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9.1 – Similar Right Triangles

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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.

C

B

A

N

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.

C

AN

CN

=

CN

BN

B

A

N

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.

C

AB

AC

=

AC

AN

B

A

N

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.

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

C

AB

AB

AC

BC

=

=

AC

BC

AN

BN

B

A

N

Set up Proportions

C

B

A

N

C

B

A

N

y

z

x

6

3

w

6 + 3 = 9

w = 9

C

w

x

A

9

K

z

y

15

B

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.

a

b

c

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.

6

y

13

x

12

12

x

5

9

8

14

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

Find Area

8 in

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.

C

a

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.

C

a

b

B

A

c

121

64

36

64

81

3

1

4

5 + 6 < 12

Neither

16

+

<

+

>

+

=

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

Reminders of the past. Properties of:

ParallelogramsRectangles

1)1)

2)2)

3)Rhombus

4)1)

5)2)

6)3)

Describe the shape, Why? Use complete sentences

25

7

24

9.4 – Special Right Triangles

- Rationalize practice

Remember, 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

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

sine sin

cosine cos

Tangent tan

A device that helps is:

SOHCAHTOA

B

in ppyp os dj yp an ppdj

HYPOTENUSE

OPPOSITE

A

C

ADJACENT

B

C

A

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

Hypotenuse

Opposite

Find x

opposite, hypotenuse

USE SIN!

x

20

Pg 845

Anglesincostan

34o .5592 .8290 .6745

Or use the calculator

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

Hypotenuse

Adjacent

Find y

adjacent, hypotenuse

USE COS!

y

20

Pg 845

Anglesincostan

34o .5592 .8290 .6745

Or use the calculator

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

Opposite

Adjacent

Find x

Adjacent, Opposite, use TANGENT!

30

4

Pg 845

Angle sin cos tan

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

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

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

For word problems, drawing a picture helps.

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

ANGLE OF ELEVATION

ANGLE OF DEPRESSION

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

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?

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

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

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

Opposite

Adjacent

Find x

Adjacent, Opposite, use TANGENT!

30

4

Pg 845

Angle sin cos tan

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

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

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

Find x

Find all angles and sides, I check HW

Find all angles and sides