9.1 – Similar Right Triangles

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

BN

Set up Proportions

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

9.2 – Pythagorean Theorem

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

Given

Starfish both sides

Cross Multiplication (property of proportion)

Addition

Distributive Property =

Seg + post

Substituition prop =

• Pythagorean Triple is a set of three positive integers a, b, and c that satisfy the equation a2 + b2 = c2.

• Examples:– 3, 4, 5– 5, 12, 13– 7, 24, 25– 8, 15, 17– Multiples of those.

12

6

14

x

222 812 x

8

2

2

208

64144

x

x

x134

13

5

x

9

y

222 135 x

144

169252

2

x

x

12x

12

222 129 yy15

DON’T BE FOOLED, no right angle at top, can’t use theorems from before

8 in

Find Area

9.3 – The Converse of the Pythagorean Theorem

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

C

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

7 7, 7, 5,18 ,7 3 2, 1, 9 6, 5,

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

Reminders of the past. Properties of:Parallelograms Rectangles1) 1)2) 2)3) Rhombus4) 1)5) 2)6) 3)

Describe the shape, Why? Use complete sentences

24

725

9.4 – Special Right Triangles

• Rationalize practice

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

Find areas

9.5 – Trigonometric Ratios

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

C

BB

BA

AA

tancos

sintan

cossin

152

28

• Calculator CHECK– MODE!!!!!!!!!!! Should be in degrees– sin(30o) Test, should give you .5

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.

x20

50

68

x

x

1283

41

49

x

x20

506

8

x

y

y

40

70

x

34

17 70

x

1770cos

From the line of sight, if you look up, it’s called the ANGLE OF ELEVATION

From the line of sight, if you look down, it’s called the ANGLE OF DEPRESSION

ANGLE OF ELEVATIONANGLE OF DEPRESSION

For word problems, drawing a picture helps.

All problems pretty much involve trig in some way.

Mr. Kim’s eyes are about 5 feet two inches above the ground. The angle of elevation from his line of sight to the top of the building was 25o, and he was 20 feet away from the building. How tall is the building in feet?

25

feet20x

2025tan

x

326.9x

167.5

167.5 493.14

Mr. Kim is trying to sneak into a building. The searchlight is 15 feet off the ground with the beam nearest to the wall having an angle of depression of 80o. Mr. Kim has to crawl along the wall, but he is 2 feet wide. Can he make it through undetected?

80o

ft644.2

x

15)80tan(

Mr. Kim saw Mr. Knox across the stream. He then walked north 1200 feet and saw Mr. Knox again, with his line of sight and his path creating a 40 degree angle. How wide is the river to the nearest foot?

1200 ft

ft1007

1200)40tan(

x

The ideal angle of elevation for a roof for effectiveness and economy is 22 degrees. If the width of the house is 40 feet, and the roof forms an isosceles triangle on top, how tall should the roof be?

• DJ is at the top of a right triangular block of stone. The face of the stone is 50 paces long. The angle of depression from the top of the stone to the ground is 40 degrees (assume DJ’s eyes are at his feet). How tall is the triangular block?

9.6 – Solving Right Triangles

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.

Find x

Find all angles and sides, I check HW

Find all angles and sides