

31. (6x50) ÷ 30 300 ÷ 30 10 40 50 ? 30 6 The measurement of the missing hypotenuse is 10

32. Similarity is used to answer real life questions. Suppose that you wanted to find the height of this tree.

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

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

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

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

37. 10 feet 2 feet 6 ft

38. The tree must be 30 ft tall. 10 feet 2 feet 6 ft

39. Determine the missing side of the triangle. 3 4 5 10 7.5 ? 

40. Determine the missing side of the triangle. 3 4 5 10 7.5 12.5 

41. Determine the missing side of the triangle. 6 4 6 21  ?

42. Determine the missing side of the triangle. 6 4 6 21  14

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

44. Determine the missing sides of the triangle. 39 24 33 13 8  11

45. The miniature car is proportional to the full-size car. Determine the height of the car. 5 3 12  ?

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

47. You should now be able to do pages 171 and 172 in our book. You can check your answers against the key posted online. When you’ve completed this work, review all that we’ve done with math in the last few weeks. Good luck on the math predictor ~ I hope to be able to congratulate you on your newly earned GED very soon!

48. THE END!