/
Pythagorean Theorem  the square of the hypotenuse (the side opposite the right angle) Pythagorean Theorem  the square of the hypotenuse (the side opposite the right angle)

Pythagorean Theorem  the square of the hypotenuse (the side opposite the right angle) - PowerPoint Presentation

helene
helene . @helene
Follow
348 views
Uploaded On 2021-12-08

Pythagorean Theorem  the square of the hypotenuse (the side opposite the right angle) - PPT Presentation

OR a 2 b 2 c 2 c a b leg leg hypotenuse Pythagorean Triples A set of nonzero whole numbers a b and c that satisfy the equation a 2 b 2 c 2 Here are some common ones to remember ID: 904582

find angle side triangle angle find triangle side missing degrees angles area triangles nearest sides hundredth ind math law

Share:

Link:

Embed:

Download Presentation from below link

Download The PPT/PDF document "Pythagorean Theorem  the square of the ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

Pythagorean Theorem

 the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides

. OR a2 + b2 = c2

c

a

b

leg

leg

hypotenuse

Slide2

Pythagorean Triples

A set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2Here are some common ones to remember:

3, 4, 55, 12, 138, 15, 177, 24, 25

Slide3

Using the Pythagorean Theorem to solve for a missing side

Solve for the missing side.

30a24

Slide4

Using the Pythagorean Theorem to solve for a missing

side

1218

c

Solve for the missing side.

Slide5

Converse of the Pythagorean Theorem

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

How will we know which is the hypotenuse?Given the sides of a triangle equal to 16, 30, 34. Does this form a right triangle?

Slide6

Which of the following are right triangles?

,

, 5 3, 4, 621, 72, 75

2.5, 6.25, 43.3125

Slide7

Showing Triangles are Obtuse or Acute

Obtuse Triangles – A triangle where one angle is greater than 90 degrees.If c2

> a2 + b2, where c is the longest side of the triangle then the triangle is an obtuse triangle.Acute Triangles – A triangle where all angles are less than 90 degrees.If c2 < a2 + b2, where c is the longest side of the triangle then the triangle is an acute triangle.

Slide8

Classify the following triangles acute, obtuse, or right.12, 16, 20

11, 12, 1531, 23, 12.3, .4, .5

Slide9

End of Day 1P 361 2-38 evens

Slide10

Special Right Triangles

Some right triangles are more common and have some short cuts we can use instead of having to use the Pythagorean Theorem.

45 – 45 – 90 Right Triangle9045

45

x

 

x

x

Slide11

Solve for the missing side(s).

10

10y

10

y

z

Slide12

30 – 60 – 90 Right Triangle

30

6090

x

 

2x

x

Slide13

Solve for the missing side(s).

30

6090

x

 

2x

25

Slide14

30

60

9018y

x

Solve for the missing side(s).

Slide15

End of Day 2P 3692-20 even and 24-29 all

Slide16

Math 2

Unit 4: "Right Triangle Trigonometry”Title

: Using Trigonometric Ratios to find Sides.Objective: To use the trigonometric ratios to find the lengths of sides in right triangles.

Slide17

Setting Up Triangles

NEVER USE RIGHT ANGLE TO SET UP!

1. Label the side across from 90° angle as Hyp for hypotenuse.2. Circle the angle you need to use.3. Draw a line from the angle to the opposite side. Label that side 



 


Opp for the side opposite the 



 given angle. 



 





4. Label the remaining side

Adj for the side adjacent to the given

angle.Using Angle

A.

A

B

C

Slide18

Setting Up Triangles

Using Angle C

.

A

B

C

NEVER USE RIGHT ANGLE

TO SET UP!

1. Label the side across from 90°

angle as

Hyp

for hypotenuse.

2. Circle the angle you need to use.

3. Draw a line from the angle to the

opposite side. Label that side 





 


Opp

for the side opposite the 







given angle. 





 







4. Label the remaining side

Adj

for

the side adjacent to the given

angle.

Slide19

D

K

L

Setting Up Triangles

Using Angle

D

.

Slide20

Trigonometric Ratios

Sine of a Angle = Leg Opposite

Hypotenusesin (θ) =

Opp

Hyp

Cosine of a Angle = Leg Adjacent

  Hypotenuse

cos

(

θ

) =

Adj

Hyp

Tangent of a Angle = Leg Opposite

  Leg Adjacent

tan (

θ

) =

Opp

Adj

"SOH CAH TOA"

Slide21

Find the following trigonometric ratios

. 1. sin A

2. cos A 3. tan A

A

B

C

3

4

5

Slide22

A

B

C

3

4

5

Find the following trigonometric ratios

.

1. sin C

2.

cos

C

3. tan C

Slide23

T

G

R

Find the following.

1. sin G

2.

cos G3. tan G

4. sin T

5. cos T

6. tan T

24

10

26

Slide24

Every angle has it's own sine, cosine, and tangent ratio!

In Geometry Textbook p. 731

Slide25

Degrees

STOP

Check 
your 
mode

Degrees

Degrees

Officer Be N. Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Slide26

40°

x

6

How to find Missing Sides

1. Circle given angle and label

sides.2. Decide on set up.

SOH CAH TOA3. Set up the proportion.

4. Solve by cross multiplication. Round answers to three decimals

Find the missing side.

Slide27

50°

x

10

Find the missing side.

Slide28

30°

x

12

Find the missing side.

Slide29

34°

x

7.2

Find the missing side.

Slide30

Find the missing measures.

x

40°

18

y

20°

z

Slide31

Find the missing measures.

8

x

68°

21

y

z

Slide32

End of Day 3

P 479 1-10, 26, 28, 41-43

Slide33

Unit 4: "Right Triangle Trigonometry“

Title: Using Trigonometric Ratios to find Angles.Objective

: To use the trigonometric ratios to find the measures of angles in right triangles.

Degrees

STOP

Check 
your 
mode

Degrees

Degrees

Officer Be N. Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Degrees

Slide34

How to find Missing Angles

1. Circle missing angle and label sides


 


 2. Decide on set up.

SOH CAH TOA3. Set up proportion.4. Solve by inverse trig ratio. sin-1 cos-1 tan

-1 Round answer to nearest degree.

Find the missing angle

.

11

7

Slide35

Find the missing angle

.

8

5

Slide36

Find the missing angle.

10.4

12.6

Slide37

Find the missing angle.

20

16

Slide38

Find the missing angles.

87

63

Slide39

Find the missing angles.

7

5

13

Slide40

Find the missing angles.

8

17

19

Slide41

Find the missing angles

.6

x°8

Slide42

Honors Math 2 Assignment:

In the Geometry Textbook,pp. 473-474 #2-16 evens, 27-29

pp. 479-480 #2-16 evens, 22-24

Slide43

Find the missing angles.

7

5

13

Find the missing measures.

8

x

68°

21

y

z

Slide44

Math 2Unit 4

Lesson 2Unit 4

: "Right Triangle Trigonometry“Title: Application - Angles of Elevation and DepressionObjective: To use angles of elevation and angles of depression to find measures.

Slide45

angle of elevation

- is the angle measured from a horizontal line up

to an object.angle of depression is the angle measured from a horizontal line down to an object.

Slide46

Angles of Elevation and Depression

Describe each angle as it relates to the situation shown.

Angle 1Angle 2Angle 3Angle 4

Slide47

Example 1

Slide48

Example 3

"Line-of-Sight Distance"

Slide49

Example 4

"Line-of-Sight Distance"

Slide50

Example 5A blimp is flying 500 ft above the ground. A person on the ground sees the blimp by looking at a 25° angle. The person’s eye level is 5 ft above the ground. Find the distance from the blimp to the person to the nearest foot.

Slide51

Example 6

A surveyor sets up a 5 ft tall theodolite (an instrument for measuring angles) 300 ft from a building to measure its height. The angle measured to the top of the building using the theodolite is 35°. How tall is the building?

Slide52

Example 7

You sight a rock climber on a nearly vertical cliff at a 32° angle of elevation. The horizontal ground distance to the base of the cliff is 1,000 ft. Find the line-of-sight distance to the rock climber.

Slide53

Example 8

An airplane flying at an altitude of 3,500 ft begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent?

Slide54

Example 9

Two buildings are 30 ft apart. The angle of depression from the 
top of higher building to the top of the other building is 19°. What is the difference in the heights of the buildings?

Slide55

Honors Math 2 Assignment:

In the Geometry Textbook,pp. 484-486 #9-18, 21, 23, 33, 34

Slide56

Example 10

In a galaxy far, far away, a spaceship is orbiting the planet Obar. The ship needs to land in a large, flat crater, but the captain wants to make sure the crater is large enough to hold the ship. When the ship is 4 miles above the planet, the onboard guidance system measures the angles of depression from the ship to both sides of the crater. The angles measure 22° and 37°, respectively. What is the distance across the crater? If the spaceship is 2,500 ft long, will it fit in the crater?

Slide57

How tall is the flag?

Slide58

End of Day 4

Slide59

Example 6

Find the area of the triangle to the nearest hundredth.

Slide60

Math 2

Unit 4Lesson 4

Unit 4: "Right Triangle Trigonometry"Title: Application - Finding Area Using Trigonometric RatiosObjective: To find the area of triangles and regular polygons using trigonometry. 


















 


Slide61

Example 1

Find the area of the triangle to the nearest hundredth.

Slide62

Area of a Triangle with an Angle

Slide63

Example 2

Find the area of the triangle to the nearest hundredth.

Slide64

Example 3F

ind the area of the triangle to the nearest hundredth.

15 m

24 m

120

°

Slide65

Example 4F

ind the area of the triangle to the nearest hundredth.

14 yd

6 yd

87

°

49

°

Slide66

Example 5F

ind the area of the triangle to the nearest hundredth.

Slide67

Example 6

A triangular plot of land has two sides that measure 300 ft and 200 ft and form a 65 angle. Find the area of the plot to the nearest square foot.

Slide68

radius

- is a line segment from center to the vertex of a polygon.

apothem - is a line segment from the center of a regular polygon perpendicular to any of its sides.Area of Regular Polygons

Area = 1/2 * side * apothem * number of sides

Slide69

Example 7F

ind the area of a regular pentagon with 8 cm sides

to the nearest hundredth.

Slide70

Example 8F

ind the area of a regular decagon with a perimeter

of 120 ft to the nearest hundredth.

Slide71

Example 9F

ind the area of a regular hexagon with a apothem

of 8 cm to the nearest hundredth.

Slide72

Example 10F

ind the area of a regular nonagon with a apothem

of 14 in to the nearest hundredth.

Slide73

Example 11F

ind the area of a regular octagon with a radius

of 16 m to the nearest hundredth.

Slide74

Example 12F

ind the area of a regular dodecagon with a radius

of 26 in to the nearest hundredth.

Slide75

Honors Math 2 Assignment:

In the Geometry Textbook,

pp. 500-501 #1-4, 6-9, 11-18

Slide76

End of Day 5

Slide77

Math 2Unit 4

Lesson 4Title: Law of

SinesObjectives: To learn to use the law of Sines to find missing sides and angles in triangles.

Slide78

Slide79

Example: Finding a side of a triangle

Slide80

Example: Finding a side of a triangle

Slide81

Example: Finding an angle of a triangle

Slide82

Example: Finding an angle of a triangle

Slide83

Slide84

Assignment:

Law of Sines Handout

Slide85

Law of Sines: "

The Ambiguous Case

"

Slide86

Law of Sines: "The Ambiguous Case

"

Slide87

Law of Sines: "The Ambiguous Case

"

Slide88

Questions?

Slide89

Math 2Unit 4

Lesson 5Title: Law of Cosines

Objectives: To learn to use the law of cosines to find missing sides and angles in triangles.

Slide90

Find the missing sides and angles of the triangles below

.

Slide91

Slide92

Example: Finding a side of a triangle

Slide93

Example: Finding a side of a triangle

Slide94

Example: Finding an angle of a triangle

Slide95

Example: Finding an angle of a triangle

Slide96

Assignment:

Law of Cosines Handout

Slide97

End of Day 6

Slide98

Math 2 Warm Up

1. Find the area of each triangle to the nearest tenth. a. 11 yd, 14

yd, 16 yd b. 100 ft, 80 ft, 140 ft c. 6 m, 12.8 m, 13 m2. In the Geometry Practice Workbook, Reteaching 11-5 (p. 139) #1, 3, 5-10, 12Practice 11-5 (p. 140) #13, 14

Slide99

Unit 4 Test Review: Trigonometry

trigonometric ratios sin (angle) = cos

(angle) =

tan (angle) =

F

ind the length of

sides

of a

right triangle

.

F

ind the measure of

angles

of a

right triangle

.

 

Slide100

Unit 4 Test Review: Trigonometry

angles of elevation

measured from horizon up to objectangles of depression measured from horizon down

to object

Slide101

Unit 4 Test Review: Trigonometry

Area of Regular Polygons

Use ½ measure of the central angle for the right triangle to find measures. A = ½ ∙ side ∙ apothem ∙ # of trianglesArea of Oblique Triangles

A = ½ ∙ side ∙ side ∙ sin(Included Angle)

A =

s = ½(a + b + c)

Equations of Circles

 

Slide102

Math 2 Assignment

In the Geometry Textbook,pp. 505-507 #8-25, 40-46

Slide103

Math 2 Test Day Warm Up

In the Geometry Textbook, p. 508 #

9-12, 13-16 , 26, 28

Slide104

TESTING!

No talking during test!When finished, turn in test and test review that you finished for homework to appropriate folders on the front table!

Slide105

Test Review: Trigonometry

Law

of Sines =

=

Law of Cosines

b

c