Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all

Chapter 9 Review

Theorem 9.1: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

CNB~ANC~ACB:Then

CN altitude ACB; rt with ABC :Given

A

C

BN

Theorem 9.2 (Geo mean altitude): When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.

CN altitude ACB; rt with ABC :Given

A

C

BN

AN CNCN BN

=

Theorem 9.3 (Geo mean legs): When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

CN altitude ACB; rt with ABC :Given

A

C

BN

AB ACAC AN

=

Theorem 9.3 (Geo mean legs): When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

CN altitude ACB; rt with ABC :Given

A

C

BN

AB ACAC AN

=AB BCBC BN

=

One way to help remember is thinking of it as a car and you draw the wheels.

Another way is hypotenuse to hypotenuse, leg to leg

A

C

BN6 3

xy

w

z

6 + 3 = 9

w = 9

altGeo

x

x

x

x

23

18

3

6

2

legsGeo

y

y

y

y

63

54

6

9

2

legsGeo

z

z

z

z

33

27

3

9

2

A

C

B

K

x

9

y z

w

15

16

259

x

x

legsGeo

z

z

z

z

20

400

16

25

2

altGeo

y

y

y

y

12

144

9

16

2

legsGeo

w

w

w

25

22599

15

15

The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

222 cba :Then

ACB rt with ABC :Given

a

c

b

8 in

Find Area

Converse of Pythagorean Theorem: If the square of the hypotenuse is equal to the sum of the squares of the legs, then the triangle is a right triangle.

ert triangl a is ABC :Then

cba with ABC :Given 222

a

c

b

B A

Cacute is ABC ;90CmThen

bac If 222

obtuse is ABC ;90CmThen

bac If 222

12 6, 5, 2 ,1 ,3 9 8, 6, 8 11, 4,

neither)?(or obtuseor right, acute,it Is

16 64121 36 64 81 3 1 4 5 + 6 < 12

Neither

+ < + > + =

Obtuse Acute Right

Watch out, if the sides are not in order, or are on a picture, c is ALWAYS the longest side and should be by itself

leg a as long as times2 is

hypotenuse the triangle,904545 aIn

904545

Theorem

legshort the times3 is leglonger

theand leg,short theas long as times2 is

hypotenuse the triangle,906030 aIn

906030

Theorem

45

45

x

x 2x

60

30

x2x

3xRemember, small side with small angle.

Common Sense: Small to big, you multiply (make bigger)

Big to small, you divide (make smaller)

For 30 – 60 – 90, find the smallest side first (Draw arrow to locate)

Lots of examples

sine sin

cosine cos

Tangent tan

These are trig ratios that describe the ratio between the side lengths given an angle.

ADJACENT

OP

PO

SIT

E

HYPOTENUSE

adjacent

OppositeA

Hypotenuse

adjacentA

Hypotenuse

OppositeA

tan

cos

sin

A

B

C

A device that helps is:

SOHCAHTOAin pp yp os dj yp an pp dj

A

B

C14

539

B

B

B

A

A

A

tan

cos

sin

tan

cos

sin39

5

39

14

14

5

39

14

39

5

5

14

x

y

20

3434sin

Find xHypotenuse

Look at what they want and what they give you, then use the correct trig ratio.

Opposite

opposite, hypotenuse

USE SIN!

hypotenuse

opposite x

20

Pg 845

Angle sin cos tan

34o .5592 .8290 .6745

Or use the calculator

205592.

x

x184.11

x

y

20

3434cos

Find yHypotenuse

Look at what they want and what they give you, then use the correct trig ratio.

Adjacent

adjacent, hypotenuse

USE COS!

hypotenuse

adjacent y

20

Pg 845

Angle sin cos tan

34o .5592 .8290 .6745

Or use the calculator

208290.

y

y58.16

4

30

x

Find x

Look at what they want and what they give you, then use the correct trig ratio.

AdjacentOpposite

Adjacent, Opposite, use TANGENT!

adjacent

oppositex tan

30

4

5.7tan x

Pg 845

Angle sin cos tan

81o .9877 .1564 6.3138 82o .9903 .1392 7.1154 83o .9925 .1219 8.1443

82x

If you use the calculator, you would put tan-1(7.5) and it will give you an angle back.

Word Problems

• Hills, Buildings, Trees

• Pg 586-587: 1 - 21 all; Pg 582-584: 1 - 24 all

• 14-23

• Geo mean legs, alt, pythag

• Pythag area of triangle

• 45-45-90, 30-60-90

• State trig ratios

• Trig word prob

• VECTORS!!