Unit 8 Right Triangles

1
Unit 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.

2
Lecture 1 (8-1)

• 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.

3
The Geometric Mean

• x is the geometric mean between a and b if

4
Example
What is the geometric mean between 3 and 6?
5
Simplifying Radical Expressions

• No perfect squares under the radical
• No fractions under the radical
• No radicals in the denominator

6
Theorem 8-1

• 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.

a
b
h
m
n
7
Corollary 1

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the length of the altitude
  is the geometric mean between the segments on the
  hypotenuse.

h
m
n
8
Corollary 2

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the length of each leg is
  the geometric mean between the hypotenuse and the
  segment on the hypotenuse closest to that leg.

a
b
m
n
9
Construction 14
Given two segments, construct their geometric
mean. Given Construct Steps
10
Homework Set 8.1

• 8-1 1-39
• WS Construction 14

11
Lecture 2 (8-2)

• Objectives
• State and apply the Pythagorean Theorem.
• Examine two proofs of the Pythagorean Theorem.
• Determine several sets of Pythagorean numbers.

12
Th. 8-2 The Pythagorean Th.

• The sum of the squares of the lengths of the legs
  of a right triangle is equal to the square of the
  length of the hypotenuse.

c
a
b
Proof
13
Pythagorean Sets

• A set of numbers is considered to be Pythagorean
  if they satisfy the Pythagorean Theorem.
• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

See more sets
14
Homework Set 8.2

• 8-2 1-29
• WS PM 43

15
Lecture 3 (8-3)

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

16
Th 8-3

• 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.

c
a
b
17
Th 8-4

• 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.

c
a
b
18
Th 8-5

• 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.

c
a
b
Sketch
19
Homework Set 8.3

• 8-3 1-17

20
Lecture 4 (8-4)

• Objectives
• Use the ratios of the sides of special right
  triangles

21
Theorem 8-6 (45-45-90)
45
a
a
45
Sketch
22
Theorem 8-7 (30-60-90)
60
2a
a
30
Sketch
23
Homework Set 8.4

• WS PM 44
• 8-4 1-29
• Quiz

24
Lecture 5 (8-5)

• Objectives
• Define the tangent ratio for a right triangle

25
Trigonometry
B
Sides are named relative to the acute angle.
Hypotenuse
Opposite side
A
C
Adjacent side
26
Trigonometry
B
Sides are named relative to the acute angle.
Hypotenuse
Adjacent side
A
C
Opposite side
27
The Tangent Ratio

• The tangent of an acute angle is defined as the
  ratio of the length of the opposite side to the
  adjacent side of the right triangle that contains
  the acute angle.

Sketch
28
Homework Set 8.5

• 8-5 1-23

29
Lecture 6 (8-6)

• Objectives
• Define the sine and cosine ratio

30
The Cosine Ratio

• The cosine of an acute angle is defined as the
  ratio of the length of the adjacent side to the
  hypotenuse of the right triangle that contains
  the acute angle.

Sketch
31
The Sine Ratio

• The sine of an acute angle is defined as the
  ratio of the length of the opposite side to the
  hypotenuse of the right triangle that contains
  the acute angle.

Sketch
32
Homework Set 8.6

• 8-6 1-17
• WS PM 45

33
Lecture 7 (8-7)

• Objectives
• Apply the trigonometric ratios to solve problems

34
Problem Solving

• Read the question.
• Assign a variable.
• Write the equation.
• Solve the equation.
• Write your answer.

35
Homework Set 8.7

• 8-7 1-9 all
• WS PM 46