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
Lecture 2 (8-2)

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

10
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
11
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
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Lecture 3 (8-3)

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

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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
14
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
15
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
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Lecture 4 (8-4)

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

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Theorem 8-6 (45-45-90)
45
a
a
45
Sketch
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Theorem 8-7 (30-60-90)
60
2a
a
30
Sketch
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Lecture 5 (8-5)

• Objectives
• Define the tangent ratio for a right triangle

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Trigonometry
B
Sides are named relative to the acute angle.
Hypotenuse
Opposite side
A
C
Adjacent side
21
Trigonometry
B
Sides are named relative to the acute angle.
Hypotenuse
Adjacent side
A
C
Opposite side
22
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
23
Lecture 6 (8-6)

• Objectives
• Define the sine and cosine ratio

24
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
25
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
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Lecture 7 (8-7)

• Objectives
• Apply the trigonometric ratios to solve problems

27
Problem Solving

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