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 = The cross products are equal! The # in the middle is the GEOMETRIC MEAN

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

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

REMEMBER THE PARTS OF A RIGHT TRIANGLE?

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

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. 925 h² = 225 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. h1 h 2 mean

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

V. The Pythagorean Theorem a 2 + b 2 = 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.

Try P 401: A.  PTG  PGA  GTA B. <PAG <TAG 9.X=  10 Y =    Yes 13.A. Yes B. Each is a multiple of C. Each is a multiple of D. Yes: sides are multiples of the primitive triple 14. About feet

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

II.The isosceles right triangle ( ) RATIO:

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 b

3. Find a and b. a b 10

4. Find x and y. x y 19

III. The right triangle RATIO: 1 : : 2

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

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

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

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

c a a b Find the indicated measures. a = c = a = 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

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

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. II. Trig Ratios 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. SOLUTION Large triangleSmall triangle sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent       Trigonometric ratios are frequently expressed as decimal approximations. A B C A B C

2. Find the sine, the cosine, and the tangent of the indicated angle. SS R TS 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  = 5 13 cos S  = tan S  = 5 12 opp. adj. hyp. R T S opp. hyp. = adj. hyp. = opp. adj. =

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 º hyp. tan 45º = 1 = 1111 sin 45º  cos 45º opp. hyp. = adj. hyp. =  opp. adj. = From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is 2. = 1 2 = = 1 2 = 2

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

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

Your turn! 6. Find the angle. a.b X ° X °

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 45 tan 59° = h 45( )  h  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

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 = Substitute 0.5 for sin 30°. 30° 76 ft d

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

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

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