Graphs: The Distance and Midpoint Formulas in the Rectangular Coordinate System

Chapter 2: Graphs

Section 2.1: The Distance and Midpoint Formulas

This section introduces the foundational concepts of the rectangular (Cartesian) coordinate system, and explains how to calculate the distance between two points and the midpoint of a line segment in the plane. These tools are essential for graphing and analyzing geometric relationships in precalculus.

Rectangular Coordinates

The rectangular coordinate system (or Cartesian coordinate system) is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection is called the origin, denoted as O. The plane defined by these axes is called the xy-plane, and the axes themselves are referred to as the coordinate axes.

• Any point P in the xy-plane is represented by an ordered pair (x, y), where x is the x-coordinate (abscissa) and y is the y-coordinate (ordinate).

• The coordinate axes divide the plane into four regions called quadrants:

• Quadrant I: x > 0, y > 0

• Quadrant II: x < 0, y > 0

• Quadrant III: x < 0, y < 0

• Quadrant IV: x > 0, y < 0

Distance Formula

The distance formula allows us to find the distance between two points in the plane. If the points are P_1 = (x_1, y_1) and P_2 = (x_2, y_2), the distance d between them is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• This formula is derived from the Pythagorean Theorem, treating the difference in x-values and y-values as the legs of a right triangle.

Example: Finding the Distance Between Two Points

Find the distance between the points (0, 2) and (4, 5).

1. Plot the points and connect them with a straight line.

2. Draw a horizontal and a vertical line to form a right triangle.

3. Apply the distance formula:

$d = \sqrt{(4 - 0)^2 + (5 - 2)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Applications: Using Algebra to Solve Geometry Problems

The distance formula can be used to determine the lengths of sides in geometric figures, such as triangles, and to classify triangles (e.g., as right triangles using the converse of the Pythagorean Theorem).

• To check if a triangle is a right triangle, calculate the lengths of all sides and verify if the sum of the squares of two sides equals the square of the third side.

Midpoint Formula

The midpoint formula finds the coordinates of the point exactly halfway between two given points P_1 = (x_1, y_1) and P_2 = (x_2, y_2):

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• The midpoint is useful for finding centers, bisecting segments, and solving geometric problems.

Example: Finding the Midpoint of a Line Segment

Given points P1 = (–4, 4) and P2 = (4, 2), the midpoint M is:

$M = \left( \frac{-4 + 4}{2}, \frac{4 + 2}{2} \right) = (0, 3)$

Additional info: The rectangular coordinate system and the distance and midpoint formulas are foundational for all subsequent topics in analytic geometry and precalculus, including graphing equations, analyzing geometric figures, and solving real-world problems involving distance and location.