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Section 2.1 The Rectangular Coordinate System

Objectives

1. Plotting Ordered Pairs in the Cartesian Plane

2. Graphing Equations by Plotting Points

3. Finding the Midpoint of a Line Segment using the Midpoint Formula

4. Finding the Distance between Two Points using the Distance Formula

Throughout chapter one, we solved several types of equations including linear equations, quadratic equations, rational equations, etc. Each of these equations had something in common. They were all examples of equations in one variable. In this chapter, we will study equations involving two variables. A solution to an equation involving two variables consists of a pair of numbers, an x-value and a y-value for which the equation is true.

Each pair of values is called an ordered pair because the order does matter. We use the notation [pic]to represent an ordered pair. Notice that the x-coordinate or abscissa is first, followed by the y-coordinate or ordinate listed second. To represent an ordered pair graphically, we use the rectangular coordinate system also called the Cartesian coordinate system named after the French mathematician René Descartes. The plane used in this system is called the coordinate plane or Cartesian plane. The horizontal axis (x-axis) and the vertical axis (the y-axis) intersect at the origin O and naturally divide the Cartesian plane into 4 quadrants labeled quadrants I, II, III and IV.

Quadrant II Quadrant I

Quadrant III Quadrant IV

Objective 1: Plotting Ordered Pairs in the Cartesian Plane

To plot the point [pic], go 2 units to the left of the origin on the x-axis then move 3 units up parallel to the y-axis. The point corresponding to the ordered pair [pic]is labeled in the figure below and is located in Quadrant II.

Objective 2: Graphing Equations by Plotting Points

One way to sketch the graph of an equation is to find several ordered pairs which satisfy the equation, plot those ordered pairs then connect the points with a smooth curve. We choose arbitrary values for one of the coordinates then solve the equation for the other coordinate.

Objective 3: Finding the Midpoint of a Line Segment using the Midpoint Formula

Suppose we wish to find the midpoint [pic] of the line segment from [pic] and [pic]. To find this midpoint, we simply “average” the x and y coordinates respectively. In other words, the x coordinate of the midpoint is [pic] while the y-coordinate of the midpoint is [pic].

Midpoint Formula

The midpoint of the line segment from [pic] to [pic] is

[pic].

Objective 4: Finding the Distance between Two Points using the Distance Formula

Recall that the Pythagorean Theorem states that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse or [pic].

The Pythagorean

Theorem [pic]

We can use the Pythagorean Theorem to find the distance between any two points in a plane. To find the length [pic] of the line segment AB, consider the point [pic] which is on the same vertical line segment as point A and the same horizontal line segment as point B. The triangle formed by points A, B and C is a right triangle whose hypotenuse has length[pic]. The horizontal leg of the triangle has length [pic] while the vertical leg of the triangle has length [pic].

[pic] Use the Pythagorean Theorem.

[pic] Use the Square Root Property.

But since distance cannot be negative, exclude [pic].

[pic] Use the positive square root only.

[pic] For any quantity A, [pic].

This formula is known as the distance formula.

The Distance Formula

The distance between any two points[pic] and [pic]is given by the formula

[pic][pic]

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[pic]

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How to do Cartesian plane equations? In order to solve the problem you need to: Choose your x-values to input into the formula. Usually you’ll want to choose integers (whole numbers plus zero and negatives like -10, 3, 1, 7…) because they are easier to ... Create a few (x, y) coordinates. Plot the points on a Cartesian plane. Draw the curve or line of the graph.