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Mary Pyo

Mary Pyo

Geometric Mean

Slide Duration:

Table of Contents

I. Tools of Geometry
Coordinate Plane

16m 41s

Intro
0:00
The Coordinate System
0:12
Coordinate Plane: X-axis and Y-axis
0:15
Quadrants
1:02
Origin
2:00
Ordered Pair
2:17
Coordinate Plane
2:59
Example: Writing Coordinates
3:01
Coordinate Plane, cont.
4:15
Example: Graphing & Coordinate Plane
4:17
Collinear
5:58
Extra Example 1: Writing Coordinates & Quadrants
7:34
Extra Example 2: Quadrants
8:52
Extra Example 3: Graphing & Coordinate Plane
10:58
Extra Example 4: Collinear
12:50
Points, Lines and Planes

17m 17s

Intro
0:00
Points
0:07
Definition and Example of Points
0:09
Lines
0:50
Definition and Example of Lines
0:51
Planes
2:59
Definition and Example of Planes
3:00
Drawing and Labeling
4:40
Example 1: Drawing and Labeling
4:41
Example 2: Drawing and Labeling
5:54
Example 3: Drawing and Labeling
6:41
Example 4: Drawing and Labeling
8:23
Extra Example 1: Points, Lines and Planes
10:19
Extra Example 2: Naming Figures
11:16
Extra Example 3: Points, Lines and Planes
12:35
Extra Example 4: Draw and Label
14:44
Measuring Segments

31m 31s

Intro
0:00
Segments
0:06
Examples of Segments
0:08
Ruler Postulate
1:30
Ruler Postulate
1:31
Segment Addition Postulate
5:02
Example and Definition of Segment Addition Postulate
5:03
Segment Addition Postulate
8:01
Example 1: Segment Addition Postulate
8:04
Example 2: Segment Addition Postulate
11:15
Pythagorean Theorem
12:36
Definition of Pythagorean Theorem
12:37
Pythagorean Theorem, cont.
15:49
Example: Pythagorean Theorem
15:50
Distance Formula
16:48
Example and Definition of Distance Formula
16:49
Extra Example 1: Find Each Measure
20:32
Extra Example 2: Find the Missing Measure
22:11
Extra Example 3: Find the Distance Between the Two Points
25:36
Extra Example 4: Pythagorean Theorem
29:33
Midpoints and Segment Congruence

42m 26s

Intro
0:00
Definition of Midpoint
0:07
Midpoint
0:10
Midpoint Formulas
1:30
Midpoint Formula: On a Number Line
1:45
Midpoint Formula: In a Coordinate Plane
2:50
Midpoint
4:40
Example: Midpoint on a Number Line
4:43
Midpoint
6:05
Example: Midpoint in a Coordinate Plane
6:06
Midpoint
8:28
Example 1
8:30
Example 2
13:01
Segment Bisector
15:14
Definition and Example of Segment Bisector
15:15
Proofs
17:27
Theorem
17:53
Proof
18:21
Midpoint Theorem
19:37
Example: Proof & Midpoint Theorem
19:38
Extra Example 1: Midpoint on a Number Line
23:44
Extra Example 2: Drawing Diagrams
26:25
Extra Example 3: Midpoint
29:14
Extra Example 4: Segment Bisector
33:21
Angles

42m 34s

Intro
0:00
Angles
0:05
Angle
0:07
Ray
0:23
Opposite Rays
2:09
Angles
3:22
Example: Naming Angle
3:23
Angles
6:39
Interior, Exterior, Angle
6:40
Measure and Degrees
7:38
Protractor Postulate
8:37
Example: Protractor Postulate
8:38
Angle Addition Postulate
11:41
Example: Angle addition Postulate
11:42
Classifying Angles
14:10
Acute Angle
14:16
Right Angles
14:30
Obtuse Angle
14:41
Angle Bisector
15:02
Example: Angle Bisector
15:04
Angle Relationships
16:43
Adjacent Angles
16:47
Vertical Angles
17:49
Linear Pair
19:40
Angle Relationships
20:31
Right Angles
20:32
Supplementary Angles
21:15
Complementary Angles
21:33
Extra Example 1: Angles
24:08
Extra Example 2: Angles
29:06
Extra Example 3: Angles
32:05
Extra Example 4 Angles
35:44
II. Reasoning & Proof
Inductive Reasoning

19m

Intro
0:00
Inductive Reasoning
0:05
Conjecture
0:06
Inductive Reasoning
0:15
Examples
0:55
Example: Sequence
0:56
More Example: Sequence
2:00
Using Inductive Reasoning
2:50
Example: Conjecture
2:51
More Example: Conjecture
3:48
Counterexamples
4:56
Counterexample
4:58
Extra Example 1: Conjecture
6:59
Extra Example 2: Sequence and Pattern
10:20
Extra Example 3: Inductive Reasoning
12:46
Extra Example 4: Conjecture and Counterexample
15:17
Conditional Statements

42m 47s

Intro
0:00
If Then Statements
0:05
If Then Statements
0:06
Other Forms
2:29
Example: Without Then
2:40
Example: Using When
3:03
Example: Hypothesis
3:24
Identify the Hypothesis and Conclusion
3:52
Example 1: Hypothesis and Conclusion
3:58
Example 2: Hypothesis and Conclusion
4:31
Example 3: Hypothesis and Conclusion
5:38
Write in If Then Form
6:16
Example 1: Write in If Then Form
6:23
Example 2: Write in If Then Form
6:57
Example 3: Write in If Then Form
7:39
Other Statements
8:40
Other Statements
8:41
Converse Statements
9:18
Converse Statements
9:20
Converses and Counterexamples
11:04
Converses and Counterexamples
11:05
Example 1: Converses and Counterexamples
12:02
Example 2: Converses and Counterexamples
15:10
Example 3: Converses and Counterexamples
17:08
Inverse Statement
19:58
Definition and Example
19:59
Inverse Statement
21:46
Example 1: Inverse and Counterexample
21:47
Example 2: Inverse and Counterexample
23:34
Contrapositive Statement
25:20
Definition and Example
25:21
Contrapositive Statement
26:58
Example: Contrapositive Statement
27:00
Summary
29:03
Summary of Lesson
29:04
Extra Example 1: Hypothesis and Conclusion
32:20
Extra Example 2: If-Then Form
33:23
Extra Example 3: Converse, Inverse, and Contrapositive
34:54
Extra Example 4: Converse, Inverse, and Contrapositive
37:56
Point, Line, and Plane Postulates

17m 24s

Intro
0:00
What are Postulates?
0:09
Definition of Postulates
0:10
Postulates
1:22
Postulate 1: Two Points
1:23
Postulate 2: Three Points
2:02
Postulate 3: Line
2:45
Postulates, cont..
3:08
Postulate 4: Plane
3:09
Postulate 5: Two Points in a Plane
3:53
Postulates, cont..
4:46
Postulate 6: Two Lines Intersect
4:47
Postulate 7: Two Plane Intersect
5:28
Using the Postulates
6:34
Examples: True or False
6:35
Using the Postulates
10:18
Examples: True or False
10:19
Extra Example 1: Always, Sometimes, or Never
12:22
Extra Example 2: Always, Sometimes, or Never
13:15
Extra Example 3: Always, Sometimes, or Never
14:16
Extra Example 4: Always, Sometimes, or Never
15:03
Deductive Reasoning

36m 3s

Intro
0:00
Deductive Reasoning
0:06
Definition of Deductive Reasoning
0:07
Inductive vs. Deductive
2:51
Inductive Reasoning
2:52
Deductive reasoning
3:19
Law of Detachment
3:47
Law of Detachment
3:48
Examples of Law of Detachment
4:31
Law of Syllogism
7:32
Law of Syllogism
7:33
Example 1: Making a Conclusion
9:02
Example 2: Making a Conclusion
12:54
Using Laws of Logic
14:12
Example 1: Determine the Logic
14:42
Example 2: Determine the Logic
17:02
Using Laws of Logic, cont.
18:47
Example 3: Determine the Logic
19:03
Example 4: Determine the Logic
20:56
Extra Example 1: Determine the Conclusion and Law
22:12
Extra Example 2: Determine the Conclusion and Law
25:39
Extra Example 3: Determine the Logic and Law
29:50
Extra Example 4: Determine the Logic and Law
31:27
Proofs in Algebra: Properties of Equality

44m 31s

Intro
0:00
Properties of Equality
0:10
Addition Property of Equality
0:28
Subtraction Property of Equality
1:10
Multiplication Property of Equality
1:41
Division Property of Equality
1:55
Addition Property of Equality Using Angles
2:46
Properties of Equality, cont.
4:10
Reflexive Property of Equality
4:11
Symmetric Property of Equality
5:24
Transitive Property of Equality
6:10
Properties of Equality, cont.
7:04
Substitution Property of Equality
7:05
Distributive Property of Equality
8:34
Two Column Proof
9:40
Example: Two Column Proof
9:46
Proof Example 1
16:13
Proof Example 2
23:49
Proof Example 3
30:33
Extra Example 1: Name the Property of Equality
38:07
Extra Example 2: Name the Property of Equality
40:16
Extra Example 3: Name the Property of Equality
41:35
Extra Example 4: Name the Property of Equality
43:02
Proving Segment Relationship

41m 2s

Intro
0:00
Good Proofs
0:12
Five Essential Parts
0:13
Proof Reasons
1:38
Undefined
1:40
Definitions
2:06
Postulates
2:42
Previously Proven Theorems
3:24
Congruence of Segments
4:10
Theorem: Congruence of Segments
4:12
Proof Example
10:16
Proof: Congruence of Segments
10:17
Setting Up Proofs
19:13
Example: Two Segments with Equal Measures
19:15
Setting Up Proofs
21:48
Example: Vertical Angles are Congruent
21:50
Setting Up Proofs
23:59
Example: Segment of a Triangle
24:00
Extra Example 1: Congruence of Segments
27:03
Extra Example 2: Setting Up Proofs
28:50
Extra Example 3: Setting Up Proofs
30:55
Extra Example 4: Two-Column Proof
33:11
Proving Angle Relationships

33m 37s

Intro
0:00
Supplement Theorem
0:05
Supplementary Angles
0:06
Congruence of Angles
2:37
Proof: Congruence of Angles
2:38
Angle Theorems
6:54
Angle Theorem 1: Supplementary Angles
6:55
Angle Theorem 2: Complementary Angles
10:25
Angle Theorems
11:32
Angle Theorem 3: Right Angles
11:35
Angle Theorem 4: Vertical Angles
12:09
Angle Theorem 5: Perpendicular Lines
12:57
Using Angle Theorems
13:45
Example 1: Always, Sometimes, or Never
13:50
Example 2: Always, Sometimes, or Never
14:28
Example 3: Always, Sometimes, or Never
16:21
Extra Example 1: Always, Sometimes, or Never
16:53
Extra Example 2: Find the Measure of Each Angle
18:55
Extra Example 3: Find the Measure of Each Angle
25:03
Extra Example 4: Two-Column Proof
27:08
III. Perpendicular & Parallel Lines
Parallel Lines and Transversals

37m 35s

Intro
0:00
Lines
0:06
Parallel Lines
0:09
Skew Lines
2:02
Transversal
3:42
Angles Formed by a Transversal
4:28
Interior Angles
5:53
Exterior Angles
6:09
Consecutive Interior Angles
7:04
Alternate Exterior Angles
9:47
Alternate Interior Angles
11:22
Corresponding Angles
12:27
Angles Formed by a Transversal
15:29
Relationship Between Angles
15:30
Extra Example 1: Intersecting, Parallel, or Skew
19:26
Extra Example 2: Draw a Diagram
21:37
Extra Example 3: Name the Figures
24:12
Extra Example 4: Angles Formed by a Transversal
28:38
Angles and Parallel Lines

41m 53s

Intro
0:00
Corresponding Angles Postulate
0:05
Corresponding Angles Postulate
0:06
Alternate Interior Angles Theorem
3:05
Alternate Interior Angles Theorem
3:07
Consecutive Interior Angles Theorem
5:16
Consecutive Interior Angles Theorem
5:17
Alternate Exterior Angles Theorem
6:42
Alternate Exterior Angles Theorem
6:43
Parallel Lines Cut by a Transversal
7:18
Example: Parallel Lines Cut by a Transversal
7:19
Perpendicular Transversal Theorem
14:54
Perpendicular Transversal Theorem
14:55
Extra Example 1: State the Postulate or Theorem
16:37
Extra Example 2: Find the Measure of the Numbered Angle
18:53
Extra Example 3: Find the Measure of Each Angle
25:13
Extra Example 4: Find the Values of x, y, and z
36:26
Slope of Lines

44m 6s

Intro
0:00
Definition of Slope
0:06
Slope Equation
0:13
Slope of a Line
3:45
Example: Find the Slope of a Line
3:47
Slope of a Line
8:38
More Example: Find the Slope of a Line
8:40
Slope Postulates
12:32
Proving Slope Postulates
12:33
Parallel or Perpendicular Lines
17:23
Example: Parallel or Perpendicular Lines
17:24
Using Slope Formula
20:02
Example: Using Slope Formula
20:03
Extra Example 1: Slope of a Line
25:10
Extra Example 2: Slope of a Line
26:31
Extra Example 3: Graph the Line
34:11
Extra Example 4: Using the Slope Formula
38:50
Proving Lines Parallel

25m 55s

Intro
0:00
Postulates
0:06
Postulate 1: Parallel Lines
0:21
Postulate 2: Parallel Lines
2:16
Parallel Postulate
3:28
Definition and Example of Parallel Postulate
3:29
Theorems
4:29
Theorem 1: Parallel Lines
4:40
Theorem 2: Parallel Lines
5:37
Theorems, cont.
6:10
Theorem 3: Parallel Lines
6:11
Extra Example 1: Determine Parallel Lines
6:56
Extra Example 2: Find the Value of x
11:42
Extra Example 3: Opposite Sides are Parallel
14:48
Extra Example 4: Proving Parallel Lines
20:42
Parallels and Distance

19m 48s

Intro
0:00
Distance Between a Points and Line
0:07
Definition and Example
0:08
Distance Between Parallel Lines
1:51
Definition and Example
1:52
Extra Example 1: Drawing a Segment to Represent Distance
3:02
Extra Example 2: Drawing a Segment to Represent Distance
4:27
Extra Example 3: Graph, Plot, and Construct a Perpendicular Segment
5:13
Extra Example 4: Distance Between Two Parallel Lines
15:37
IV. Congruent Triangles
Classifying Triangles

28m 43s

Intro
0:00
Triangles
0:09
Triangle: A Three-Sided Polygon
0:10
Sides
1:00
Vertices
1:22
Angles
1:56
Classifying Triangles by Angles
2:59
Acute Triangle
3:19
Obtuse Triangle
4:08
Right Triangle
4:44
Equiangular Triangle
5:38
Definition and Example of an Equiangular Triangle
5:39
Classifying Triangles by Sides
6:57
Scalene Triangle
7:17
Isosceles Triangle
7:57
Equilateral Triangle
8:12
Isosceles Triangle
8:58
Labeling Isosceles Triangle
9:00
Labeling Right Triangle
10:44
Isosceles Triangle
11:10
Example: Find x, AB, BC, and AC
11:11
Extra Example 1: Classify Each Triangle
13:45
Extra Example 2: Always, Sometimes, or Never
16:28
Extra Example 3: Find All the Sides of the Isosceles Triangle
20:29
Extra Example 4: Distance Formula and Triangle
22:29
Measuring Angles in Triangles

44m 43s

Intro
0:00
Angle Sum Theorem
0:09
Angle Sum Theorem for Triangle
0:11
Using Angle Sum Theorem
4:06
Find the Measure of the Missing Angle
4:07
Third Angle Theorem
4:58
Example: Third Angle Theorem
4:59
Exterior Angle Theorem
7:58
Example: Exterior Angle Theorem
8:00
Flow Proof of Exterior Angle Theorem
15:14
Flow Proof of Exterior Angle Theorem
15:17
Triangle Corollaries
27:21
Triangle Corollary 1
27:50
Triangle Corollary 2
30:42
Extra Example 1: Find the Value of x
32:55
Extra Example 2: Find the Value of x
34:20
Extra Example 3: Find the Measure of the Angle
35:38
Extra Example 4: Find the Measure of Each Numbered Angle
39:00
Exploring Congruent Triangles

26m 46s

Intro
0:00
Congruent Triangles
0:15
Example of Congruent Triangles
0:17
Corresponding Parts
3:39
Corresponding Angles and Sides of Triangles
3:40
Definition of Congruent Triangles
11:24
Definition of Congruent Triangles
11:25
Triangle Congruence
16:37
Congruence of Triangles
16:38
Extra Example 1: Congruence Statement
18:24
Extra Example 2: Congruence Statement
21:26
Extra Example 3: Draw and Label the Figure
23:09
Extra Example 4: Drawing Triangles
24:04
Proving Triangles Congruent

47m 51s

Intro
0:00
SSS Postulate
0:18
Side-Side-Side Postulate
0:27
SAS Postulate
2:26
Side-Angle-Side Postulate
2:29
SAS Postulate
3:57
Proof Example
3:58
ASA Postulate
11:47
Angle-Side-Angle Postulate
11:53
AAS Theorem
14:13
Angle-Angle-Side Theorem
14:14
Methods Overview
16:16
Methods Overview
16:17
SSS
16:33
SAS
17:06
ASA
17:50
AAS
18:17
CPCTC
19:14
Extra Example 1:Proving Triangles are Congruent
21:29
Extra Example 2: Proof
25:40
Extra Example 3: Proof
30:41
Extra Example 4: Proof
38:41
Isosceles and Equilateral Triangles

27m 53s

Intro
0:00
Isosceles Triangle Theorem
0:07
Isosceles Triangle Theorem
0:09
Isosceles Triangle Theorem
2:26
Example: Using the Isosceles Triangle Theorem
2:27
Isosceles Triangle Theorem Converse
3:29
Isosceles Triangle Theorem Converse
3:30
Equilateral Triangle Theorem Corollaries
4:30
Equilateral Triangle Theorem Corollary 1
4:59
Equilateral Triangle Theorem Corollary 2
5:55
Extra Example 1: Find the Value of x
7:08
Extra Example 2: Find the Value of x
10:04
Extra Example 3: Proof
14:04
Extra Example 4: Proof
22:41
V. Triangle Inequalities
Special Segments in Triangles

43m 44s

Intro
0:00
Perpendicular Bisector
0:06
Perpendicular Bisector
0:07
Perpendicular Bisector
4:07
Perpendicular Bisector Theorems
4:08
Median
6:30
Definition of Median
6:31
Median
9:41
Example: Median
9:42
Altitude
12:22
Definition of Altitude
12:23
Angle Bisector
14:33
Definition of Angle Bisector
14:34
Angle Bisector
16:41
Angle Bisector Theorems
16:42
Special Segments Overview
18:57
Perpendicular Bisector
19:04
Median
19:32
Altitude
19:49
Angle Bisector
20:02
Examples: Special Segments
20:18
Extra Example 1: Draw and Label
22:36
Extra Example 2: Draw the Altitudes for Each Triangle
24:37
Extra Example 3: Perpendicular Bisector
27:57
Extra Example 4: Draw, Label, and Write Proof
34:33
Right Triangles

26m 34s

Intro
0:00
LL Theorem
0:21
Leg-Leg Theorem
0:25
HA Theorem
2:23
Hypotenuse-Angle Theorem
2:24
LA Theorem
4:49
Leg-Angle Theorem
4:50
LA Theorem
6:18
Example: Find x and y
6:19
HL Postulate
8:22
Hypotenuse-Leg Postulate
8:23
Extra Example 1: LA Theorem & HL Postulate
10:57
Extra Example 2: Find x So That Each Pair of Triangles is Congruent
14:15
Extra Example 3: Two-column Proof
17:02
Extra Example 4: Two-column Proof
21:01
Indirect Proofs and Inequalities

33m 30s

Intro
0:00
Writing an Indirect Proof
0:09
Step 1
0:49
Step 2
2:32
Step 3
3:00
Indirect Proof
4:30
Example: 2 + 6 = 8
5:00
Example: The Suspect is Guilty
5:40
Example: Measure of Angle A < Measure of Angle B
6:06
Definition of Inequality
7:47
Definition of Inequality & Example
7:48
Properties of Inequality
9:55
Comparison Property
9:58
Transitive Property
10:33
Addition and Subtraction Properties
12:01
Multiplication and Division Properties
13:07
Exterior Angle Inequality Theorem
14:12
Example: Exterior Angle Inequality Theorem
14:13
Extra Example 1: Draw a Diagram for the Statement
18:32
Extra Example 2: Name the Property for Each Statement
19:56
Extra Example 3: State the Assumption
21:22
Extra Example 4: Write an Indirect Proof
25:39
Inequalities for Sides and Angles of a Triangle

17m 26s

Intro
0:00
Side to Angles
0:10
If One Side of a Triangle is Longer Than Another Side
0:11
Converse: Angles to Sides
1:57
If One Angle of a Triangle Has a Greater Measure Than Another Angle
1:58
Extra Example 1: Name the Angles in the Triangle From Least to Greatest
2:38
Extra Example 2: Find the Longest and Shortest Segment in the Triangle
3:47
Extra Example 3: Angles and Sides of a Triangle
4:51
Extra Example 4: Two-column Proof
9:08
Triangle Inequality

28m 11s

Intro
0:00
Triangle Inequality Theorem
0:05
Triangle Inequality Theorem
0:06
Triangle Inequality Theorem
4:22
Example 1: Triangle Inequality Theorem
4:23
Example 2: Triangle Inequality Theorem
9:40
Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle
12:00
Extra Example 2: Finding the Third Side of a Triangle
13:34
Extra Example 3: Always True, Sometimes True, or Never True
18:18
Extra Example 4: Triangle and Vertices
22:36
Inequalities Involving Two Triangles

29m 36s

Intro
0:00
SAS Inequality Theorem
0:06
SAS Inequality Theorem & Example
0:25
SSS Inequality Theorem
4:33
SSS Inequality Theorem & Example
4:34
Extra Example 1: Write an Inequality Comparing the Segments
6:08
Extra Example 2: Determine if the Statement is True
9:52
Extra Example 3: Write an Inequality for x
14:20
Extra Example 4: Two-column Proof
17:44
VI. Quadrilaterals
Parallelograms

29m 11s

Intro
0:00
Quadrilaterals
0:06
Four-sided Polygons
0:08
Non Examples of Quadrilaterals
0:47
Parallelograms
1:35
Parallelograms
1:36
Properties of Parallelograms
4:28
Opposite Sides of a Parallelogram are Congruent
4:29
Opposite Angles of a Parallelogram are Congruent
5:49
Angles and Diagonals
6:24
Consecutive Angles in a Parallelogram are Supplementary
6:25
The Diagonals of a Parallelogram Bisect Each Other
8:42
Extra Example 1: Complete Each Statement About the Parallelogram
10:26
Extra Example 2: Find the Values of x, y, and z of the Parallelogram
13:21
Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram
16:35
Extra Example 4: Slope of Parallelogram
23:15
Proving Parallelograms

42m 43s

Intro
0:00
Parallelogram Theorems
0:09
Theorem 1
0:20
Theorem 2
1:50
Parallelogram Theorems, Cont.
3:10
Theorem 3
3:11
Theorem 4
4:15
Proving Parallelogram
6:21
Example: Determine if Quadrilateral ABCD is a Parallelogram
6:22
Summary
14:01
Both Pairs of Opposite Sides are Parallel
14:14
Both Pairs of Opposite Sides are Congruent
15:09
Both Pairs of Opposite Angles are Congruent
15:24
Diagonals Bisect Each Other
15:44
A Pair of Opposite Sides is Both Parallel and Congruent
16:13
Extra Example 1: Determine if Each Quadrilateral is a Parallelogram
16:54
Extra Example 2: Find the Value of x and y
20:23
Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram
24:05
Extra Example 4: Two-column Proof
30:28
Rectangles

29m 47s

Intro
0:00
Rectangles
0:03
Definition of Rectangles
0:04
Diagonals of Rectangles
2:52
Rectangles: Diagonals Property 1
2:53
Rectangles: Diagonals Property 2
3:30
Proving a Rectangle
4:40
Example: Determine Whether Parallelogram ABCD is a Rectangle
4:41
Rectangles Summary
9:22
Opposite Sides are Congruent and Parallel
9:40
Opposite Angles are Congruent
9:51
Consecutive Angles are Supplementary
9:58
Diagonals are Congruent and Bisect Each Other
10:05
All Four Angles are Right Angles
10:40
Extra Example 1: Find the Value of x
11:03
Extra Example 2: Name All Congruent Sides and Angles
13:52
Extra Example 3: Always, Sometimes, or Never True
19:39
Extra Example 4: Determine if ABCD is a Rectangle
26:45
Squares and Rhombi

39m 14s

Intro
0:00
Rhombus
0:09
Definition of a Rhombus
0:10
Diagonals of a Rhombus
2:03
Rhombus: Diagonals Property 1
2:21
Rhombus: Diagonals Property 2
3:49
Rhombus: Diagonals Property 3
4:36
Rhombus
6:17
Example: Use the Rhombus to Find the Missing Value
6:18
Square
8:17
Definition of a Square
8:20
Summary Chart
11:06
Parallelogram
11:07
Rectangle
12:56
Rhombus
13:54
Square
14:44
Extra Example 1: Diagonal Property
15:44
Extra Example 2: Use Rhombus ABCD to Find the Missing Value
19:39
Extra Example 3: Always, Sometimes, or Never True
23:06
Extra Example 4: Determine the Quadrilateral
28:02
Trapezoids and Kites

30m 48s

Intro
0:00
Trapezoid
0:10
Definition of Trapezoid
0:12
Isosceles Trapezoid
2:57
Base Angles of an Isosceles Trapezoid
2:58
Diagonals of an Isosceles Trapezoid
4:05
Median of a Trapezoid
4:26
Median of a Trapezoid
4:27
Median of a Trapezoid
6:41
Median Formula
7:00
Kite
8:28
Definition of a Kite
8:29
Quadrilaterals Summary
11:19
A Quadrilateral with Two Pairs of Adjacent Congruent Sides
11:20
Extra Example 1: Isosceles Trapezoid
14:50
Extra Example 2: Median of Trapezoid
18:28
Extra Example 3: Always, Sometimes, or Never
24:13
Extra Example 4: Determine if the Figure is a Trapezoid
26:49
VII. Proportions and Similarity
Using Proportions and Ratios

20m 10s

Intro
0:00
Ratio
0:05
Definition and Examples of Writing Ratio
0:06
Proportion
2:05
Definition of Proportion
2:06
Examples of Proportion
2:29
Using Ratio
5:53
Example: Ratio
5:54
Extra Example 1: Find Three Ratios Equivalent to 2/5
9:28
Extra Example 2: Proportion and Cross Products
10:32
Extra Example 3: Express Each Ratio as a Fraction
13:18
Extra Example 4: Fin the Measure of a 3:4:5 Triangle
17:26
Similar Polygons

27m 53s

Intro
0:00
Similar Polygons
0:05
Definition of Similar Polygons
0:06
Example of Similar Polygons
2:32
Scale Factor
4:26
Scale Factor: Definition and Example
4:27
Extra Example 1: Determine if Each Pair of Figures is Similar
7:03
Extra Example 2: Find the Values of x and y
11:33
Extra Example 3: Similar Triangles
19:57
Extra Example 4: Draw Two Similar Figures
23:36
Similar Triangles

34m 10s

Intro
0:00
AA Similarity
0:10
Definition of AA Similarity
0:20
Example of AA Similarity
2:32
SSS Similarity
4:46
Definition of SSS Similarity
4:47
Example of SSS Similarity
6:00
SAS Similarity
8:04
Definition of SAS Similarity
8:05
Example of SAS Similarity
9:12
Extra Example 1: Determine Whether Each Pair of Triangles is Similar
10:59
Extra Example 2: Determine Which Triangles are Similar
16:08
Extra Example 3: Determine if the Statement is True or False
23:11
Extra Example 4: Write Two-Column Proof
26:25
Parallel Lines and Proportional Parts

24m 7s

Intro
0:00
Triangle Proportionality
0:07
Definition of Triangle Proportionality
0:08
Example of Triangle Proportionality
0:51
Triangle Proportionality Converse
2:19
Triangle Proportionality Converse
2:20
Triangle Mid-segment
3:42
Triangle Mid-segment: Definition and Example
3:43
Parallel Lines and Transversal
6:51
Parallel Lines and Transversal
6:52
Extra Example 1: Complete Each Statement
8:59
Extra Example 2: Determine if the Statement is True or False
12:28
Extra Example 3: Find the Value of x and y
15:35
Extra Example 4: Find Midpoints of a Triangle
20:43
Parts of Similar Triangles

27m 6s

Intro
0:00
Proportional Perimeters
0:09
Proportional Perimeters: Definition and Example
0:10
Similar Altitudes
2:23
Similar Altitudes: Definition and Example
2:24
Similar Angle Bisectors
4:50
Similar Angle Bisectors: Definition and Example
4:51
Similar Medians
6:05
Similar Medians: Definition and Example
6:06
Angle Bisector Theorem
7:33
Angle Bisector Theorem
7:34
Extra Example 1: Parts of Similar Triangles
10:52
Extra Example 2: Parts of Similar Triangles
14:57
Extra Example 3: Parts of Similar Triangles
19:27
Extra Example 4: Find the Perimeter of Triangle ABC
23:14
VIII. Applying Right Triangles & Trigonometry
Pythagorean Theorem

21m 14s

Intro
0:00
Pythagorean Theorem
0:05
Pythagorean Theorem & Example
0:06
Pythagorean Converse
1:20
Pythagorean Converse & Example
1:21
Pythagorean Triple
2:42
Pythagorean Triple
2:43
Extra Example 1: Find the Missing Side
4:59
Extra Example 2: Determine Right Triangle
7:40
Extra Example 3: Determine Pythagorean Triple
11:30
Extra Example 4: Vertices and Right Triangle
14:29
Geometric Mean

40m 59s

Intro
0:00
Geometric Mean
0:04
Geometric Mean & Example
0:05
Similar Triangles
4:32
Similar Triangles
4:33
Geometric Mean-Altitude
11:10
Geometric Mean-Altitude & Example
11:11
Geometric Mean-Leg
14:47
Geometric Mean-Leg & Example
14:18
Extra Example 1: Geometric Mean Between Each Pair of Numbers
20:10
Extra Example 2: Similar Triangles
23:46
Extra Example 3: Geometric Mean of Triangles
28:30
Extra Example 4: Geometric Mean of Triangles
36:58
Special Right Triangles

37m 57s

Intro
0:00
45-45-90 Triangles
0:06
Definition of 45-45-90 Triangles
0:25
45-45-90 Triangles
5:51
Example: Find n
5:52
30-60-90 Triangles
8:59
Definition of 30-60-90 Triangles
9:00
30-60-90 Triangles
12:25
Example: Find n
12:26
Extra Example 1: Special Right Triangles
15:08
Extra Example 2: Special Right Triangles
18:22
Extra Example 3: Word Problems & Special Triangles
27:40
Extra Example 4: Hexagon & Special Triangles
33:51
Ratios in Right Triangles

40m 37s

Intro
0:00
Trigonometric Ratios
0:08
Definition of Trigonometry
0:13
Sine (sin), Cosine (cos), & Tangent (tan)
0:50
Trigonometric Ratios
3:04
Trig Functions
3:05
Inverse Trig Functions
5:02
SOHCAHTOA
8:16
sin x
9:07
cos x
10:00
tan x
10:32
Example: SOHCAHTOA & Triangle
12:10
Extra Example 1: Find the Value of Each Ratio or Angle Measure
14:36
Extra Example 2: Find Sin, Cos, and Tan
18:51
Extra Example 3: Find the Value of x Using SOHCAHTOA
22:55
Extra Example 4: Trigonometric Ratios in Right Triangles
32:13
Angles of Elevation and Depression

21m 4s

Intro
0:00
Angle of Elevation
0:10
Definition of Angle of Elevation & Example
0:11
Angle of Depression
1:19
Definition of Angle of Depression & Example
1:20
Extra Example 1: Name the Angle of Elevation and Depression
2:22
Extra Example 2: Word Problem & Angle of Depression
4:41
Extra Example 3: Word Problem & Angle of Elevation
14:02
Extra Example 4: Find the Missing Measure
18:10
Law of Sines

35m 25s

Intro
0:00
Law of Sines
0:20
Law of Sines
0:21
Law of Sines
3:34
Example: Find b
3:35
Solving the Triangle
9:19
Example: Using the Law of Sines to Solve Triangle
9:20
Extra Example 1: Law of Sines and Triangle
17:43
Extra Example 2: Law of Sines and Triangle
20:06
Extra Example 3: Law of Sines and Triangle
23:54
Extra Example 4: Law of Sines and Triangle
28:59
Law of Cosines

52m 43s

Intro
0:00
Law of Cosines
0:35
Law of Cosines
0:36
Law of Cosines
6:22
Use the Law of Cosines When Both are True
6:23
Law of Cosines
8:35
Example: Law of Cosines
8:36
Extra Example 1: Law of Sines or Law of Cosines?
13:35
Extra Example 2: Use the Law of Cosines to Find the Missing Measure
17:02
Extra Example 3: Solve the Triangle
30:49
Extra Example 4: Find the Measure of Each Diagonal of the Parallelogram
41:39
IX. Circles
Segments in a Circle

22m 43s

Intro
0:00
Segments in a Circle
0:10
Circle
0:11
Chord
0:59
Diameter
1:32
Radius
2:07
Secant
2:17
Tangent
3:10
Circumference
3:56
Introduction to Circumference
3:57
Example: Find the Circumference of the Circle
5:09
Circumference
6:40
Example: Find the Circumference of the Circle
6:41
Extra Example 1: Use the Circle to Answer the Following
9:10
Extra Example 2: Find the Missing Measure
12:53
Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle
15:51
Extra Example 4: Find the Circumference of Each Circle
19:24
Angles and Arc

35m 24s

Intro
0:00
Central Angle
0:06
Definition of Central Angle
0:07
Sum of Central Angles
1:17
Sum of Central Angles
1:18
Arcs
2:27
Minor Arc
2:30
Major Arc
3:47
Arc Measure
5:24
Measure of Minor Arc
5:24
Measure of Major Arc
6:53
Measure of a Semicircle
7:11
Arc Addition Postulate
8:25
Arc Addition Postulate
8:26
Arc Length
9:43
Arc Length and Example
9:44
Concentric Circles
16:05
Concentric Circles
16:06
Congruent Circles and Arcs
17:50
Congruent Circles
17:51
Congruent Arcs
18:47
Extra Example 1: Minor Arc, Major Arc, and Semicircle
20:14
Extra Example 2: Measure and Length of Arc
22:52
Extra Example 3: Congruent Arcs
25:48
Extra Example 4: Angles and Arcs
30:33
Arcs and Chords

21m 51s

Intro
0:00
Arcs and Chords
0:07
Arc of the Chord
0:08
Theorem 1: Congruent Minor Arcs
1:01
Inscribed Polygon
2:10
Inscribed Polygon
2:11
Arcs and Chords
3:18
Theorem 2: When a Diameter is Perpendicular to a Chord
3:19
Arcs and Chords
5:05
Theorem 3: Congruent Chords
5:06
Extra Example 1: Congruent Arcs
10:35
Extra Example 2: Length of Arc
13:50
Extra Example 3: Arcs and Chords
17:09
Extra Example 4: Arcs and Chords
19:45
Inscribed Angles

27m 53s

Intro
0:00
Inscribed Angles
0:07
Definition of Inscribed Angles
0:08
Inscribed Angles
0:58
Inscribed Angle Theorem 1
0:59
Inscribed Angles
3:29
Inscribed Angle Theorem 2
3:30
Inscribed Angles
4:38
Inscribed Angle Theorem 3
4:39
Inscribed Quadrilateral
5:50
Inscribed Quadrilateral
5:51
Extra Example 1: Central Angle, Inscribed Angle, and Intercepted Arc
7:02
Extra Example 2: Inscribed Angles
9:24
Extra Example 3: Inscribed Angles
14:00
Extra Example 4: Complete the Proof
17:58
Tangents

26m 16s

Intro
0:00
Tangent Theorems
0:04
Tangent Theorem 1
0:05
Tangent Theorem 1 Converse
0:55
Common Tangents
1:34
Common External Tangent
2:12
Common Internal Tangent
2:30
Tangent Segments
3:08
Tangent Segments
3:09
Circumscribed Polygons
4:11
Circumscribed Polygons
4:12
Extra Example 1: Tangents & Circumscribed Polygons
5:50
Extra Example 2: Tangents & Circumscribed Polygons
8:35
Extra Example 3: Tangents & Circumscribed Polygons
11:50
Extra Example 4: Tangents & Circumscribed Polygons
15:43
Secants, Tangents, & Angle Measures

27m 50s

Intro
0:00
Secant
0:08
Secant
0:09
Secant and Tangent
0:49
Secant and Tangent
0:50
Interior Angles
2:56
Secants & Interior Angles
2:57
Exterior Angles
7:21
Secants & Exterior Angles
7:22
Extra Example 1: Secants, Tangents, & Angle Measures
10:53
Extra Example 2: Secants, Tangents, & Angle Measures
13:31
Extra Example 3: Secants, Tangents, & Angle Measures
19:54
Extra Example 4: Secants, Tangents, & Angle Measures
22:29
Special Segments in a Circle

23m 8s

Intro
0:00
Chord Segments
0:05
Chord Segments
0:06
Secant Segments
1:36
Secant Segments
1:37
Tangent and Secant Segments
4:10
Tangent and Secant Segments
4:11
Extra Example 1: Special Segments in a Circle
5:53
Extra Example 2: Special Segments in a Circle
7:58
Extra Example 3: Special Segments in a Circle
11:24
Extra Example 4: Special Segments in a Circle
18:09
Equations of Circles

27m 1s

Intro
0:00
Equation of a Circle
0:06
Standard Equation of a Circle
0:07
Example 1: Equation of a Circle
0:57
Example 2: Equation of a Circle
1:36
Extra Example 1: Determine the Coordinates of the Center and the Radius
4:56
Extra Example 2: Write an Equation Based on the Given Information
7:53
Extra Example 3: Graph Each Circle
16:48
Extra Example 4: Write the Equation of Each Circle
19:17
X. Polygons & Area
Polygons

27m 24s

Intro
0:00
Polygons
0:10
Polygon vs. Not Polygon
0:18
Convex and Concave
1:46
Convex vs. Concave Polygon
1:52
Regular Polygon
4:04
Regular Polygon
4:05
Interior Angle Sum Theorem
4:53
Triangle
5:03
Quadrilateral
6:05
Pentagon
6:38
Hexagon
7:59
20-Gon
9:36
Exterior Angle Sum Theorem
12:04
Exterior Angle Sum Theorem
12:05
Extra Example 1: Drawing Polygons
13:51
Extra Example 2: Convex Polygon
15:16
Extra Example 3: Exterior Angle Sum Theorem
18:21
Extra Example 4: Interior Angle Sum Theorem
22:20
Area of Parallelograms

17m 46s

Intro
0:00
Parallelograms
0:06
Definition and Area Formula
0:07
Area of Figure
2:00
Area of Figure
2:01
Extra Example 1:Find the Area of the Shaded Area
3:14
Extra Example 2: Find the Height and Area of the Parallelogram
6:00
Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices
10:11
Extra Example 4: Find the Area of the Figure
14:31
Area of Triangles Rhombi, & Trapezoids

20m 31s

Intro
0:00
Area of a Triangle
0:06
Area of a Triangle: Formula and Example
0:07
Area of a Trapezoid
2:31
Area of a Trapezoid: Formula
2:32
Area of a Trapezoid: Example
6:55
Area of a Rhombus
8:05
Area of a Rhombus: Formula and Example
8:06
Extra Example 1: Find the Area of the Polygon
9:51
Extra Example 2: Find the Area of the Figure
11:19
Extra Example 3: Find the Area of the Figure
14:16
Extra Example 4: Find the Height of the Trapezoid
18:10
Area of Regular Polygons & Circles

36m 43s

Intro
0:00
Regular Polygon
0:08
SOHCAHTOA
0:54
30-60-90 Triangle
1:52
45-45-90 Triangle
2:40
Area of a Regular Polygon
3:39
Area of a Regular Polygon
3:40
Are of a Circle
7:55
Are of a Circle
7:56
Extra Example 1: Find the Area of the Regular Polygon
8:22
Extra Example 2: Find the Area of the Regular Polygon
16:48
Extra Example 3: Find the Area of the Shaded Region
24:11
Extra Example 4: Find the Area of the Shaded Region
32:24
Perimeter & Area of Similar Figures

18m 17s

Intro
0:00
Perimeter of Similar Figures
0:08
Example: Scale Factor & Perimeter of Similar Figures
0:09
Area of Similar Figures
2:44
Example:Scale Factor & Area of Similar Figures
2:55
Extra Example 1: Complete the Table
6:09
Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures
8:56
Extra Example 3: Find the Unknown Area
12:04
Extra Example 4: Use the Given Area to Find AB
14:26
Geometric Probability

38m 40s

Intro
0:00
Length Probability Postulate
0:05
Length Probability Postulate
0:06
Are Probability Postulate
2:34
Are Probability Postulate
2:35
Are of a Sector of a Circle
4:11
Are of a Sector of a Circle Formula
4:12
Are of a Sector of a Circle Example
7:51
Extra Example 1: Length Probability
11:07
Extra Example 2: Area Probability
12:14
Extra Example 3: Area Probability
17:17
Extra Example 4: Area of a Sector of a Circle
26:23
XI. Solids
Three-Dimensional Figures

23m 39s

Intro
0:00
Polyhedrons
0:05
Polyhedrons: Definition and Examples
0:06
Faces
1:08
Edges
1:55
Vertices
2:23
Solids
2:51
Pyramid
2:54
Cylinder
3:45
Cone
4:09
Sphere
4:23
Prisms
5:00
Rectangular, Regular, and Cube Prisms
5:02
Platonic Solids
9:48
Five Types of Regular Polyhedra
9:49
Slices and Cross Sections
12:07
Slices
12:08
Cross Sections
12:47
Extra Example 1: Name the Edges, Faces, and Vertices of the Polyhedron
14:23
Extra Example 2: Determine if the Figure is a Polyhedron and Explain Why
17:37
Extra Example 3: Describe the Slice Resulting from the Cut
19:12
Extra Example 4: Describe the Shape of the Intersection
21:25
Surface Area of Prisms and Cylinders

38m 50s

Intro
0:00
Prisms
0:06
Bases
0:07
Lateral Faces
0:52
Lateral Edges
1:19
Altitude
1:58
Prisms
2:24
Right Prism
2:25
Oblique Prism
2:56
Classifying Prisms
3:27
Right Rectangular Prism
3:28
4:55
Oblique Pentagonal Prism
6:26
Right Hexagonal Prism
7:14
Lateral Area of a Prism
7:42
Lateral Area of a Prism
7:43
Surface Area of a Prism
13:44
Surface Area of a Prism
13:45
Cylinder
16:18
Cylinder: Right and Oblique
16:19
Lateral Area of a Cylinder
18:02
Lateral Area of a Cylinder
18:03
Surface Area of a Cylinder
20:54
Surface Area of a Cylinder
20:55
Extra Example 1: Find the Lateral Area and Surface Are of the Prism
21:51
Extra Example 2: Find the Lateral Area of the Prism
28:15
Extra Example 3: Find the Surface Area of the Prism
31:57
Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder
34:17
Surface Area of Pyramids and Cones

26m 10s

Intro
0:00
Pyramids
0:07
Pyramids
0:08
Regular Pyramids
1:52
Regular Pyramids
1:53
Lateral Area of a Pyramid
4:33
Lateral Area of a Pyramid
4:34
Surface Area of a Pyramid
9:19
Surface Area of a Pyramid
9:20
Cone
10:09
Right and Oblique Cone
10:10
Lateral Area and Surface Area of a Right Cone
11:20
Lateral Area and Surface Are of a Right Cone
11:21
Extra Example 1: Pyramid and Prism
13:11
Extra Example 2: Find the Lateral Area of the Regular Pyramid
15:00
Extra Example 3: Find the Surface Area of the Pyramid
18:29
Extra Example 4: Find the Lateral Area and Surface Area of the Cone
22:08
Volume of Prisms and Cylinders

21m 59s

Intro
0:00
Volume of Prism
0:08
Volume of Prism
0:10
Volume of Cylinder
3:38
Volume of Cylinder
3:39
Extra Example 1: Find the Volume of the Prism
5:10
Extra Example 2: Find the Volume of the Cylinder
8:03
Extra Example 3: Find the Volume of the Prism
9:35
Extra Example 4: Find the Volume of the Solid
19:06
Volume of Pyramids and Cones

22m 2s

Intro
0:00
Volume of a Cone
0:08
Volume of a Cone: Example
0:10
Volume of a Pyramid
3:02
Volume of a Pyramid: Example
3:03
Extra Example 1: Find the Volume of the Pyramid
4:56
Extra Example 2: Find the Volume of the Solid
6:01
Extra Example 3: Find the Volume of the Pyramid
10:28
Extra Example 4: Find the Volume of the Octahedron
16:23
Surface Area and Volume of Spheres

14m 46s

Intro
0:00
Special Segments
0:06
Radius
0:07
Chord
0:31
Diameter
0:55
Tangent
1:20
Sphere
1:43
Plane & Sphere
1:44
Hemisphere
2:56
Surface Area of a Sphere
3:25
Surface Area of a Sphere
3:26
Volume of a Sphere
4:08
Volume of a Sphere
4:09
Extra Example 1: Determine Whether Each Statement is True or False
4:24
Extra Example 2: Find the Surface Area of the Sphere
6:17
Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters
7:25
Extra Example 4: Find the Surface Area and Volume of the Solid
9:17
Congruent and Similar Solids

16m 6s

Intro
0:00
Scale Factor
0:06
Scale Factor: Definition and Example
0:08
Congruent Solids
1:09
Congruent Solids
1:10
Similar Solids
2:17
Similar Solids
2:18
Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither
3:35
Extra Example 2: Determine if Each Statement is True or False
7:47
Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume
10:14
Extra Example 4: Find the Volume of the Larger Prism
12:14
XII. Transformational Geometry
Mapping

14m 12s

Intro
0:00
Transformation
0:04
Rotation
0:32
Translation
1:03
Reflection
1:17
Dilation
1:24
Transformations
1:45
Examples
1:46
Congruence Transformation
2:51
Congruence Transformation
2:52
Extra Example 1: Describe the Transformation that Occurred in the Mappings
3:37
Extra Example 2: Determine if the Transformation is an Isometry
5:16
Extra Example 3: Isometry
8:16
Reflections

23m 17s

Intro
0:00
Reflection
0:05
Definition of Reflection
0:06
Line of Reflection
0:35
Point of Reflection
1:22
Symmetry
1:59
Line of Symmetry
2:00
Point of Symmetry
2:48
Extra Example 1: Draw the Image over the Line of Reflection and the Point of Reflection
3:45
Extra Example 2: Determine Lines and Point of Symmetry
6:59
Extra Example 3: Graph the Reflection of the Polygon
11:15
Extra Example 4: Graph the Coordinates
16:07
Translations

18m 43s

Intro
0:00
Translation
0:05
Translation: Preimage & Image
0:06
Example
0:56
Composite of Reflections
6:28
Composite of Reflections
6:29
Extra Example 1: Translation
7:48
Extra Example 2: Image, Preimage, and Translation
12:38
Extra Example 3: Find the Translation Image Using a Composite of Reflections
15:08
Extra Example 4: Find the Value of Each Variable in the Translation
17:18
Rotations

21m 26s

Intro
0:00
Rotations
0:04
Rotations
0:05
Performing Rotations
2:13
Composite of Two Successive Reflections over Two Intersecting Lines
2:14
Angle of Rotation: Angle Formed by Intersecting Lines
4:29
Angle of Rotation
5:30
Rotation Postulate
5:31
Extra Example 1: Find the Rotated Image
7:32
Extra Example 2: Rotations and Coordinate Plane
10:33
Extra Example 3: Find the Value of Each Variable in the Rotation
14:29
Extra Example 4: Draw the Polygon Rotated 90 Degree Clockwise about P
16:13
Dilation

37m 6s

Intro
0:00
Dilations
0:06
Dilations
0:07
Scale Factor
1:36
Scale Factor
1:37
Example 1
2:06
Example 2
6:22
Scale Factor
8:20
Positive Scale Factor
8:21
Negative Scale Factor
9:25
Enlargement
12:43
Reduction
13:52
Extra Example 1: Find the Scale Factor
16:39
Extra Example 2: Find the Measure of the Dilation Image
19:32
Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation
26:18
Extra Example 4: Graphing Polygon, Dilation, and Scale Factor
32:08
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Related Articles:

Geometric Mean

  • Geometric Mean: The geometric mean between two positive numbers a and b is x, where
  • Similar Triangles: If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other
  • The measures of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse

Geometric Mean


Right triangle ABC, BD ⊥AC , write all the pairs of similar triangles.
∆ABC  ∼  ∆ADB, ∆ABC  ∼  ∆BDC, ∆ADB  ∼  ∆BDC.
Find the geometric mean between 6 and 10.
x = √{6*10} = √{60} = 2√{15} .
Find the geometric mean between 13 and 15.
x = √{13*15} = √{195} .
Determine whether the following statement is true or false.

Right triangle ABC, if BD ⊥AC , then BD is the geometric mean between AC and AD .
False.
Determine whether the following statement is true or false.

Right triangle ABC, if BD ⊥AC , then [AC/AB] = [AB/AD].
True.

Right triangle ABC, BD ⊥AC , AD = 3, CD = 9, find BD.
  • [AD/BD] = [BD/CD]
  • [3/BD] = [BD/9]
BD = 3√3 .

find the values of x and y.
  • [3/x] = [x/5]
  • x = √{15}
  • [3/y] = [y/(3 + 5)]
y = 2√6 .
Find the geometric mean between 2 and 8.
x = √{2*8} = 4.
Find the geometric mean between 9 and 16.
x = √{9*16} = √{144} = 12.
Find the geometric mean between 10 and 20.
x = √{10*20} = √{200} = 10√2

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Geometric Mean

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Geometric Mean 0:04
    • Geometric Mean & Example
  • Similar Triangles 4:32
    • Similar Triangles
  • Geometric Mean-Altitude 11:10
    • Geometric Mean-Altitude & Example
  • Geometric Mean-Leg 14:47
    • Geometric Mean-Leg & Example
  • Extra Example 1: Geometric Mean Between Each Pair of Numbers 20:10
  • Extra Example 2: Similar Triangles 23:46
  • Extra Example 3: Geometric Mean of Triangles 28:30
  • Extra Example 4: Geometric Mean of Triangles 36:58

Transcription: Geometric Mean

Welcome back to Educator.com.0000

We are going to go over the geometric mean for our next lesson.0002

For the geometric mean, here, we have a/x, which is equal to x/b.0006

Now, if you remember, a few lessons ago, we went over ratios and proportions.0019

When we have a proportion, we have two ratios that are equal to each other.0025

Here, these numbers right here are called the means, and these numbers here are called extremes.0032

So, when we talk about the geometric mean, we are talking about this number and this number.0049

The geometric mean between two positive numbers, a and b, is x.0060

If we are given two numbers, a and b, those numbers would be considered the extremes.0067

And then, x, which is the geometric mean, is going to go here and here.0073

Now, when we solve proportions, remember: we cross-multiply; so we are going to do this number times this number, is equal to a times b.0082

If we are asked to find the geometric mean between two positive numbers--let's say 3 and 8--find the geometric mean between 3 and 8...0094

Now, actually, before we continue with that problem, let's go back to this.0114

If we cross-multiply, then we are going to get x2 = a times b; then how would you find x?0121

x is going to be the square root of ab.0134

The geometric mean x is going to be the square root of this number times that number.0140

This is how you would find the geometric mean between the two numbers.0148

Now, back to this: 3 and 8--find the geometric mean between 3 and 8.0152

What you can do is go ahead and solve it like this: plug it into here and find the square root of 3 times 8.0159

Or you can just first set it up as a proportion; and this actually makes it a lot easier to see, especially if you are given0168

different versions of this problem, like if you are given the geometric mean and asked to find one of these numbers.0179

It is always easiest to put it into this proportion first, because you know that these two numbers here have to be the geometric mean.0186

If they are asking you to find the geometric mean, that means that that would be this number down here, and that number up there, for your proportion.0196

And then, 3 and 8 are going to go here and here; so this is how you would set it up.0209

Find the geometric mean--that means that these numbers right here are what you are looking for.0216

To solve it out, cross-multiply: x2 is equal to 3 times 8, which is 24, so x is equal to √24.0220

And that is going to be 2√6, or you can use your calculator to change that to a decimal.0235

But this would be the answer, right here: this is the geometric mean between 3 and 8.0247

Again, set it up as a proportion; put the two numbers up here and down there.0254

Make sure that the geometric mean goes here and here, because those are the means; and then, just cross-multiply to solve.0262

Similar triangles: now, here we have a right triangle, and remember: an altitude is a segment0274

starting from a vertex and going to the side opposite that vertex so that it is perpendicular to it.0286

If we draw an altitude from this vertex (and it has to be drawn from that right angle)--0299

so then here is our right angle, and you are going to draw an altitude so that it is perpendicular to the side opposite--0307

in this case, it is the hypotenuse, because it is coming from that right angle--then the two triangles formed0318

are similar to the given triangle and to each other.0325

Before this altitude was drawn, we only had one triangle.0330

Now, after the altitude, we have three triangles: we have the big one; we have this one right here; and we have this one right here--three triangles.0335

Now, this theorem is saying that, once that altitude is drawn from that right angle to the hypotenuse, all three triangles are now similar.0345

Remember: similar triangles are triangles that have congruent angles, but proportional sides.0356

"Similar" just means that they have the same shape, but a different size; that is "similar" or "similarity."0368

So, here we know that we have three triangles--the same shape, but slightly different sizes.0377

To state all three triangles so that the corresponding parts are in order, so that we can say that they are similar,0386

I can name all three; it doesn't matter which order, so if I want to say the big one first:0397

the big one is going to be triangle ABC; it is similar to...and then we are going to just name another triangle.0406

Now, remember: it has to be corresponding to the order; so AB in the big triangle0419

(and that is kind of hard to see)...the easiest way to see their corresponding parts is to look at it from long leg0429

to short leg, and then hypotenuse; those are the three parts that make up a right triangle.0445

It is always easiest to determine whatever you are stating--which side it would be considered from that triangle.0453

And then, you can just look at the corresponding part of the other triangle.0464

So, AB from the big triangle is considered the long leg; this would be AB.0468

AB is the long leg; BC is the short leg; and then, AC is the hypotenuse.0479

If AB is the long leg, then I have to mention that one first.0487

And remember: B is the right angle; so then, the right angle is going to go in the middle of the next triangle that I am stating.0491

Then, let's say I am going to name this triangle--not the very small one, but the medium-sized one, this one right here.0499

I am going to label this D; so remember, I have to state the long first.0510

The long leg from this triangle would be AD, because here is the right angle; AD is the long leg.0516

Now, does it matter if I say AD or DA?0525

Well, I know that, since B is my right angle for the first triangle, I have to state the right angle in that same order.0528

So, it is going to be AD, triangle AD, because AD is my long leg, and D is my right angle; and then, it would be AD, and then B.0540

So again, from the big triangle, it is triangle ABC, with AB as a long leg, BC as a short leg, and AC as the hypotenuse.0556

For my second triangle that I am naming, AD is my long leg; BD is my short leg; and AB is my hypotenuse.0571

And then, see how D is the right angle; the middle angle that I am listing is the right angle.0583

So then, the third triangle has to be in the same order, so from this one right here,0593

which one is my long leg? Which one is my short leg? And which one is my hypotenuse?0600

My long leg from here...this is the right angle...would be BD or DB.0606

Now, from this triangle, where is my right angle?--this one, D.0613

I know that that letter D is going to go in the middle.0618

So then, BD is my long leg; D is my right angle; C is my missing vertex.0622

BD is my long leg; DC is my short leg; and BC is my hypotenuse.0634

So now, I have listed all three triangles within their corresponding parts.0642

Just keep in mind that, when the altitude is drawn from the right angle, you are actually splitting this up into three similar triangles--not congruent, but similar.0653

Here, we have a right triangle, again, with the altitude drawn from the right angle of that triangle down to the hypotenuse.0672

When you have this diagram, the measure of the altitude, BD, is going to be the geometric mean between AD and DC.0685

Remember how, when we went over geometric mean, we found the geometric mean between two numbers.0710

So then, the two numbers that are given are a and b, and then, this is a geometric mean and a geometric mean.0719

This right here, BD, would be the geometric mean between the two parts of the hypotenuse.0731

Just to read this to you: the measures of the altitude drawn from the vertex of the right angle of the right triangle0739

to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.0748

So, because this right here--see how it splits up the hypotenuse into two parts, AD and DC--those two parts,0756

then it is as if AD is a here; DC is b; so then, those are the two numbers; and BD, the altitude, would be the geometric mean.0765

So then, if you were to write this out, this would be like BD is the geometric mean between AD and DC, these two parts.0778

Then, AD would go up here, and DC would go right there; that is like a and b--those are the two numbers.0788

And then, the geometric mean, which is the altitude, BD, is going to go here and here.0798

So, whenever they say that something is the geometric mean, that something goes in here and here,0808

because remember, again: these two would be the means.0816

Let's say that this is x, and we want to find the geometric mean between...let's say AD is 20 and DC is 5.0823

So then, we are trying to find the altitude.0838

Now, we know, because of this theorem, that the altitude, BD, is the geometric mean between AD and DC, these two parts.0840

And this would be the geometric mean; so to solve it out, I am going to make it into my proportion.0855

The geometric mean is BD; that is going to go here and here.0863

My two numbers, a and b, are going to go here and here; so when I solve it out, it is going to become x2 = 100, so x is 10.0867

So then, this right here...the measure of the altitude is 10.0879

Now, one more: here, for this one, if the altitude is drawn to the hypotenuse of a right triangle,0888

then the measure of a leg of the triangle (we have two legs: it is AB and BC, the two legs of the big triangle)0897

is the geometric mean between the measures of the hypotenuse0909

(this whole thing, AC) and the segment of the hypotenuse adjacent to that leg.0917

"Adjacent" means "next to"; so then, to find the geometric mean, we need two numbers.0925

The two numbers would be the hypotenuse and the segment of the hypotenuse adjacent to that leg--0936

the part of the hypotenuse that is close to that leg that we are trying to find.0944

Here, for example, we are looking for this leg; this is the geometric mean between...the two numbers would be0950

the whole thing, the hypotenuse, AC, and the part of the hypotenuse (remember, this altitude0961

can divide this hypotenuse into two parts, AD and BC) that is adjacent (meaning closest to, next) to that leg that we are trying to find.0971

It would be right here--that part; that means that we make our proportion...0983

Again, the geometric mean...BC goes here and here; or you can write BC, BC.0992

And then, let's write that so that you know what we are looking for: BC is going to go here and here.1000

And then, the two numbers that we are going to use are AC, the whole hypotenuse, and then this part right here; that is DC.1012

OK, and then you would solve it that way.1028

Now, let's say that we are looking for this leg.1030

If this is the leg that we are looking for--let's say y--then it is going to be the geometric mean1033

between, again, the whole thing, AC, and the part of the hypotenuse adjacent to that leg that we are looking for.1039

In this case, if we are looking for y, this leg, we are not going to be using DC, because that is not the one adjacent to that.1048

Then, we are going to be using AC and AD, because that is the part of the hypotenuse that is adjacent to this leg that we are looking for.1056

Again, y, y...or you can write AB, AB...that is going to be the whole thing, AC, and AD.1068

It is the geometric mean between those two numbers.1083

Let's say that this whole thing is 20, and DC is 5.1086

Now, let's look for BC; BC is the geometric mean between 20 and 5; we put 20 up here, x, and 5.1098

It is going to be x2 = 100; x = 10; so this right here is 10.1114

To find y, if this is 5, and this whole thing is 20, I know that AD has to be 15, because 15 + 5 makes up this whole thing, which is 20.1124

So, AC is 20, over y, equals y over this part, which is 15.1140

y2 (I am going to cross-multiply) equals...this would be 300, so y is going to be √300, which is 10√3.1154

That is how you would find the geometric mean.1181

Now, the one before (remember, the altitude one)--the altitude is the geometric mean between part and part of the hypotenuse.1185

The leg is the geometric mean between the whole and this part close to it.1197

Again, this leg is the geometric mean between the whole and the part close to it.1203

Those are the two theorems, there; now let's go over our examples.1210

Find the geometric mean between each pair of numbers.1215

Find the geometric mean: to use our proportion, the geometric mean goes here and here;1220

and then, the two numbers that we are going to use, that are given to us--those are the extremes, 8 and 11.1231

Make sure that whatever the geometric mean is...it has to go here and here.1241

Then, cross-multiply; I get x2 = 88, so x = √88, and then that would simplify to...let's see...2√22.1247

You can simplify it, if your teacher wants you to round it to several decimal places, or whatever it is.1287

And you would have to use your calculator to figure that out in decimals.1293

But this is how it would simplify.1299

The next one: 5 and 2/3--again, 5/x = x/(2/3).1303

Now, I am doing it this way, just so that it is easier to set up.1320

But you also know that x is going to equal the square root of AB; so you can just use that if you want to.1325

That is where you just take the two numbers, multiply it, and then take the square root of it to find the geometric mean.1333

Or you can just set it up so that this is easier to understand, because you know that these two numbers make up the means.1341

So, you know that whatever it says the mean is, you write that here.1350

In this case, we are looking for it; that is why we have x's there.1354

And then, the two numbers that are given would go there.1360

Then, x2 =...here, this becomes...5 times 2/3 is going to be 10/3, so x is the square root of 10/3.1365

And then, just use your calculator for that.1380

If you have to leave it in radical form, it would just be √10/√3.1383

And then, I would have to rationalize this denominator, so it is going to be...see how √3/√3 is just equal to 1.1391

So, this would be √30/3, because √3 times √3 equals 3.1400

Now again, if you don't understand this, just go ahead and use your calculator; just do 10/3, and then you can just take the square root of that.1408

Or if your calculator will allow you, just do the square root of 10/3.1416

Name the three similar triangles: again, we have the altitude from the right angle of this big triangle.1428

The altitude is from the right angle to the hypotenuse of the big triangle.1440

We are going to name the three triangles that are similar.1447

Now, again, we can start with whatever triangle we want.1451

If you do this on your own, then your triangles will probably be different; it is probably going to be listed differently than how I list it.1458

But that is fine, just as long as whatever you wrote, the three triangles, are corresponding with each other in the parts.1466

The first triangle that I want to name is the big one.1475

I want to name the big one first, and I am going to say that triangle...let's do the hypotenuse first:1478

MOP...that P, my right angle, would be last; so MO, and then P is my right angle.1486

And that is going to be similar to triangle...make sure that...MO is my hypotenuse, and then OP would be1505

the short side; so triangle...and then, let's see, let's do this one...MPN--let's try that,1524

because again, P is my right angle; so then, N, which is the right angle for this triangle, has to be listed last.1544

So, is MP my hypotenuse?--yes, so that is right.1551

How about PN--is PN my short side?--yes, so this is correct.1555

And then, it is similar to...what is my other one?--the small triangle right here, right?1571

So then, this is the right angle; that is going to go last; and the triangle's hypotenuse would be...1581

let's see, OPN...OP is my hypotenuse; N is my right angle; and now, let's look at PN.1590

For all of these other ones, OP was my short leg; PN was also my short leg of this triangle.1604

Is PN the short side of this small triangle? It is actually not, so this is wrong.1615

I know that OP is my hypotenuse, but then, instead of saying OPN, I would have to say PON.1629

N is in the correct position, because that is the right angle, and that has to go last.1641

I know that, instead of OP, like how we had it, it would have to go PO.1645

PON: that way, PO is my hypotenuse, and then ON is my short side.1654

Those are my three triangles that are similar.1667

Now, again, if you had your first triangle listed out differently, and if you used another triangle, that is fine;1670

just make sure that the other two triangles that you list out are similar to that triangle.1680

It is always easiest to just maybe write some symbols like this, like how I did it.1688

And then, just make sure that you know that your hypotenuse has to go hypotenuse, short leg, long leg, right angle...1694

use those to help you list it in the corresponding order.1704

The next example: we are going to find the values of x and y.1712

We are actually going to do a few of these for the next example, too.1714

For the first one, let's see: it is a right angle's altitude; that means that I know that this altitude is the geometric mean between these parts of the hypotenuse.1720

So, that means that the altitude is the geometric mean, x and x, between 4 and 7.1743

Solve this out; this is going to be x2 = 28, so x is √28; does that simplify?1757

Let's see: it is going to be 2√7.1770

Again, if you want to change it to decimals, then use your calculator.1779

2√7 is this value right here.1784

And then, to find y, you can do two things: you can use that second theorem1787

that says that the leg of the big triangle is the geometric mean between the whole hypotenuse1796

and the part that is adjacent to it, and solve it out this way; or you can just, using this right triangle,1807

now that you know what this side is, and you have this side as 7, use the Pythagorean theorem.1819

This would be a2 + b2 = c2 (or y2).1825

Either way, it does not matter; you will still get the same answer.1835

Let's just go ahead and solve it this way, with the geometric mean.1840

So, we are going to write y and y there; this is the geometric mean between the whole thing--1847

what is the whole thing?--4 + 7 is the whole thing; the whole hypotenuse would be 11;1856

and then, the part of the hypotenuse that is close to this leg that we are looking for is 7.1864

So, y2 = 77; y = √77; and that is it; so we have here and here.1871

The next one: Let's see, this altitude is the geometric mean between this part and this part.1886

Now, this one is given to me; I am not looking for the altitude.1896

But the theorem does not change; the theorem stays the same.1901

It is still saying that the altitude is a geometric mean between this part and this part.1904

So, when I write it in my proportion, I still have to keep this altitude, whatever it is,1910

whether it is x or whether it is given--I still have to write it as my mean, there, and then between what?-- between this, which is x, and 5x.1917

So, it still stays the same, just like this; whatever this is right here, that is going to go here and here.1934

In the same way, whatever the altitude is, the altitude is the geometric mean; so then, that is going to go here and here.1943

When you solve it out, it is going to be 252 = 5x2; you are just multiplying through.1950

And then, 252 is 625; that is equal to 5x2.1964

If you divide the 5, that is going to give you 125 = x2, so √125 is going to be equal to x, which is 5√5.1975

And that is a 5 right there; so 5√5 is x, so this is going to be 5√5.2009

This is going to be 5 times x, which is 5 times 5√5, is 25√5.2020

And then, y: again, you can use the Pythagorean theorem: 252 + (5√5)2 = y2.2032

Or you can just use this: y is a geometric mean between the whole thing--2042

the whole thing would be 25√5 + 5√5; that is going to be 30√5--2058

and this part that is close to the leg, 5√5.2068

y2 is equal to...this is going to be 30 times 5, is going to be 150; and then, √5 times √5 becomes 5.2076

So then, it is 150 times 5; so again, √5 times √5 is (√5)2.2104

Remember: that square root and the square cancel each other out, so it just becomes 5.2114

And then, this becomes 750; and then, y becomes √750, and to simplify that out, it becomes...2119

let's see...25...OK, well, here, y equals 5√30; I believe that is correct.2158

Or you can just, again, change it to decimals with your calculator.2191

It is kind of a large number, so just go ahead and use your calculator.2196

And that is it for this problem; here is x, and here is y, for this.2205

Now, we are going to go on to the next problem; and for the next one,2214

we are going to go over a couple more, just so you can get more familiar with these types of problems.2217

Again, here we have the geometric mean, the altitude being the geometric mean between this part and this part.2223

So then, the geometric mean is going to go here and here, between these two numbers, x and 8.2236

8x = 9; x is equal to 9/8; and then, for the y, again, you can use the same concept,2250

the theorem that says that this is the geometric mean between the whole thing and then this part.2268

Or we can use the Pythagorean theorem; let's go ahead and just use the Pythagorean theorem this time.2274

So then, a2, the leg squared, plus the other leg squared, is equal to the hypotenuse squared.2280

32 + 82 = y2.2288

32 is 9, plus 64, is equal to y2; 73 = y2; y = √73.2294

And you can just leave it like that; that does not simplify.2311

The next one: the same thing: this is the geometric mean between this part and that part.2320

So then, 12...12...the geometric between 8 and x...8x = 144; divide the 8; x is equal to 18.2331

And then, here, to find the y, the same thing happens.2359

We are going to say that this...we know that, since this is 18, we can use the Pythagorean theorem.2378

122 + 182 is equal to y2.2385

Or we can use the theorem; let's just use the theorem: y...y; that is the geometric mean between the whole thing...2390

18 + 8 is 26; that is the whole thing; this whole thing would be 26...and then this part that is close to it is 18.2400

Those are my extremes, and then my mains would be this: y2 is equal to--2418

and you can just use your calculator for this part...it is 26 times 18, which is 468;2427

and then, y is equal to the square root of 468, and again, you can just use your calculator to simplify that out.2440

That is it for this lesson; thank you for watching Educator.com.2456

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