Functions and Graphs

Distance and Midpoint Formulas; Circles

This section covers essential geometric formulas and concepts in the coordinate plane, including how to calculate the distance between two points, find the midpoint of a segment, and understand the equations and properties of circles. These tools are foundational for analytic geometry and are widely used in precalculus and beyond.

Distance Formula

The distance formula allows you to find the distance between two points in the rectangular coordinate system.

• Formula: The distance d between points and is given by:

• To compute the distance, subtract the x-coordinates and y-coordinates, square each difference, add the results, and take the principal square root.

Example: Find the distance between and .

Rounded to two decimal places:

Midpoint Formula

The midpoint formula finds the point exactly halfway between two endpoints of a line segment.

• Formula: The midpoint of the segment with endpoints and is:

• Take the average of the x-coordinates and the average of the y-coordinates.

Example: Find the midpoint of the segment with endpoints and .

Definition of a Circle

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance is the radius.

• Center: The fixed point

• Radius: The fixed distance

Standard Form of the Equation of a Circle

The standard form of a circle’s equation with center and radius is:

• All points that satisfy this equation are on the circle.

Example: Write the standard form of the equation of a circle with center and radius $10$.

Finding the Center and Radius from Standard Form

Given a circle’s equation in standard form, you can identify the center and radius directly:

• Center:

• Radius: (where is the constant on the right side of the equation)

Example: For :

• Center:

• Radius: $5$

General Form of the Equation of a Circle

The general form of a circle’s equation is:

• Here, , , and are real numbers.

• To convert to standard form, complete the square for both and terms.

Converting General Form to Standard Form

To rewrite the general form as standard form:

1. Group and terms:

2. Complete the square for and :

3. Write in standard form:

Example: Convert to standard form.

Group and complete the square:

• Center:

• Radius: $5$