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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