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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 I. Geometric Mean • This is the geometric mean: a x x b • So x x a b x a b 2 x ab The geometric mean has to be a positive number! Example 1: Find the geometric means for: 1 and 25 7 and 2 1 x x 25 7 x 3 x 1 x 2 x 3 X² = 25 X=5 x² = 14 X = 14 3 and 1/3 x² = 1 X=1 REMEMBER THE PARTS OF A RIGHT TRIANGLE? II. Similar Triangles Q R This is the geometric mean! S P QPS QRP PRS III. Altitude Formula • In a right triangle, the altitude is the geometric mean of the two parts of the hypotenuse mean h1 h2 Example 2: Find h. 9 h h 25 h² = 225 9 25 h = 15 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. mean h1 h2 Example 3: Find the value of a and b 6 a a 4 4 a 2 a² = 24 A=2 6 b 6 b b 2 b² = 12 B=2 3 V. The Pythagorean Theorem • a 2 + b2 = c 2 • Pythagorean Triples: whole number side lengths that fit the theorem. Example 4: 6. Do 8,18, and 20 form a right triangle? 7. Name two other Pythagorean triples you can think of. http://www.pisgah.us/organiz/geometry/accessoryinfo/Pyth-2.html Try P 401: 7 - 14 7. 10 8. A. PTG PGA GTA B. <PAG <TAG 9. X= 10 Y = 14 10. 213 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 7-3 Special Right Triangles • I. Review • What is the geometric mean of two numbers a and b? 5 • Solve for x. 25 X II. RATIO: The isosceles right triangle ( 45-45-90) Looking for the hypotenuse? Multiply the leg by √2 Looking for the leg? Divide the hypotenuse by √2 Examples • 1. Find AB and AC for isosceles triangle ABC. 3 • 2. Find a and b. a 7 2 b • 3. Find a and b. 10 a b 4. Find x and y. 19 x y III. The 30-60-90 right triangle RATIO: 1: 3 : 2 You know the longest leg! 15 60° DIVIDE BY √3 DIVIDE BY √3 AND MULTIPLY BY 2 You know the shortest leg! 30° MULTIPLY BY MULTIPLY BY 2 √3 18 You know the hypotenuse! DIVIDE BY 2, 30° 40 MULTIPLY BY √3 DIVIDE BY 2 5. Find b and c. 30 c 8 3 60 b You know the longer leg! 6. Find the indicated measures. 30 c a 10 a 30 60 60 9 b a= a= c= b= 7. The measures of both legs of a right triangle are 4. What is the measure of the hypotenuse? 8. Find x. CHALLENGE: FIND THE AREA OF THE TRIANGLE! • 9. The length of a diagonal of a square is 20 centimeters. Find the length of a side of a square 7-4 Special Ratios I. NAMING SIDES IN A RIGHT TRIANGLE II. Trig Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. A. THE SINE RATIO B. THE COSINE RATIO C. THE TANGENT RATIO 1. Compare the sine, the cosine, and the tangent ratios for A in each triangle below. 8.5 B 17 A 8 15 A 7.5 B 4 C C SOLUTION Large triangle sin A = cos A = tan A = Small triangle opposite hypotenuse 8 0.4706 17 4 0.4706 8.5 adjacent hypotenuse opposite adjacent 15 0.8824 17 8 0.5333 15 7.5 0.8824 8.5 4 0.5333 7.5 Trigonometric ratios are frequently expressed as decimal approximations. 2. Find the sine, the cosine, and the tangent of the indicated angle. S R 5 SOLUTION T 13 12 S 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. opp. 5 = hyp. 13 0.3846 R adj. 12 0.9231 cos S = = hyp. 13 opp.5 sin S tan S = opp. 5 0.4167 = = adj. 12 T 13 12 adj. hyp. S 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. From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2 . sin 45º = opp. = hyp. 1 = 2 2 2 0.7071 2 hyp. 1 cos 45º = adj. = hyp. tan 45º = opp. 1 = adj. 1 1 = 2 =1 2 2 0.7071 45º 1 4. Find the given length. a. b. 35 ° 20 15 X 53 ° X III. Finding the angle. A. If you know the side lengths, and need to find the angle, you just use the inverse button. Tan X ° = opp adj = 6 6 20 X° Press tan-1 (6 / 20)= 20 Your turn! • 6. Find the angle. • a. b. 18 X° 14 32 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 II. Angle of Depression Down from the point of reference - the Horizon 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 Write ratio. tan 59° = opposite h adjacent 45 Substitute. 45 tan 59° = h Multiply each side by 45. 45(1.6643) h Use a calculator or table to find tan 59°. 74.9 h Simplify. The tree is about 75 feet tall. 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 Write ratio for sine of 30°. sin 30° = 76opposite hypotenuse d Substitute. d sin 30° = 76 76 d= sin 30° 76 d= 0.5 d = 152 Multiply each side by d. d 76 ft Divide each side by sin 30°. 30° Substitute 0.5 for sin 30°. Simplify. A person travels 152 feet on the escalator stairs. 3. Find how high the plane is from the ground. 16 km 12° 4. How far is the base of the tower from the fire? 5° 43 ft 5. Find the angle of elevation. 24 ft 11 ft