1. A geometric sequence is found by multiplying the previous number by a given factor, or number. 5, 15, 45, 135,… Set up a proportion to compare the first 3 numbers 5 = 15 15 45 The cross products are equal! The # in the middle is the GEOMETRIC MEAN

2. I. Geometric Mean This is the geometric mean: So The geometric mean has to be a positive number!

3. Example 1: Find the geometric means for: 1 and 257 and 23 and 1/3 X² = 25 X = 5 x² = 14 X = x² = 1 X = 1

4. REMEMBER THE PARTS OF A RIGHT TRIANGLE?

5. II. Similar Triangles P Q R S  QPS  QRP  PRS This is the geometric mean!

6. III. Altitude Formula In a right triangle, the altitude is the geometric mean of the two parts of the hypotenuse mean h1 h2

7. Example 2: Find h. 925 h² = 225 h = 15

8. IV. Leg Formula In a right triangle, the leg is the geometric mean of the hypotenuse and the part of the hypotenuse adjacent to that leg. h1 h 2 mean

9. Example 3: Find the value of a and b 4 2 a b a² = 24 A = 2 b² = 12 B = 2

10. V. The Pythagorean Theorem a 2 + b 2 = c 2 Pythagorean Triples: whole number side lengths that fit the theorem.

11. Example 4: 6. Do 8,18, and 20 form a right triangle? 7. Name two other Pythagorean triples you can think of.

12. http://www.pisgah.us/organiz/geometry/accessoryinfo/Pyth-2.html

13. Try P 401: 7 - 14 7.10 8.A.  PTG  PGA  GTA B. <PAG <TAG 9.X=  10 Y =  14 10.2  13 11.  51 12.Yes 13.A. Yes 3 4 5 B. Each is a multiple of 3 4 5 C. Each is a multiple of 3 4 5 D. Yes: sides are multiples of the primitive triple 14. About 179.29 feet

14. 7-3 Special Right Triangles I. Review What is the geometric mean of two numbers a and b? Solve for x. X 5 25

16. II.The isosceles right triangle ( 45-45-90) RATIO:

17. Looking for the hypotenuse? Multiply the leg by √2

18. Looking for the leg? Divide the hypotenuse by √2

19. Examples 1. Find AB and AC for isosceles triangle ABC. 3

20. 2. Find a and b. a b

21. 3. Find a and b. a b 10

22. 4. Find x and y. x y 19

23. III. The 30-60-90 right triangle RATIO: 1 : : 2

24. You know the longest leg! 15 60° DIVIDE BY √3 AND MULTIPLY BY 2 DIVIDE BY √3

25. You know the shortest leg! 18 30° MULTIPLY BY 2 MULTIPLY BY √3

26. You know the hypotenuse! 40 DIVIDE BY 2 DIVIDE BY 2, MULTIPLY BY √3 30°

27. 5. Find b and c. c b 60 30 You know the longer leg!

28. c a 9 30 60 10 a b 30 60 6. Find the indicated measures. a = c = a = b =

29. 7. The measures of both legs of a right triangle are 4. What is the measure of the hypotenuse?

30. 8. Find x. CHALLENGE: FIND THE AREA OF THE TRIANGLE!

31. 9. The length of a diagonal of a square is 20 centimeters. Find the length of a side of a square

32. I. NAMING SIDES IN A RIGHT TRIANGLE 7-4 Special Ratios

33. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. II. Trig Ratios A. THE SINE RATIO

34. B. THE COSINE RATIO

35. C. THE TANGENT RATIO

36. 1. Compare the sine, the cosine, and the tangent ratios for  A in each triangle below. SOLUTION Large triangleSmall triangle sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent  0. 4706 8 17  0. 4706 48.548.5  0. 8824 15 17  0. 8824 7.58.57.58.5  0. 5333 8 15  0. 5333 47.547.5 Trigonometric ratios are frequently expressed as decimal approximations. A B C 17 8 15 A B C 8.58.5 4 7.57.5

37. 2. Find the sine, the cosine, and the tangent of the indicated angle. SS R TS 5 13 12 SOLUTION The length of the hypotenuse is 13. For  S, the length of the opposite side is 5, and the length of the adjacent side is 12. sin S  0. 3846 = 5 13 cos S  0. 9231 = 12 13 tan S  0. 4167 = 5 12 opp. adj. hyp. R T S 5 12 13 opp. hyp. = adj. hyp. = opp. adj. =

38. 3. Find the sine, the cosine, and the tangent of 45º. SOLUTION Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. 1 1 45º hyp. tan 45º = 1 = 1111 sin 45º  0. 7071 cos 45º opp. hyp. = adj. hyp. =  0. 7071 opp. adj. = From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2. = 1 2 = = 1 2 = 2

39. 4. Find the given length. a.b. X 20 53 ° 15 X 35 °

40. III. Finding the angle. A. If you know the side lengths, and need to find the angle, you just use the inverse button. 20 6 X°X° Tan X ° = opp adj = 6 20 Press tan -1 (6 / 20)=

41. Your turn! 6. Find the angle. a.b. 32 14 X ° 42 18 X °

42. 7-5 Angles of Elevation and Depression I.Angle of Elevation Up from the point of reference - the Horizon Perspective to the Horizon

44. II. Angle of Depression Down from the point of reference - the Horizon

45. 1. FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet. tan 59° = opposite adjacent 45 tan 59° = h 45(1. 6643)  h 74. 9  h The tree is about 75 feet tall. Write ratio. Substitute. Multiply each side by 45. Use a calculator or table to find tan 59°. Simplify. tan 59° = opposite adjacent h 45

46. 2. ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet. sin 30° = opposite hypotenuse d sin 30° = 76 d = 152 A person travels 152 feet on the escalator stairs. Write ratio for sine of 30°. Substitute. Multiply each side by d. Divide each side by sin 30°. Simplify. sin 30° = opposite hypotenuse 76 d d = 76 sin 30° d = 76 0. 5 Substitute 0.5 for sin 30°. 30° 76 ft d

47. 3. Find how high the plane is from the ground. 12° 16 km

48. 4. How far is the base of the tower from the fire? 5°5° 43 ft

49. 5. Find the angle of elevation. 24 ft 11 ft