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Sunday 19th of May
Calculations with Negative Numbers
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Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
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Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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The Distance and Midpoint Formulas

Recall from the Pythagorean Theorem that, in a right triangle, the hypotenuse c and sides a and b are related by a2 + b2 = c2. Conversely, if a2 + b2 = c2, the triangle is a right triangle (see the figure below).

Suppose you want to determine the distance d between the two points (x1, y1) and (x2, y2) in the plane. If the points lie on a horizontal line, then y1 = y2 and the distance between the points is | x2 - x1 |. If the points lie on a vertical line, then x1 = x2 and the distance between the points is | y2 - y1 |. If the two points do not lie on a horizontal or vertical line, they can be used to form a right triangle, as shown in the figure below.

The length of the vertical side of the triangle is | y2 - y1 | and the length of the horizontal side is | x2 - x1 |. By the Pythagorean Theorem, it follows that

Replacing | x2 - x1 | 2 and |y2 - y1 | 2 by the equivalent expressions (x2 - x1) 2 and (y2 - y1) 2 produces the following result.

Distance Formula

The distance d between the points (x1, y1) and (x2, y2) in the plane is given by



Example 1

Finding the Distance Between Two Points

Find the distance between the points (-2, 1) and (3, 4).



Example 2

Verifying a Right Triangle

Verify that the points (2, 1), (4, 0), and (5, 7) form the vertices of a right triangle.


The figure below shows the triangle formed by the three points.

The lengths of the three sides are as follows.


d12 + d22 = 45 + 5 = 50 Sum of squares of sides


d32 = 50 Square of hypotenuse

you can apply the Pythagorean Theorem to conclude that the triangle is a right triangle.


Example 3

Using the Distance Formula

Find x such that the distance between (x, 3) and (2, -1) is 5.


Using the Distance Formula, you can write the following.

Distance Formula
25 = (x2 - 4x + 4) + 16 Square both sides.
0 = x2 - 4x - 5 Write in standard form.
0 = (x - 5)(x + 1) Factor.

Therefore, x = 5 or x = -1, and you can conclude that there are two solutions. That is, each of the points (5, 3) and (-1, 3) lies five units from the point as shown in the following figure.

The coordinates of the midpoint of the line segment joining two points can be found by “averaging” the x-coordinates of the two points and “averaging” the y-coordinates of the two points. That is, the midpoint of the line segment joining the points (x1, y1) and (x2, y2) in the plane is

For instance, the midpoint of the line segment joining the points (-5, -3) and (9, 3) is

as shown in the figure below

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