#### Activity Overview

Explore right triangles and the Pythagorean Theorem to develop the equation of a circle.

#### Objectives

- Students will understand the definition of a circle as a set of all points that are equidistant from a given point.
- Students will understand that the coordinates of a point on a circle must satisfy the equation of that circle.
- Students will relate the Pythagorean Theorem and Distance Formula to the equation of a circle.
- Given the equation of a circle (x – h)
^{2}+ (y – k)^{2}= r^{2}, students will identify the radius r and center (h, k).

#### Vocabulary

- Pythagorean Theorem
- Distance Formula
- Radius

#### About the Lesson

This lesson involves plotting points that are a fixed distance from the origin, dilating a circle entered on the origin, translating a circle away from the origin, and dilating and translating a circle while tracing a point along its circumference. As a result students will:

- Visualize the definition of a circle.
- Visualize the relationship between the radius and the hypotenuse of a right triangle.
- Observe the consequence of this manipulation on the equation of the circle.
- Infer the relationship between the equation of a circle and the Pythagorean Theorem.
- Infer the relationship between the equation of a circle and the Distance Formula.
- Identify the radius
*r*and center (*h*,*k*) of the circle (*x*−*h*)^{2}+ (*y*−*k*)^{2}=*r*^{2}. - Deduce that the coordinates of a point on the circle must satisfy the equation of that circle.