Geometry Chapter 9 “Right Triangles and Trigonometry”

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Page Problems

1 9.1 Similar Right Triangles 531 11-15, 17-29 odd, 32

2 9.2 The Pythagorean Theorem 538 7 – 15 odd, 19-23 odd,31

Worksheet 9.2

3 9.3 The Converse of the Pythagorean Theorem

*Need NSPIRE 546 9-25 odd, and on page 549 Quiz 1:1-8

4 QUIZ (9.1-9.3) TBD

5 9.4 Special Right Triangles 554 9-23odd

6 9.4 Special Right Triangles 555 24 – 34, 37, 38, and 47-49, WS

7 Semester 2 Cumulative Review Review

8 Target Test (C8 – C9.4)

9 9.5 Trigonometric Ratios 562 11 – 43 odd

10 9.5 Trigonometric Ratios 562 10 – 42 even, 53, 54, 56, and Quiz 2: 1 – 7

11 9.6 Trigonometric Applications 570 11 – 27 odd

12 9.6 Solving Right Triangles

570 22 – 26 even, 29 – 33 odd, 38, 39

13 QUIZ (9.5-9.6) Chapter 9 Review Packet

14 Chapter 9 Review 582 1 – 4, 6, and 8 - 21

15 Chapter 9 Test

~ ~

A 18

5

24

x

C

B D

9.1 Similar Right Triangles

Learning Targets: Solve problems involving similar right triangles formed by the altitude drawn to the

hypotenuse of a right triangle.

Use the geometric mean to solve problems.

THEOREM

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.

∆_______∼∆_______~∆________

Write similarity statements for the three triangles in each diagram and then find the indicated measure.

ex1) Find AD ex 2) Find NQ

Redraw triangles Redraw the triangles

∆_______∼∆_______~∆________ ∆_______∼∆_______~∆________

Ex 3) A ramp has a cross section that is a right triangle. The diagram shows the approximate dimensions of

this cross section. Find the height of the ramp.

A D

C

B

4.7

ft. h

11 ft.

11.7 ft.

10 26

M

Q

P

N

y

3

6 x

4 9

y

9

6

x

3 12

4

z

z

6

5

13 z

ex 4) ex 5)

ex 6) ex 4)

ex 5) ex 6)

3 c

4

13

5

b 5

9.2 The Pythagorean Theorem

LEARNING TARGETS: Find the unknown side lengths using the Pythagorean Theorem.

Use families of right triangles to solve for missing lengths.

PYTHAGOREAN THEOREM

In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of

the hypotenuse. (The hypotenuse is always the longest side, across from the right angle and is “c”.

_______+_______=________

A Pythagorean Triple is a set of three positive integers, such that a2 + b

2 = c

2.

ex1) ex 2)

The most commonly used Pythagorean Triples are:

( ________________ ) ( ________________ ) ( ________________ ) ( ________________ )

THESE SHOULD BE MEMORIZED.

Pythagorean Triples that are multiples of other Pythagorean triples are said to be in the same family.

Use the families of Pythagorean triples to find the lengths of right triangles.

Steps: 1) Look for a common factor, or denominator.

2) Rewrite in factored (or fraction) form.

3) Determine the family.

4) Use the pattern (multiple) to find the third side.

Ex 3) Find x. ex 4) Find x.

35 x

21

3

x

11

4

a

c

b

.7 2.5

x

x 45

51

3 4

x

List the 4 families: , , ,

Ex 5) Solve for the missing side. Ex 6) Solve for the missing side.

family= family=

side = side=

scale factor= scale factor=

Ex 7) Solve for the missing side.

family=

side=

scale factor=

9.3 The Converse of the Pythagorean Theorem

LEARNING TARGET: Use side lengths to classify triangles by their angle measures.

Warm-up

Ex 1: Ex 2: Find the value of x. Find the value of x.

NSPIRE:

1) START A NEW PAGE (NEW DOC, ADD GEOMETRY)

2) MAKE A TRIANGLE (MENU, SHAPES, TRIANGLE, ENTER TO SET A POINT, HIT ESC WHEN

DONE)

3) MEASURE EACH SIDE (MENU, MEASUREMENT, LENGTH, TAB TO SIDE, ENTER, ESC)

4) MEASURE EACH ANGLE (MENU, MEASUREMENT, ANGLE, SELECT 3 POINTS BUT

VERTEX MUST BE 2ND, ENTER, ESC)

5) MAKE A TEXT BOX (MENU, ACTIONS, TEXT, ENTER, 2 2a b , ESC)

6) MAKE ANOTHER TEXT BOX (MENU, ACTIONS, TEXT, ENTER, 2c , ESC)

7) CALCULATE THE VALUES OF EACH TEXT BOX (MENU, ACTIONS, CALCULATE, CLICK

ON THE TEXT BOX, SELECT THE SIDE YOU WANT)

8) DRAG ONE POINT OF THE TRIANGLE AND MOVE IT AROUND

(MOVE CURSOR TO A POINT, CLICK & HOLD UNTIL HAND SQUEEZES THE POINT)

WHAT TO LOOK FOR: 2 2 2a b c ? WHAT TYPE OF TRIANGLE IS THAT?

2 2 2a b c ? WHAT TYPE OF TRIANGLE IS THAT?

x 8.5

4

26

x

24

Family:

Side:

Scale Factor:

Family:

Side:

Scale Factor:

CLASSIFYING TRIANGLES

Theorems

1) If c2 = a

2 + b

2, then ABC is

2) If c2 < a

2 + b

2, then ABC is

3) If c2 > a

2 + b

2, then ABC is

Ex 1 Determine the type of triangle formed by the given sides (MAKE SURE IT CAN BE A TRIANGLE FIRST!!!)

Remember: If a triangle can be formed, the sum of the two shortest sides must be greater than the longest side.

a. side lengths: 13, 5, 9 b. side lengths: 5, 11, 6 c. side lengths: 10, 7, 8

9.4 Special Right Triangles LEARNING TARGET: Find the side lengths of special right triangles.

Given the equilateral triangle shown, find the length of the altitude.

Theorem:

In a 30o – 60

o – 90

o triangle, the sides are in the ratio of x : 3x : 2x

Ex1) Find the missing sides Ex2) Find the missing sides

Ex3) Find the missing sides

2x

30o

60o

30o

7 3 30

o

9

60o

9

Given the square shown, find the length of the diagonal.

Theorem:

In a 45o – 45

o – 90

o triangle, the sides are in the ratio of x : x : 2x

Ex4) Find the diagonal of a square with ex 5) Find the missing sides.

perimeter = 28

ex 6) Find the missing sides.

x

45o

45o

11

26

A 1 C

B

60o

9.5 Trigonometric Ratios LEARNING TARGETS: Find the sine, the cosine, and the tangent of an acute angle.

Use trig ratios to solve real life problems.

What is trigonometry? It comes from Greek and means triangle measurement. Basically, it is the study of

relationships between angles and sides of a right triangle.

∆ABC is a right triangle.

Begin by naming the sides relative to angle A.

Opposite from angle A = o

Adjacent to angle A = a

Hypotenuse = h

Using these terms we are introduced to the 3 fundamental ratios of trigonometry.

Sine- sinA =

Cosine- cosA =

Tangent- tanA =

Ex 1) Using the given triangle find the following:

sinA = _______=________ sinB = _______=________

cosA =_______=________ cosB = _______=________

tanA =_______=________ tanB = _______=________

Ex 2) Find the sine, the cosine, and the tangent of 60o.

sin 60o =

cos60o =

tan60o =

Use calculator.

sin 60o = cos60

o = tan60

o =

A 8 C

B

10

A

a

9 b

47o

Ex 3) Find the value of each variable. *Hint- label sides relative to angle shown.

a) b)

Ex 4) If 7

sin25

D , what is the cos∠D?

Ex 5) Use a calculator to find the following trig values to 4 decimal places.

a) sin21o= b) tan52

o= c) cos60

o =

ANGLE OF ELEVATION AND ANGLE OF DEPRESSION

Ex 6) You are measuring the height of a flag pole. You stand 19 ft away from the base of the pole. You

measure an angle of elevation from a point on the ground to the top of the pole to be 64o. Estimate the height of

the pole.

78o

n

m 14

4

8

A C

B

9.6 Solving Right Triangles Day 1

LEARNING TARGETS: Solve right triangles.

Use right triangles for real-life applications.

To solve a right triangle means to determine the measures of all and the lengths of all .

This chapter we have practiced:

Pythagorean Theorem Triples Special triangles Trig Functions

To solve a right triangle, you need to know:

ONE and ONE OR 2

Trig functions can be used to find angles as well as sides.

INVERSE TRIG FUNCTIONS:

(used to find the angle measure)

If sin∠A = x, then sin-1

x = A

If cos∠A = x, then cos-1

x = A

If tan∠A = x, then tan-1

x = A

Ex 1) Use your calculator to find the measure of these angles to the nearest degree.

a) sin∠A = .6947 b. cos∠A = .8988 c. tan∠A = 28.636 d. sin∠A = 5

13

Ex 2) Solve the right triangle. Ex 3) Solve the right triangle.

D

E F

e f

8 32°

9.6 Solving Right Triangles Day 2

WARM UP: Solve the right triangle when given a side length and an acute angle.

a. b.

Ex 1) When a hockey player is 35 feet from the goal line, he shoots the puck directly at the goal. The

angle of elevation at which the puck leaves the ice is 7°. The height of the goal is 4 feet. Will

the player score a goal?

Ex 2) During a flight, a hot air balloon is observed by two persons standing at points A and B as illustrated

in the diagram. The angle of elevation of point A is 28o. Point A is 1.8 miles from the balloon as

measured along the ground.

a) What is the height x of the balloon?

b) Point B is 2.8 miles from point A. Find the angle of elevation of point B.

35 ft

7o

D

E F

e f

8 32°

D

E F

e 30

d

53°

B A

x