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:

• So

b

x

x

a

baxx

bax 2

bax The geometric mean has to be a positive number!

Example 1: Find the geometric means for:

1 and 25 7 and 2 3 and 1/3

25

1 x

x

X² = 25 X = 5

31

3

2

7 x

x

x

x

x² = 14X = 14

x² = 1X = 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.

9 25

25

9 h

h

h² = 225

h = 15

IV. Leg Formula

In a right triangle, the leg is the geometric mean ofthe 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

2a

b

4

6 a

a

a² = 24A = 2 6

2

6 b

b b² = 12 B = 2 3

V. The Pythagorean Theorem

• a2 + b2 = c2

• 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. 108. A. PTG PGA GTA B. <PAG <TAG9. X= 10 Y = 1410. 21311. 5112. Yes13. 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 triple14. About 179.29 feet

7-3 Special Right Triangles7-3 Special Right Triangles

• I. Review

• What is the geometric mean of two numbers a and b?

• Solve for x.

X

525

II.II. The isosceles right The isosceles right triangletriangle

( 45-45-90)( 45-45-90)

RATIO:

Looking for the Looking for the hypotenuse?hypotenuse?

Multiply the leg

by √2

Looking for the leg?Looking for the leg?

Divide the hypotenuse

by √2

ExamplesExamples

• 1. Find AB and AC for isosceles triangle ABC.

3

• 2. Find a and b.

a

b

27

• 3. Find a and b.

a

b

10

4. Find x and y.4. Find x and y.

x

y

19

III. The 30-60-90 right III. The 30-60-90 right triangletriangle

RATIO:

1 : : 23

You know the longest leg!You know the longest leg!

15

60°

DIVIDE BY √3 AND

MULTIPLY BY 2

DIVIDE BY

√3

You know the shortest leg!You know the shortest leg!

18

30° MULTIPLY

BY 2MULTIPLY BY

√3

You know the hypotenuse!You know the hypotenuse!

40

DIVIDE BY 2

DIVIDE BY 2,

MULTIPLY BY √3

30°

5. Find b and c.5. Find b and c.

c

b

60

3038

You know the longer leg!

ca

9

30

60

10

a

b

30

60

6. 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 triangle Small triangle

sin A =opposite

hypotenuse

cos A =adjacent

hypotenuse

tan A =oppositeadjacent

0.47068

17 0.47064

8.5

0.88241517 0.8824

7.58.5

0.53338

15 0.53334

7.5

Trigonometric ratios are frequently expressed as decimal approximations.

A

B

C

178

15

A

B

C

8.5 4

7.5

2. Find the sine, the cosine, and the tangent of the indicated angle.

S R

T S

5 13

12SOLUTION

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=1213

tan S

0.4167=5

12

opp.

adj.

hyp.

R

T S

5

12

13

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.

1

1

45º

hyp.

tan 45º = 1=11

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

= 22

= 1 2

= 22

2

4. Find the given length.

a. b.

X20

53 °

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.

20

6

X°

Tan X ° = opp

adj

= 6

20

Press tan-1 (6 / 20)=

Your turn!

•6. Find the angle.•a. b.

32

14

X °

42

18

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 DepressionDown 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° =oppositeadjacent

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° =oppositeadjacenth45

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

hypotenuse76d

d =76

sin 30°

d =760.5

Substitute 0.5 for sin 30°.

30°

76 ftd

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°43 ft

5. Find the angle of elevation.

24 ft

11 ft