In this lesson, students move from finding distances between specific points in the coordinate plane to generalizing an equation or formula to find the distance between any two points in the plane. They begin by connecting to previous work with the Pythagorean Theorem to explain how the difference in - coordinates and in -coordinates of the points represents the lengths of the legs of a right triangle. Next, students apply this reasoning repeatedly as they make observations about a set of points (MP8). Finally, students use their observations and generalizations to build an equation that describes the distance, , between any two points,  and .

In future courses, students will represent the equation of a circle on the plane using a similar equation. This work provides a strong connection point between the distance formula, right triangles, and circles, which will benefit students in making future connections. It is not necessary at this time that students recognize that the set of all points a given distance from a point is a circle.