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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Midpoint and Distance Formulas

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (9)

0 answers

Post by julius mogyorossy on March 26, 2014

Forget what I said above, Dr. Carleen did not leave out a factor, 2, I apologize to Dr. Carleen, and to my fellow Educators. My head is really messed up right now, PTSD, but every now and then I get a sense of the God I shall be, it is so awesome, I wish I could record that state of mind on a DVD, and let you experience it.

0 answers

Post by julius mogyorossy on March 24, 2014

It seems Dr. Carleen left out a factor, 2, when finding the distance for the second example, in the first term of the first term, when she simplifies before squaring the two terms under the radical, but don't even take my word for it, I have bad PTSD, very stressed out, can't wait to be perfect soon and then take the Algebra Clep test. I so love, solving, writing equations, on my tablet.  

1 answer

Last reply by: Dr Carleen Eaton
Wed Nov 2, 2011 9:20 PM

Post by Jonathan Taylor on October 30, 2011

1st example of distance formula 9+25 is not 36 it was 34 so the square root shold have been 34

1 answer

Last reply by: Dr Carleen Eaton
Wed Nov 2, 2011 9:17 PM

Post by Jonathan Taylor on October 25, 2011

I did not understand this question did u do the example as shown are did u make up a example

2 answers

Last reply by: Dr Carleen Eaton
Thu Oct 13, 2011 9:05 PM

Post by Manuela Fridman on October 6, 2011

I found 2 minor mistake in this video that I think should be corrected and reposted. The 1st one: in "example:distance" Dr. Eaton put the answer was 6. I believe that is incorrect because sq root 9+25= sq root 34 not 36 as Dr. Eaton put. Therefore the answer would be 5.83. The 2nd thing, i'm not sure if this makes a difference in the overall answer but in the "example 2:midpoint and" section Dr Eaton says to first put "y2" first but actually rights it using the "x2". It made it a little confusing, but of course i realized it was just a simple mistake. Thank you very much for all of your help though! Dr Eaton and the rest of the tutors are great! Just wish i was able to sign up for the year long membership without having to pay upfront for all the classes. $35 is a little high every month.

Midpoint and Distance Formulas

  • Make sure that you know and understand how to use both of these formulas. They will be used in a lot of later work.
  • Remember that the order of the x coordinates in the distance formula does not matter. So take whichever difference is easier to compute. The same comment applies to the y coordinates.
  • After squaring the differences in the distance formula, be sure to take the positive square root of their sum.

Midpoint and Distance Formulas

Find the midpoint and distance of the segment with endpoints ( − 4, − 1),(2,4)
  • Use the midpoint formula M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) and distance formula D = √{(x2 − x1)2 + (y2 − y1)2}
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [( − 4 + 2)/2],[( − 1 + 4)/2] ) = ( [( − 2)/2],[3/2] ) = ( − 1,[3/2] )
D = √{(x2 − x1)2 + (y2 − y1)2} = √{(2 − ( − 4))2 + (4 − ( − 1))2} = √{(6)2 + (5)2} = √{36 + 25} = √{61} ≈ 7.81
Find the midpoint and distance of the segment with endpoints ( − 3, − 5),(1,7)
  • Use the midpoint formula M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) and distance formula D = √{(x2 − x1)2 + (y2 − y1)2}
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [( − 3 + 1)/2],[( − 5 + 7)/2] ) = ( [( − 2)/2],[2/2] ) = ( − 1,1 )
D = √{(x2 − x1)2 + (y2 − y1)2} = √{(1 − ( − 3))2 + (7 − ( − 5))2} = √{(4)2 + (12)2} = √{16 + 144} = √{160} ≈ 12.65
Find the midpoint and distance of the segment with endpoints (3, − 2),( − 2,5)
  • Use the midpoint formula M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) and distance formula D = √{(x2 − x1)2 + (y2 − y1)2}
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [(3 − 2)/2],[( − 2 + 5)/2] ) = ( [1/2],[3/2] )
D = √{(x2 − x1)2 + (y2 − y1)2} = √{( − 2 − (3))2 + (5 − ( − 2))2} = √{( − 5)2 + (7)2} = √{25 + 49} = √{74} ≈ 8.60
Find the midpoint and distance of the segment with endpoints ( − 3, − 5),(1,7)
  • Use the midpoint formula M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) and distance formula D = √{(x2 − x1)2 + (y2 − y1)2}
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [( − 3 + 1)/2],[( − 5 + 7)/2] ) = ( [( − 2)/2],[2/2] ) = ( − 1,1 )
D = √{(x2 − x1)2 + (y2 − y1)2} = √{(1 − ( − 3))2 + (7 − ( − 5))2} = √{(4)2 + (12)2} = √{16 + 144} = √{160} ≈ 12.65
Triangle xyz has vertices x(5,9), y(1,1), and z(9,1). Find the length of the median from x to line yz.
  • Step 1 - Find the midpoint M between the points y and z.
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [(1 + 9)/2],[(1 + 1)/2] ) = ( 5,1 )
  • Step 2 - Find the Distance between the midpoint and point x.
  • D = √{(x2 − x1)2 + (y2 − y1)2} = √{(5 − (5))2 + (1 − (9))2} = √{(0)2 + ( − 8)2} = √{64} = 8
The length of the median from x to line yz is 8.
Triangle xyz has vertices x(8,8), y(2,4), and z(10,0). Find the length of the median from x to line yz.
  • Step 1 - Find the midpoint M between the points y and z.
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [(2 + 10)/2],[(4 + 0)/2] ) = ( 6,2 )
  • Step 2 - Find the Distance between the midpoint and point x.
  • D = √{(x2 − x1)2 + (y2 − y1)2} = √{(6 − (8))2 + (2 − (8))2} = √{(2)2 + ( − 6)2} = √{4 + 36} = √{40} ≈ 6.32
The length of the median from x to line yz is ≈ 6.32
Triangle xyz has vertices x( − 4, − 8), y(1,2), and z(9, − 4). Find the length of the median from x to line yz.
  • Step 1 - Find the midpoint M between the points y and z.
  • M = ( [(x1 + x2)/2],[(y1 + y2)/2] ) = ( [(1 + 9)/2],[(2 + − 4)/2] ) = ( 5, − 1 )
  • Step 2 - Find the Distance between the midpoint and point x.
  • D = √{(x2 − x1)2 + (y2 − y1)2} = √{(5 − ( − 4))2 + ( − 1 − ( − 8))2} = √{(9)2 + (7)2} = √{81 + 49} = √{130} ≈ 11.40
The length of the median from x to line yz is ≈ 11.40
Find the perimeter and area of the triangle with vertices at A(2,4) B(5,2) C(6,10).
  • Step 1. Find lenths for sides AC ,AB andBC and add them to find the perimeter.
  • AC : (2,4)(6,10)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(6 − (2))2 + (10 − (4))2} = √{(4)2 + (6)2} = √{16 + 36} = √{52} ≈ 7.21
  • AB : (2,4)(5,2)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(5 − (2))2 + (2 − (4))2} = √{(3)2 + ( − 2)2} = √{9 + 4} = √{13} ≈ 3.61.
  • BC : (5,2),(6,10)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(6 − (5))2 + (10 − (2))2} = √{(1)2 + (8)2} = √{1 + 64} = √{65} ≈ 8.06
  • Perimeter = AC +AB +BC = 7.21 + 3.61 + 8.06 = 18.88
  • Step 2: Find the area of the triangle using the formula A = [1/2]bh
  • For this formula to work, you must have a Right Triangle.
  • Recall that perpendicular lines form a 90 degree angle, and the product of their slopes is − 1.
  • Check the slopes of AB and AC to see if their product of their slopes equals − 1.
  • mAB = [rise/run] =
  • mAC = [rise/run] =
  • mAB = [rise/run] = [( − 2)/3] = − [2/3]
  • mAC = [rise/run] = [6/4] = [3/2]
  • Check the product of the slopes
  • mAB*mAC = [( − 2)/3]*[3/2] = − 1
  • Find the area
A = [1/2]bh = [1/2](3.61)(7.21) = 13
Find the perimeter and area of the triangle with vertices at A(2,4) B(4, - 2) C(11,7).
  • Step 1. Find lenths for sides AC ,AB andBC and add them to find the perimeter.
  • AC : (2,4)(11,7)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(11 − (2))2 + (7 − (4))2} = √{(9)2 + (3)2} = √{81 + 9} = √{90} ≈ 9.49
  • AB : (2,4)(4, − 2)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(4 − (2))2 + ( − 2 − (4))2} = √{(2)2 + ( − 6)2} = √{4 + 36} = √{40} ≈ 6.32
  • BC : (4, − 2),(11,7)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(11 − (4))2 + (7 − ( − 2))2} = √{(7)2 + (9)2} = √{49 + 81} = √{130} ≈ 11.40
  • Perimeter = AC +AB +BC = 9.49 + 6.32 + 11.40 = 27.21
  • Step 2: Find the area of the triangle using the formula A = [1/2]bh
  • For this formula to work, you must have a Right Triangle.
  • Recall that perpendicular lines form a 90 degree angle, and the product of their slopes is − 1.
  • Check the slopes of AB and AC to see if their product of their slopes equals − 1.
  • mAB = [rise/run] =
  • mAC = [rise/run] =
  • mAB = [rise/run] = [( − 6)/2] = − 3
  • mAC = [rise/run] = [3/9] = [1/3]
  • Check the product of the slopes
  • mAB*mAC = − 3*[1/3] = − 1
  • Find the area
A = [1/2]bh = [1/2](6.32)(9.49) = 30
Find the perimeter and area of the triangle with vertices at A(1, - 1) B(2,2) C(7, - 3).
  • Step 1. Find lenths for sides AC ,AB and BC and add them to find the perimeter.
  • AB : (1, − 1)(2,2)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(2 − (1))2 + (2 − ( − 1))2} = √{(1)2 + (3)2} = √{1 + 9} = √{10} ≈ 3.16
  • AC : (1, − 1)(7, − 3)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(7 − (1))2 + ( − 3 − ( − 1))2} = √{(6)2 + ( − 2)2} = √{36 + 4} = √{40} ≈ 6.32
  • BC : (2,2),(7, − 3)
  • D = √{(x2 − x1)2 + (y2 − y1)2} =
  • √{(7 − (2))2 + ( − 3 − (2))2} = √{(5)2 + ( − 5)2} = √{25 + 25} = √{50} ≈ 7.07
  • Perimeter = AC +AB +BC = 3.16 + 6.32 + 7.07 = 16.55
  • Step 2: Find the area of the triangle using the formula A = [1/2]bh
  • For this formula to work, you must have a Right Triangle.
  • Recall that perpendicular lines form a 90 degree angle, and the product of their slopes is − 1.
  • Check the slopes of AB and AC to see if their product of their slopes equals − 1.
  • mAB = [rise/run] =
  • mAC = [rise/run] =
  • mAB = [rise/run] = [3/1] = 3
  • mAC = [rise/run] = [( − 2)/6] = − [1/3]
  • Check the product of the slopes
  • mAB*mAC = 3*[( − 1)/3] = − 1
  • Find the area
A = [1/2]bh = [1/2](3.16)(6.32) = 10

*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

Midpoint and Distance Formulas

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
  • Midpoint Formula 0:15
    • Example: Midpoint
  • Distance Formula 2:30
    • Example: Distance
  • Example 1: Midpoint and Distance 4:58
  • Example 2: Midpoint and Distance 8:07
  • Example 3: Median Length 18:51
  • Example 4: Perimeter and Area 23:36

Transcription: Midpoint and Distance Formulas

Welcome to Educator.com.0000

Today is the first in a series of lectures on conic sections.0002

And we are going to start out with a review of two formulas that you will be applying later on, when we work with circles and hyperbolas and other conic sections.0005

Starting out with the midpoint formula: this formula gives you the midpoint of line segments with endpoints (x1,y1) and (x2,y2).0015

So, let's take an example, just to illustrate this: let's say you were asked to find the midpoint of a line segment with endpoints at (2,3) and (5,1).0028

If these were given as the endpoints, and you were asked to find the midpoint, you could go ahead and apply this formula.0044

So, to visualize this, let's draw out this line segment: this is at (2,3); that is one endpoint, and (5,1) is the other endpoint.0053

I draw a line between these two; and I am looking for the midpoint, which is somewhere around here.0065

Looking at this formula, it is actually somewhat intuitive, because what you are doing is finding0072

the average of the x-values and the average of the y-values, which will give you the middle of each (the midpoint).0078

So, x1 + x2 would give me 2 + 5 (the two x-values), divided by 2.0085

That is going to give me the x-coordinate of the midpoint of this line segment.0095

For the y-coordinate, I am going to add the two y's and divide by 2; so I will average those two y-values, which are 3 and 1.0099

2 + 5 gives me 7/2; 3 + 1 is 4/2; I can simplify to (7/2,2); and I could rewrite this, even, as (3 1/2,2) to make it a little easier to visualize on the graph.0110

This is the midpoint; it is going to be at 3 1/2 (that is going to be my x-coordinate); and 2 is going to be my y-coordinate.0128

And that is the midpoint; this is a fairly straightforward formula; you will need to apply it in a little while, when we start working with circles.0138

Also, to review the distance formula: if you recall, the distance formula is based on the Pythagorean theorem.0151

And the distance formula tells us that we can find the distance between two sets of points0159

if we take the square root of x2 - x1, squared, plus y2 - y1, squared.0163

Now recall: if I am given two points; if I am asked to find the distance between a set of two points, (2,1) and (5,6),0172

I can assign either one to be...I could assign this as (x1,y1), (x2,y2);0192

or I could do it the other way around: I could say this is (x2,y2); this is (x1,y1).0198

It doesn't matter, as long as you assign it and stick with that; you don't mix and match and say this is (x1,y2).0202

Just assign either set; it doesn't matter which set you call which; just stay consistent with the order that you use the points in.0209

OK, so let's look at what this would graph out to.0216

(2,1) and (5,6) would be right about up there; so I have a line segment, and I am asked to find the distance.0221

So, the distance is going to be equal to the square root of...I am going to call this (x1,y1) and this (x2,y2).0231

So, x2 is 5, minus x1, which is 2; take that squared,0239

and add it to y2, which is 6, minus y1, which is 1, squared.0247

Therefore, distance equals 32, plus 6 minus 1 is 52.0253

The distance equals the square root of 32, which is 9, plus 25.0259

Therefore, distance equals √36; distance equals 6.0269

The distance from this point to this point is equal to 6.0277

And again, this is review from an earlier lecture, when we discussed the Pythagorean theorem in Algebra I0282

and talked about how the distance formula comes from that.0288

So, you can always go back and review that information; but this is the application of the distance formula.0291

In the first example, we are asked to find the midpoint and distance of the segments with these endpoints (-9,-7) and (-3,-1).0299

So, recall that the midpoint formula just involves finding the average of the x-coordinates0310

of the two points, and the average of the y-coordinates of the two points.0315

Applying these values to this set of equations gives me -9 + -3, divided by 2, and -7 + -1, divided by 2.0324

Adding -9 and -3 gives me -12, divided by 2; and that is -8 divided by 2.0349

This becomes -6 (-12/2 is -6), and then -8/2 is -4.0359

So, that is the midpoint; that was the midpoint formula.0367

Recall that the distance formula is (x2 - x1)2 + (y2 - y1)2.0377

I am going to assign this as (x1,y1) and this as (x2,y2).0392

Again, it doesn't matter; you can do it the other way around.0404

Distance equals...I have x2 as -3, minus -9, all of this squared; plus y2, which is -1, minus -7, and that whole thing squared.0406

This gives me -3; a negative and a negative is a positive, so plus 9, squared, plus -1...and a negative and a negative gives me + 7...squared.0433

Therefore, distance equals the square root of...9 - 3 is 6, squared, and then 7 - 1 is also 6, squared.0444

Distance equals √(36 + 36), or radic;72.0453

But I can take this a step farther, and say, "OK, this is the square root of 36 times 2," and then simplify that,0461

because this is a perfect square, to 6√2.0467

So, I found the midpoint; it is (-6,-4); that is the midpoint of this segment with these endpoints.0475

And the distance between these endpoints is 6√2.0481

In Example 2, we are asked to find the midpoint and distance of the segment with these endpoints.0488

We go about it as we usually do; but this time we are working with radicals,0493

so we have to be really careful that we keep everything straight.0496

Recall that the midpoint formula is the average of the x-coordinates, and then the average of the y-coordinates.0499

Therefore, the midpoint equals x1...the square root of 2 + 3, plus x2, 2√2 - 4√3, divided by 2;0512

for the y-coordinate of the midpoint, we are going to get √3 - 5, plus 4√3, plus 2√5.0526

And then, we simplify this as much as we can.0536

Here, I have two like radicals: they have the same radicand, so I can combine those to make this 3√2.0540

I have a constant, and then this -4√3.0549

So, this is the x-coordinate of the midpoint; the y-coordinate is √3 here and 4√3; those can be combined into 5√3;0555

and then, let's see: this actually should have a radical over it, because that is a radical up there;0567

I have -√5 + 2√5; that is going to leave me with just + √5, divided by 2.0577

And the midpoint is given by this; it is a little bit messy-looking, but it is correct--you can't simplify, really, any farther.0585

So, we are just going to leave it as it is.0592

Now, we are also asked to find the distance; so I am going to go ahead and work that out here.0594

We found the midpoint; we are working with these same endpoints, but we are finding the distance.0599

Recall that the distance formula is the square root of (x2 - x1)2 + (x2 - y1)2.0603

So, I am going to let this be (x1,y1), (x2,y2).0617

I could have done it the other way around; it doesn't matter, as long as you are consistent.0625

Therefore, the distance is going to be equal to y2, which is 2√2, minus 4√3;0630

and I am going to take that, and from that I am going to subtract y1, which is √2 + 3.0645

So, this covers my (x2 - x1)2.0657

I am going to add that to (y2 - y1): I go over here, and I have y2; that is 4√3 + 2√5.0664

And I am going to subtract y1 from that, which is over here; and that is √3 - √5; and this whole thing is also going to be squared.0678

Let's apply these negative signs to everything inside the parentheses, so we can start doing some combining.0692

This is going to give me 2√2 - 4√3; negative...that is going to give me -√2;0700

apply the negative to that 3--it is going to give me -3; all of this is going to be squared.0713

Plus 4√3, plus 2√5...apply the negative to each term inside the parentheses to get -√3;0718

and this negative, and then the negative √5, gives me a positive √5; and this whole thing is squared.0728

All right, so let's see if I can do some combining to simplify a little bit before I start squaring everything,0736

because that is going to be the most difficult step.0742

All right, this gives me 2√2, and this is minus √2; so I can combine these two to get √2.0747

-4√3, minus 3; this whole thing squared, plus 4√3 - √3--that simplifies to 3√3.0757

2√5 + √5 gives me 3√5; squared.0773

Now, I need to just square everything, and then combine it.0780

There is no easy way around this first one; what I am going to do is multiply √2 times itself, times this, times this,0784

and then go on with the second, and finally the third, term.0792

So, this √2 times √2 is simply going to give me 2; √2 times -4√3 is going to give me -4... 2 times 3 is √6.0795

Then, √2 times -3 is going to give me -3√2.0817

OK, now I take -4√3 and multiply it by this first term, √2, to get -4; 3 times 2 is 6.0828

When I multiply this times itself, I am going to get -4 times -4; that is going to give me 16,0840

times √3 times √3 is going to give me 3.0846

Then, I multiply this times -3 to get -4 times -3 is 12√3.0851

Finally, multiplying -3 times √2 gives me -3√2; -3 times -4 is 12√3; -3 times -3 is 12√3.0861

And then, -3 times -3 is 9; all of this is the trinomial squared; now let's square the binomial.0878

3√3 times 3√3 is going to be...3 times 3 is 9; and then √3 times √3 is just going to be 3.0887

So, looking up here, just to make this a little clearer: 3√3 + 3√5, times itself (squared)...0902

I am going to do my first terms, and I get this; then I do the outer terms, plus the inner terms,0919

which would just be this times this times 2; so that is going to give me 2 times 3 times 3, which is 9,0924

times 3 times 5, with a radical over it; so that is this.0941

So, what I did is took...really, it is just outer plus inner, which is going to give me the outer, which is 9√15, plus the inner, which also 9√15.0951

And this is going to end up giving me 18√15, which is the same as what I have here.0969

And then, finally, the last terms are going to give me 3 times 3 is 9, and then √5 times √5 is just going to give me 5.0976

OK, now combining what we can in order to just make this a bit simpler: I have a constant here;0989

I am just going to cluster all of my constants together in the beginning; that is 2; 16 times 3 is 48, so that is 48;1007

what other constants do I have?--I have a 9, and then I have 27, and I have 45.1016

All right, so those are my constants.1035

Now, for terms with a √6 in them, I have 2 of those: -4√6 and -4√6 is going to give me -8√6.1037

So, I took care of the constants and the terms with a radicand of 6.1048

Next, let's look at terms with a radicand of 2: -3√2, and I have -3√2 here; and that is it.1052

I add those two together, and I am going to get -6√2.1062

So, these are all taken care of: √2, √6, constant: now I have √3.1066

12√3--do I have any other terms like that?--yes: 12√3, so I have 2 of those; and that is going to be...12 and 12 is 24√3.1072

Let's see what else I have left: I took care of those, those...that just leaves me with...2 times 9 is 18 times √15.1083

Therefore, the distance equals...putting this all together, if you added these up, you will get 131 - 8√6 - 6√2 + 24 √3 + 18 √15.1094

So, this is the distance; and this was really a lot of practice just working with multiplying and adding and subtracting radicals.1115

We found the midpoint in the previous slide; and here is the distance for the segment with these endpoints, applying the distance formula.1123

Example 3: Triangle XYZ has vertices X (4,9), Y (8,-9), and Z (-5,2).1132

Find the length of the median from X to line YZ.1143

Definitely, a sketch would help us to solve this; so let's sketch this triangle.1146

And it has...we will call this X at (4,9); at (8,-9), we are going to have Y; and then, Z is going to be over here at (-5,2).1152

The median is going to be a line going here from here right to this midpoint.1173

So, what we are asked to find is the length: we can use the distance formula to find the length.1179

But in order to use the distance formula, I need this endpoint and this endpoint.1185

But I don't have this endpoint; however, I can find it, because if you look at this, this is going to draw out right to here, right in the middle.1190

So, this is the midpoint; if I find the midpoint of YZ, then I have the endpoint of this median.1197

Using the midpoint formula: (x1 + x2)/2, and then for the y-coordinate, (y1 + y2)/2.1208

So, I am looking for the midpoint of YZ: that is going to give me...1227

for Y, the x-value is 8; for Z, it is -5; divided by 2; for Y, the y-value is -9; for Z, the y-value is 2; divided by 2.1235

This is going to give me a midpoint of 8 - 5; that is 3/2; -9 + 2 is -7/2.1252

This is the midpoint, which is (3/2, -7/2); now I can use my distance formula,1262

distance equals √((x2 - x 1)2 + (y2 - y1)2).1271

I can use that distance formula to find this.1282

And I am going to call (4,9)...I need the distance from x to this point m...I am going to call this1283

(x1,y1), and this (x2,y2), applying my distance formula.1292

So first, x2, which is 3/2, minus x1, which is 4--I am going to rewrite that as 8/2 to make my subtraction easier.1299

But this is just 4; squared; plus y2, which is -7/2, minus 9, which I am going to rewrite as 18/2.1311

Again, I am just moving on to the next step of finding the common denominator.1324

But this is my (x1,y1), (x2,y2); that is where these came from.1328

Therefore, the distance equals...3/2 - 8/2 is -5/2; and we are going to square that;1339

plus -18 and -7 combines to -25/2, squared.1347

Therefore, the distance equals the square root of...this is 25 (5 squared), over 4, plus...25 squared is actually 625, divided by 4,1357

which equals the square root of 625 and 25 is 650, and they have the common denominator of 4.1370

And you could leave it like this, or take it a step farther and write this as 650 times 1/4, which equals 1/2.1379

And you could even look and see if there are any other perfect squares that you could factor out.1389

But this does give us the length of the median; in order to find the length of the median, we had to find this other endpoint,1393

which conveniently was the midpoint of YZ, so we found the midpoint of YZ and used that as the other endpoint.1401

Then I had X as an endpoint and the midpoint as the endpoint.1408

Plug that into the distance formula to get the length.1412

Working with triangles again: find the perimeter and area of the triangle with vertices at (1,-4), (-1,-2), and (6,1).1418

So, to help visualize this, we are going to draw it out on the coordinate plane.1427

(1,-4) is my first point; (-1,-2) is my second point; and then 6 is going to be about here; this is 6; 1 will be right here.1433

Let's see what this triangle looks like, and if the graph can help me.1449

Now, I am asked to find the perimeter and area: if I want to find the area of a right triangle,1459

I know that it is going to be 1/2 the base times height.1462

The problem is that I don't know if I am working with a right triangle; I may be; but this definitely does not look like a right angle.1465

This may be a right angle, but I am not sure.1472

In order to determine if it is a right angle (let's call this A, B, and C)--is this a right angle?--1475

well, if it is a right angle, that means that this AC and BC are going to be perpendicular.1485

And recall that perpendicular lines...the product of their slope equals -1.1492

So, what I am going to do is go ahead and find the slope of these two.1510

And I am going to determine what this slope is of AC; what BC is; find their product; and if they are a right angle, then I can proceed.1515

So, this point right here is (-1,-2); this point is (1,-4); and this point is (6,1).1523

Slope is just change in y, divided by change in x.1534

So, let's find the slope of AC: that is going to be the change in y (-2 - -4), divided by change in x (which is -1 - 1).1538

So, this is going to be -2 plus 4, divided by -2; this is going to give me 4 divided by -2, which is -2.1561

I just double-check this: -2 minus -4...actually, correction: 4 minus 2 is going to give me, of course, 2; and the slope, therefore, will be -1.1579

OK, now I also need to find, over here (to keep this separate), the slope of BC.1596

So, change in y: y is 1 - -4, over change in x: x is 6 - 1; this is going to be equal to 1 + 4, divided by 6 - 1 is 5, which equals 5/5.1606

So, the slope of BC equals 1.1630

Now, this means that, if I have the slope of BC, times the slope of AC, equals 1 times -1, or -1.1634

This is a right angle; I have a right triangle; I can use my formula, 1/2 the base times the height.1649

What I need to do next is find the length of these sides, using the distance formula.1656

So, let's start out by finding the length of side AC.1661

Recall your distance formula: for the distance formula, I am going to take x2, side AC,1668

and I am going to call...it doesn't matter which one...I am going to call this, for this first one, (x1,y1), (x2,y2).1678

x2 is -1, minus x1, which is 1, squared, plus y2, which is going to be -2, minus y1, which is -4.1689

So, the length of AC equals...-1 and -1 is -2, squared; plus -2...and a negative and a negative is a positive, so -2 + 4 is going to give me 2, squared.1705

Therefore, the length of AC is going to equal the square root of 4 plus 4, which equals the square root of 8.1723

So, that is the length of AC.1733

Now, I need to find the length of another side to find the area.1735

I found AC; let's go for BC next--the length of BC.1739

I am going to make this (x1,y1),(x2,y2).1748

So, I am starting out with my x2, which is 6, minus x1, which is 1, squared,1753

plus x2, which is 1, minus -4 (that is y2 - y1), squared,1761

equals the square root of...6 minus 1 is 5, squared; 1 minus -4...this becomes 1 + 4, so that is 5, squared;1773

So, this equals the square root of 25 + 25, or the square root of 50.1783

All right, so now I have two sides: I know that this side, AC, has a length of √8; and I know that BC is √50.1789

So, the area equals 1/2 the base times the height; so the area equals 1/2 (√8)(√50), which equals 1/2√400.1804

All right, and you could go on and then simplify this, because this would give you the perfect square of 20 times 20,1828

because 202 would give you 400; so then I could make this 1/2(20) is 10.1845

All right, now we still need to find the perimeter.1856

In order to find the perimeter, I need this third side; and I can use the Pythagorean theorem, because this is the hypotenuse.1859

And I know that a2 + b2 = c2.1864

So, a2 is...one side is the square root of 8, squared, plus b2, the square root of 50 squared, equals c2.1869

Well, the square root of 8 squared is 8, plus 50 (because the square root of 50 squared is going to be 50), equals c2.1879

So, 58 = c2; therefore, c equals √58.1890

All right, so I found the area right here; now, the perimeter.1895

The perimeter equals the sum of the three sides: so that is √8 + √50 + √58.1907

And you could go on and do a little simplification.1920

You could pull the perfect squares out of here.1922

So, I could go a little farther with this and say, "OK, 8 is equal to 4 minus 2; 4 is a perfect square, so this is 2√2."1927

50 is the square root of 25 times 2, so that is going to give me 5√2, plus √58.1935

But you can't combine any of these, because they are not like radicals.1942

So, in this example, we applied the distance formula, as well as the Pythagorean theorem and some geometry,1947

to find the area and the perimeter of this triangle.1953

Thanks for visiting Educator.com; and that concludes this lesson on the midpoint and distance formulas.1957

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