9 1 similar right triangles
This presentation is the property of its rightful owner.
Sponsored Links
1 / 48

9.1 – Similar Right Triangles PowerPoint PPT Presentation


  • 74 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

9.1 – Similar Right Triangles

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


9 1 similar right triangles

9.1 – Similar Right Triangles


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


9 1 similar right triangles

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


9 1 similar right triangles

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


9 1 similar right triangles

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


9 1 similar right triangles

Set up Proportions

C

B

A

N


9 1 similar right triangles

C

B

A

N

y

z

x

6

3

w

6 + 3 = 9

w = 9


9 1 similar right triangles

C

w

x

A

9

K

z

y

15

B


9 2 pythagorean theorem

9.2 – Pythagorean Theorem


9 1 similar right triangles

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


9 1 similar right triangles

Given

Starfish both sides

Cross Multiplication (property of proportion)

Addition

Distributive Property =

Seg + post

Substituition prop =


9 1 similar right triangles

  • 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.


9 1 similar right triangles

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


9 1 similar right triangles

Find Area

8 in


9 3 the converse of the pythagorean theorem

9.3 – The Converse of the Pythagorean Theorem


9 1 similar right triangles

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


9 1 similar right triangles

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


9 1 similar right triangles

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


9 1 similar right triangles

Reminders of the past. Properties of:

ParallelogramsRectangles

1)1)

2)2)

3)Rhombus

4)1)

5)2)

6)3)


9 1 similar right triangles

Describe the shape, Why? Use complete sentences

25

7

24


9 4 special right triangles

9.4 – Special Right Triangles


9 1 similar right triangles

  • Rationalize practice


9 1 similar right triangles

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)


9 1 similar right triangles

Lots of examples


9 1 similar right triangles

Find areas


9 5 trigonometric ratios

9.5 – Trigonometric Ratios


9 1 similar right triangles

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


9 1 similar right triangles

B

C

A


9 1 similar right triangles

  • Calculator CHECK

    • MODE!!!!!!!!!!! Should be in degrees

    • sin(30o) Test, should give you .5


9 1 similar right triangles

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.


9 1 similar right triangles

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.


9 1 similar 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.


9 1 similar right triangles

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


9 1 similar right triangles

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?


9 1 similar right triangles

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


9 1 similar right triangles

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


9 1 similar right triangles

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?


9 1 similar right triangles

  • 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

9.6 – Solving Right Triangles


9 1 similar 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.


9 1 similar right triangles

Find x


9 1 similar right triangles

Find all angles and sides, I check HW


9 1 similar right triangles

Find all angles and sides


  • Login