# Coordinates, Distance, and Midpoint

## Overview

### Description

When the coordinate plane is introduced, the focus begins with learning to plot ordered pairs to help visualize a graph. In algebra, the coordinate plane is also useful for deriving formulas. Two formulas, the distance formula and the midpoint formula, show how to calculate the distance between two points and the halfway point. They are commonly used in real-world problems involving distance and for writing proofs in analytic geometry.

### At A Glance

• The coordinate plane is defined by two perpendicular lines called axes, which divide the coordinate plane into four quadrants. Any point in the coordinate plane can be located by an ordered pair of values $\left(x, y\right)$.
• The distance between two points with the same $x$- or $y$-coordinate is the length of a horizontal or vertical line segment connecting those points. This can be found by counting grid squares or subtracting the coordinates that are not equal.
• The distance between two points that do not have the same $x$- or $y$-coordinate can be found by drawing a right triangle with the line segment between the points as the hypotenuse and using the Pythagorean theorem.
• Using a right triangle, a formula can be derived for the distance between any two points in the coordinate plane.
• The midpoint between two points with the same $x$- or $y$-coordinate is the midpoint of a horizontal or vertical line segment connecting those points.
• The midpoint formula can be used to find the midpoint between any two points in the coordinate plane.