Geometry CP Lesson 8



Geometry CP Lesson 8.2 – Pythagorean Theorem and its Converse Page 1 of 2

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems.

CA Geometry Standard: 12, 14, 15

Historical Background

▪ Pythagoras was a Greek mathematician born around 569 BC and died around 475 BC. Although he is very famous, we know little about his achievements. He was the leader of a society, half religious and half scientific, that followed a code of secrecy. His followers were known as mathematikoi. One thing that we do know is that they discovered what is known as the Pythagorean Theorem.

← Key Concept: The Pythagorean Theorem

In a right triangle, where the legs are length a and b, and the

hypotenuse is length c, the following equation is true:

___________________________

A proof of the Pythagorean Theorem:

Examples: Use the Pythagorean Theorem to solve for each variable.

← Key Concept: The Converse of the Pythagorean Theorem

In a triangle where c is the longest side, if [pic], then it is a right triangle.

Geometry CP Lesson 8.2 – Pythagorean Theorem and its Converse Page 2 of 2

← Key Concept: A Pythagorean Triple is three whole numbers that satisfy the Pythagorean Theorem equation. List some common Pythagorean Triples: ______________________________________

Example: Determine whether each set of measures can be those of a right triangle. Then state if they form a Pythagorean Triple.

9, 40, 41 7, 28, 29 [pic]

Example: Determine whether (HEY is a right triangle if its vertices are: H(2, 7) E(3, 6) Y(-4, -1)

Example: An airplane lands at an airport 60 miles east and 25 miles north of where it took off. How far apart are the two airports?

← Key Concept: Pythagorean Inequalities

If [pic], then (ABC is _________________________

If [pic], then (ABC is __________________________

Example: Acute, obtuse, right or not a triangle?

4, 9, 12 8, 15, 23

Geometry CP Lesson 8.3 – Special Right Triangles Page 1 of 2

Objective: Use the properties of special right triangles to solve problems.

CA Geometry Standard: 20

There are two types of “special right triangles” in math.

▪ 45(-45(-90( Triangles ( 30(-60(-90( Triangles

These special triangles have properties that make finding missing sides a lot quicker! Memorize these properties and they will save you lots of time. You’ll have less headaches and have a much happier life in general (.

← Key Concept: The Properties of a 45(-45(-90( Triangle

In a 45-45-90 triangle, the hypotenuse is ______ times as long as each leg.

Examples: Solve for each variable.

Geometry CP Lesson 8.3 – Special Right Triangles Page 2 of 2

← Key Concept: The Properties of a 30(-60(-90( Triangle

In a 30-60-90 triangle, the hypotenuse is __________ as long as the shorter leg and

the longer leg is ______ times as long as the shorter leg.

Examples: Solve for each variable.

Geometry CP Lesson 8-4: Trigonometry Page 1 of 3

Lesson objective: Find trigonometric ratios using right triangles.

CA Geometry Standard: 18, 19

▪ Trigonometry was developed for use by astronomers and surveyors to calculate distance or height.

▪ A ratio of the lengths of sides of a right triangle is called a trigonometric ratio. The three most common trig ratios are: _____________________________________

▪ Definitions for trig ratios in a right triangle.

o sin (X = ——————————————

o cos (X = ——————————————

o tan (X = ——————————————

▪ In (ABC, what are the trig ratios for (A?

o sin (A = ______ cos (A = ______ tan (A = ______

▪ In (ABC, what are the trig ratios for (C?

o sin (C = ______ cos (C = ______ tan (C = ______

▪ Example 1: Find the sin, cos, and tan ratios for (D and (F

sin (D = ______ cos (D = ______ tan (D = ______

sin (F = ______ cos (F = ______ tan (F = ______

▪ Example 2: Find the missing sides, then find the trig ratios for each acute angle:

sin 60( = ______ cos 60( = ______ tan 60( = ______

sin 30( = ______ cos 30( = ______ tan 30( = ______

Use a scientific calculator (must be in DEGREE mode), to find these values rounded to 4 decimal places. Compare them to the values you determined above.

o sin 60( = _________ cos 60( = _________ tan 60( = _________

o sin 30( = _________ cos 30( = _________ tan 30( = _________

Geometry CP Lesson 8-4: Trigonometry Page 2 of 3

▪ Example 3:

Find ratios: sin (E = ___________ cos (E = ___________ tan (E = _______

Use calculator: sin 22.62( = _________ cos 22.62( = _________ tan 22.62( = _________

▪ Example 4:

Find ratios: sin (X = __________ cos (X = _________ tan (X = _________

Use calculator: sin 45( = _________ cos 45( = _________ tan 45( = _________

▪ Solving Trig Equations

o Step 1: Identify the “players” (Hyp? Opp? Adj?)

o Step 2: Identify the trig function that applies to the “players” (SOH? CAH? TOA?)

o Step 3: Set up an equation and solve

Example 5: Example 6: Example 7:

o Example 8: Example 9:

Geometry CP Lesson 8-4: Trigonometry Page 3 of 3

← Inverse Trigonometric Functions

o Inverse trig functions are used to find the measure of an angle. This can only be done using your calculator.

sin (X = [pic] ( m(X = [pic]

cos (X = [pic] ( m(X = [pic]

tan (X = [pic] ( m(X = [pic]

Examples: Find the measure of each angle

sin (W = [pic] ( m(W = ____________

cos (X = [pic] ( m(X = ____________

tan (Y = 1.5 ( m(Y = ____________

sin (Z = [pic] ( m(Z = ____________

← Solving for angles using trig equations.

Step 1: Identify the “players” (Hyp? Opp? Adj?)

Step 2: Identify the trig function that applies to the “players” (SOH? CAH? TOA?)

Step 3: Set up an equation, use an inverse trig function to solve

A 60-foot ramp rises from the first floor to the second floor of a parking garage. The second floor is 15.5 feet above the second floor. What angle does the ramp make with the first floor?

Lesson objective: Solve problems involving elevation and depression angles.

CA Geometry Standard: 19

[pic]

[pic]

-----------------------

a

c

b

A

C

B

a

b

c

y

x

m

3

4

y

1

[pic]

3

2

x

60(

30(

short leg

long leg

hypotenuse

45(

45(

hypotenuse

legs are congruent

45(

45(

45(

45(

45(

x

[pic]

15

w

45(

45(

10

h

k

45(

[pic]

45(

10

y

14

m

60(

30(

60(

30(

60(

30(

60(

a

h

18

60(

30(

y

x

12

60(

30(

h

b

10

The perimeter of an equilateral triangle is 39 cm. Find the length of an altitude of the triangle.

30(

40 ft

A 40 foot long escalator rises from the first floor to the second floor of a shopping mall. The escalator makes a 30( angle with the horizontal. How high above the first floor is the second floor?

These trig ratios ONLY apply to the acute angles of a right triangle.

SOH CAH TOA

A

B

C

a

c

b

D

E

6

F

10

8

4

30(

60(

5

13

H

Y

E

22.62(

6

6

X

75(

8

x

50(

x

10

35(

12

x

7

48(

x

4

55(

x

6

14

x(

9

x(

13

18

15

x(

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