Chapter 1: Graphs – The Distance and Midpoint Formulas

Chapter 1: Graphs

Section 1.1: The Distance and Midpoint Formulas

This section introduces foundational concepts for locating points in one and two dimensions, and provides formulas for calculating the distance and midpoint between points in the Cartesian plane. These concepts are essential for understanding graphs and geometric relationships in precalculus.

Rectangular (Cartesian) Coordinate System

The rectangular coordinate system (also called the Cartesian coordinate system) is used to locate points in two dimensions using two perpendicular axes:

• x-axis: The horizontal axis

• y-axis: The vertical axis

• Origin: The point where the axes intersect, with coordinates (0, 0)

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

Each point in the plane is represented by an ordered pair (x, y):

• The x-coordinate indicates horizontal position

• The y-coordinate indicates vertical position

• Points to the left of the origin have a negative x-coordinate; points to the right have a positive x-coordinate

• Points below the origin have a negative y-coordinate; points above have a positive y-coordinate

• Every point on the x-axis has coordinates (a, 0); every point on the y-axis has coordinates (0, b)

To locate a point (x, y): start at the origin, move horizontally by x units, then vertically by y units.

Distance Formula

The distance formula is used to find the length between two points in the plane. It is derived from the Pythagorean Theorem:

• Given points $A(x_1, y_1)$ and $B(x_2, y_2)$, the distance $d$ between them is:

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

• The horizontal leg of the triangle is $|x_2 - x_1|$

• The vertical leg is $|y_2 - y_1|$

• The hypotenuse (distance) is found using the Pythagorean Theorem

Key Takeaways:

• The distance between two points is never negative

• The distance is zero only if the points are identical

• The order of the points does not affect the result

Example: Find the distance between $A(1, 2)$ and $B(4, 6)$:

$d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Midpoint Formula

The midpoint formula finds the center point of a line segment between two points:

• Given points $A(x_1, y_1)$ and $B(x_2, y_2)$, the midpoint $M$ is:

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

• The midpoint is a point, so it is written as an ordered pair

• It is found by averaging the x-coordinates and y-coordinates of the endpoints

• The distance from the midpoint to each endpoint is equal

Example: Find the midpoint between $A(2, 3)$ and $B(6, 7)$:

$M = \left( \frac{2 + 6}{2}, \frac{3 + 7}{2} \right) = (4, 5)$

Applications: Geometry Problems

These formulas are used to solve geometric problems, such as:

• Finding the lengths of triangle sides

• Determining if a triangle is a right triangle (using the Pythagorean Theorem)

• Calculating the area of triangles

Example: Given three points, plot them, form a triangle, find side lengths using the distance formula, and check for a right angle.