Chapter 8Right Triangles

• Determine the geometric mean between two numbers.

• State and apply the Pythagorean Theorem.

• Determine the ratios of the sides of the special right triangles.

• Apply the basic trigonometric ratios to solve problems.

8.1 Similarity in Right Triangles

Objectives• Determine the

geometric mean between two numbers.

• State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

Means-Extremes property of proportions

• The product of the extremes equals the product of the means.

=a

b

c

d ad = cb

The Geometric Mean

“x” is the geometric mean between “a” and “b” if:

a

x b

x

or x ab

x2 = ab

√x2 = √ab

x = +/- √abTake Notice: The term said to be the

geometric mean will always be cross-multiplied w/ itself.

Take Notice: In a geometric mean problem, there are only 3 variables to account for, instead of four.

Example

What is the geometric mean between 3 and 6?

3

6

x

x

3 6 18 3 2o xr

You try it

• Find the geometric mean between 2 and 18.

6

Simplifying Radical Expressions(pg. 287)

• No “party people” under the radical

• No fractions under the radical

• No radicals in the denominator

4 4 2

3 3 3

2 3 2 3

33 3

Party People are perfect square #’s

which are?

White Board Practice

• Simplify 503

215

White Board Practice

• Simplify 147

27

White Board Practice

• Simplify

3

12

34

Find the Geometric Mean

• 2 and 3– √6

• 2 and 6– 2√3

• 4 and 25– 10

White Board Practice

• Simplify

22

8

22

Warm-up

• Simplify

545

4

3

• Find Geometric Mean of 7 and 12

White Board Practice

• Simplify

4

3

2

3

Similarity and Geometric Mean

• Similar Triangle Example

• What is special about a geometric mean proportion?

• We are now going to combine the idea of similarity with a geometric mean proportion.

SHMOOP VID

• http://www.shmoop.com/video/geometric-mean

TheoremIf the altitudealtitude is drawn to the hypotenuse of a right triangle…..

– 2 additional right triangles are created – The 3 triangles are all similar

• Their sides are in proportion to one another

b

y

g

o

p1 23Note: What one color side

represents to one triangle, represents something different in another!

Hypotenuse Big Leg Small Leg

OG Triangle

Medium

Small

Fill in the table with the letter of the color that represents each part of each different triangle.

PARTNERS: Find all of similarity proportions that would create geometric mean problems.

CorollaryWhen the altitudealtitude is drawn to the hypotenuse of a right

triangle, the length of the altitude is the geometric mean between the segments on the hypotenuse.

p

p

o

p

p

y

b

y

g

o

p

Easier way to remember… create the proportion of the legs of both smaller triangles.

Corollary 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 (closest to that leg.)

b

b

y

b

b

oy

o

g

g

oy

b

y

g

o

p

Group Practice

• Pg. 288 #17

• a. √14

• b. 3√ 2

• c. 3 √ 7

Group Practice

• If RS = 2 and SQ = 8 find PS

• PS = 4

R

P Q

S

Group Practice

• If RP = 10 and RS = 5 find RQ

• RQ = 20

R

P Q

S

Group Practice

• If RS = 4 and PS = 6, find SQ

• SQ = 9

R

P Q

S

8.2 The Pythagorean Theorem

Objectives• State and apply the

Pythagorean Theorem.• Examine proofs of the

Pythagorean Theorem.

WARM - UP

• Label the triangle with 4 letters

• Re-draw the 3 similar triangles, lining them up so that their corresponding parts are in the same position

• Write down 1 of the 3 proportions that create a geometric mean

Movie Time

• We consider the scene from the 1939 film The Wizard Of Oz in which the Scarecrow receives his “brain,”

Scarecrow:  “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

• We also consider the introductory scene from the episode of The Simpsons in which Homer finds a pair of eyeglasses in a public restroom…

Homer:  “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

Man in bathroom stall:  “That's a right triangle, you idiot!”

Homer:  “D'oh!”

•   Homer's recitation is the same as the Scarecrow's, although Homer receives a response

Think – Pair - Share

1. What are Homer and the Scarecrow attempting to recite?

• Identify the error or errors in their version of this well-known result.

• Is their statement true for any triangles at all? If so, which ones?

Think – Pair - Share

2. Is the correction from the man in the stall sufficient?

• Give a complete, correct statement of what Homer and the Scarecrow are trying to recite.

• Do this first using only English words, and a second time using mathematical notation. Use complete sentences.

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

222 bac

Brightstorm - proof

Find the value of each variable

1.

x

3

2

13x

Find the value of each variable

2.

6

4y

52y

Find the length of a diagonal of a rectangle with length 8 and width 4.

4.

4

8

8

4

Find the length of a diagonal of a rectangle with length 8 and width 4.

4.

8

4

54

Find the value of each variable

3.

4

x

x

22x

Find the value of each variable

5.

4

X + 2

x

Find the value of each variable

5.

X2 + (x+2) 2 = 10

X2 + x2 + 4x + 4 = 100

2x2 + 4x – 96 = 0

X2 + 2x – 48 = 0

(x + 8)(x – 6) = 0

X = -8 ; x = 6

10

X + 2

x

8.3 The Converse of the Pythagorean Theorem

Objectives

• Use the lengths of the sides of a triangle to determine the kind of triangle.

• Determine several sets of Pythagorean numbers.

                          

  

Given the side lengths of a triangle….

• Can we tell what type of triangle we have?YES!!

• How?

– We use c2 a2 + b2

– c always represents the longest side• Lets try… what type of triangle has sides

lengths of 3, 4, and 5?

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.

a

b

c

222 bac

Right Triangle

Pythagorean Sets• A set of numbers is considered to be

Pythagorean set if they satisfy the Pythagorean Theorem. WHAT DO I MEAN BY SATISFY THE PYTHAGOREAN THEOREM?

3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 256,8,10 10,24,269,12,1512,16,2015,20,25

This column should be memorized!!

Theorem (pg. 296)

If the square of one side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle.

a

b

c

222 bac Triangle is acute

a= 6 , b = 7, c = 8

Is it a right triangle?

Theorem (pg. 296)

If the square of one side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle.

a

b

c

222 bac Triangle is obtuse

a= 3 , b = 7, c = 9

Is it a right triangle?

Review

• We use c2 a2 + b2

•C2 = then we a right triangle

•C2 < then we have acute triangle

•C2 > then we have obtuse triangle

• Always make ‘c’ the largest number!!

The sides of a triangle have the lengths given. Is the triangle acute,

right, or obtuse?1. 20, 21, 29

• right

The sides of a triangle have the lengths given. Is the triangle acute,

right, or obtuse?2. 5, 12, 14

• obtuse

The sides of a triangle have the lengths given. Is the triangle acute,

right, or obtuse?3. 6, 7, 8

• acute

The sides of a triangle have the lengths given. Is the triangle acute,

right, or obtuse?4. 1, 4, 6

– Not possible

The sides of a triangle have the lengths given. Is the triangle acute,

right, or obtuse?5.

• acute

5,4,3

Warm – up

• Create a diagram and label it…

• An isosceles triangle has a perimeter of 38in with a base length of 10 in. The altitude to the base has a length of 12in. What are the dimensions of the right triangles within the larger isosceles triangle?

WARM - UP

• Solve for x, y, and z

xy

z

16 4

8.4 Special Right Triangles

Objectives

• Use the ratios of the sides of special right triangles

45º-45º-90º Theorem

In a 45-45-90 triangle, the hypotenuse is 2

times the length of each leg.

x

x

45

aHypotenuse = √2 ∙ leg

45

x√2

2 x- 90º

x - 45º

x - 45º

2 x- 90º

x - 45º

x - 45º• The sides opposite the 45◦ angles are congruent.

• The side opposite the 90◦ angle is the length of the leg

multiplied by √2

Look for the pattern..USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE

PROBLEMS!!!

Look for the pattern..USE THIS SET UP EVERY TIME YOU HAVE

ONE OF THESE PROBLEMS!!!

2 x- 90º

x - 45º

6 x - 45º

Look for the pattern

262 x- 90º

6 x - 45º

6 x - 45º

Look for the pattern

2 x- 90º

x - 45º

x - 45º

10

Look for the pattern

2 x- 90º

25 x - 45º

25 x - 45º

10

White Board Practice

6

x

x

Hypotenuse = √2 * leg

6 = √2 x

23x

Partner Discussion

• If we know the length of a diagonal of a square, can we determine the length of a side? If so, how?

x

x

x√2

White Board Practice

• If the length of a diagonal of a square is 4cm long, what is the perimeter of the square?

•Perimeter = 8√2cm

White Board Practice

• A square has a perimeter of 20cm, what is the length of each diagonal?

•Diagonal = 5√2 cm

30º-60º-90º Triangle

60

30

60

30A 30º-60º-90º triangle is half an equilateral

triangle

30º-60º-90º TheoremIn a 30-60-90 triangle, the hypotenuse is

twice as long as the shorter leg and the

longer leg is 3 times the shorter leg.

x2x60

30

3

Hypotenuse = 2 ∙ short leg

Long leg = √3 ∙ short leg

x 2x - 90º

3 x - 60º

x - 30º

2x - 90º

3 x - 60º

x - 30ºShort leg

hypotenuse

Long leg

Look for the pattern..USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE

PROBLEMS!!!

Look for the pattern

2x - 90º

3 x - 60º

6 x - 30º

Look for the pattern

12 2x - 90º

36 3 x - 60º

6 x - 30º

Look for the pattern

2x - 90º

8 3 x - 60º

x - 30º

Look for the pattern

3

316 2x - 90º

8 3 x - 60º

3

38 x - 30º

White Board Practice

5y

x

60º

Hypotenuse = 2 ∙ short leg

Long leg = √3 ∙ short leg

10

35

y

x

White Board Practice

9

y

x60º

30º

y = 3√3

x = 6√3

White Board Practice

• Find the length of an altitude of a equilateral triangle if the side lengths are 16cm.

•8√3 cm

Quiz Review Sec. 1 - 4

8.1• Geometric mean / simplifying radical expressions

• Corollary 1 & 2 - ** #32 p. 289 **

8.2• Pythag. Thm – rectangle problems - pg. 292 #10, 13, 14

– Isosceles triangle problems pg. 304 #7

8.3• Use side lengths to determine the type of triangle (right, obtuse, acute)

– Pg. 297 1 – 5

8.4• 45-45-90 triangles (problems using squares)

• 30-60-90 triangles (problems using equilateral triangles )

WARM-UP

• What is the one piece of information we need to prove 2 RIGHT triangles are similar? Explain in complete sentences why.

8.5 The Tangent Ratio

Objectives

• Define the tangent ratio for a right triangle

TrigonometryPg. 311

• When you have a right triangle you always have a 90◦ angle and 2 acute angles

• Based on the measurements of those acute angles you can discover the lengths of the sides of the right triangle

• Mathematicians have discovered ratios that exist for every degree from 1 to 89.

• The ratios exist, no matter what size the triangle

Trigonometry

A

B

C

Opp

osit

e le

g

Adjacent leg

Hypotenuse

Sides are named relative to an acute angle.

“Triangle measurement”

Trigonometry

A

B

COpposite leg

Adj

acen

t leg

Hypotenuse

Sides are named relative to the acute angle.

What never changes?

The Tangent Ratio

The tangent of an acute angle is defined as the ratio of the length of the opposite leg divided by the adjacent leg of the right triangle.

Tangent LA =

Adj

OppTan A

length of opposite leg

length of adjacent leg

C

oppo

site

Adjacent A

B

Find Tan AA

BC

7

2

Tan A7

2

Find Tan BA

BC

7

2

Tan B2

7

How do we use it?

1. If we know the ratio we can use it to determine the measurement of the angle

– We either look up the value of the ratio in the book on page 311

– Or we use a scientific calculator by entering the ratio and then pressing inverse TAN (TAN-1)

Find AA

BC

7

2B

Tan A7

2

Tan A ≈ .2857

- pg. 311

-.2857 (TAN-1)

AA

BC

7

2

16

Find BA

BC

7

2

74B

A 28

A

B C

17

8

Find A

How do we use it?

2. If we know the angle degree measure we can use it to find a missing side length – Look it up in the table (pg. 311) by finding the

degree and then looking under Tangent – Or we use scientific calculator by entering the degree

measure and then pressing TAN

Find the value of x to the nearest tenth

35º10

xTan 35º

10

x

.7002 10

x

0.7x

Find the value of x to the nearest tenth

21º

30

x

1.78x

Find the measure of angle y

yº

8

5

58y

Page 306#7

Brightstorm

Find the value of x to the nearest tenth

X

20

24º

9.8x

Find the measurement of angle x

Xº

68

10

37x

ON PG. 311…

WHY IS THE TANGENT RATIO FOR 45◦ 1.000?

WHY IS THE TANGENT RATIO FOR 60◦ 1.7321?

WARM-UP

8.6 The Sine and Cosine Ratios

Objectives

• Define the sine and cosine ratio

Sine and Cosine Ratios

• Both of these ratios involve the length of the hypotenuse

The Cosine RatioThe cosine of an acute angle is defined as the ratio

of the length of the adjacent leg to the hypotenuse of the right triangle.

Cosine LA =

Hyp

AdjCos A

length of adjacent leg

length of hypotenuse

C

oppo

site

Adjacent A

BHypotenuse

Find Cos AA

BC

15

12

Cos A15

99

A

BC

15

12

Cos A15

9

9

A ≈ 53▫

cos A ≈ .6

- pg. 311

-.3 (COS-1)

Find A

The Sine RatioThe sine of an acute angle is defined as the ratio of

the length of the opposite leg to the hypotenuse of the right triangle.

Sine LA =

Hyp

oppsin A

length of opposite leg

length of hypotenuse

C

oppo

site

Adjacent A

BHypotenuse

Find Sin A

A

BC

15

12

Sin A15

129

sin A15

12

A ≈ 53▫

sin A ≈ .8

- pg. 311

-.3 (SIN-1)

Find A using sine

A

BC

15

12

9

SOH-CAH-TOASineOppositeHypotenuseCosineAdjacentHypotenuseTangentOppositeAdjacent

• Some Old Horse Caught Another Horse Taking Oats Away.

• Sally Often Hears Cats Answer Her Telephone on Afternoons

• Sally Owns Horrible Cats And Hits Them On Accident.

With a partner try to come up

with a new saying.

SOHCAHTOA

So which one do I use?

• Sin

• Cos

• Tan

Label your sides and see which ratio you can use. Sometimes you can use more than one, so just choose one.

Whiteboards

• Page 313 – #7, 9

White boards - Example 2• Find xº correct to the nearest degree.

xº

30

18

x ≈ 37º

White Board

• An isosceles triangle has sides 8, 8, and 6. Find the length of the altitude from angle C to side AB.

• √55 ≈ 7.4

Brightstorm

Brightstorm

8.7 Applications of Right Triangle Trigonometry

Objectives

• Apply the trigonometric ratios to solve problems

• Every problem involves a diagram of a right triangle

An operator at the top of a lighthouse sees a sailboat

with an angle of depression of 2º

Angle of depression

Angle of elevation

Angle of depression = Angle of elevation

2º

2º

Horizontal

Horizontal

An operator at the top of a lighthouse (25m) sees a

Sailboat with an angle of depression of 2º. How far away is the boat?

Distance to light house (X)

2º

2º

Horizontal

25m

X ≈ 716m 88º

88º

Example 1

• You are flying a kite is flying at an angle of elevation of 40º. All 80 m of string have been let out. Ignoring the sag in the string, find the height of the kite to the nearest 10m.

• How would I label this diagram using these terms..

• Kite, yourself, height (h) , angle of elev.,

• 80m

WHITE BOARDS

• A kite is flying at an angle of elevation of 40º. All 80 m of string have been let out. Ignoring the sag in the string, find the height of the kite to the nearest 10m.

40º

80 x

8040

xSin

806428.

x

x4.51

WHITE BOARDS• An observer located 3 km from a rocket

launch site sees a rocket at an angle of elevation of 38º. How high is the rocket?

• Use the right triangle to first correctly label the diagram!!

Example

• An observer located 3 km from a rocket launch site sees a rocket at an angle of elevation of 38º. How high is the rocket?

38º3km

x

338

xTan

37813.

x

x34.2

• http://www.brightstorm.com/math/geometry/basic-trigonometry/angle-of-elevation-and-depression-problem-1/

Grade

• Incline of a driveway or a road

• Grade = Tangent

Example

• A driveway has a 15% grade– What is the angle of elevation?

xº

Example

• Tan = 15%

• Tan xº = .15

xº

Example

• Tan = 15%

• Tan xº = .15

9º

Example

• If the driveway is 12m long, about how much does it rise?

9º

12 x

Example

• If the driveway is 12m long, about how much does it rise?

9º

12 1.8