Equation of a Circle

Geometry Lesson Activities

Equation of a Circle The Lesson Activities will help you meet these educational goals:  Content Knowledge—You will derive the equation of a circle of given center and radius and complete the square to find the center and radius of a circle given by an equation.  Mathematical Practices—You will make sense of problems and solve them.  21st Century Skills—You will use critical-thinking and problem-solving skills.

Directions You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities. ______

Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1. Equation of a Circle The equation of a circle centered at the origin with a radius of unit length is x2 + y2 = 1. This equation changes if the center of the circle is not located at the origin or the radius is not of unit length. You will use the GeoGebra geometry tool to examine how the equation of a circle changes as you move the center and change the radius. Go to equation of a circle1, and complete each step below. If you need help, follow these instructions for using GeoGebra.

a. Move the sliders to set the values of r, h, and k, and record at least three different sets of data. Also record the equations of the corresponding circles.

Type your response here: Horizontal Vertical Coordinate Coordinate of Center (h) of Center (k) Radius (r) Equation of Circle 1 2 3

1 This work is adapted from “Graphing a Circle, Equation of a Circle,” created with GeoGebra by (awardyn2), Dec. 1, 2012, and used under a Creative Commons Attribution-ShareAlike license.

b. Move the center of the circle horizontally to the left and then to the right of the y-axis. How does the equation of the circle change as the center crosses the y-axis?

Type your response here:

c. Move the center of the circle vertically so it lies above and then below the x-axis. How does the equation of the circle change as the center crosses the x-axis?

d. How does the equation change when the radius changes? Unlike h and k, why is r always positive?

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

2. Completing the Square a. The equation of a given circle in general form is x2 + y2 − 8x + 12y + 27 = 0. Write the equation in standard form, (x − h)2 + (y − k)2 = r2, by completing the squares in the equation. Show your work in the table.

Type your response here: Step Reason

2 b. Using the standard form of the equation in part a, what are the length of the radius and the coordinates of the center for this particular circle? Watch your signs for the variables h and k.

c. Open GeoGebra. Plot the circle from part b by entering the standard form of the equation in the Input window. (Use ^2 to represent an exponent of 2.) Then plot the same circle using the general form of the equation that was given in part a. (Placing your cursor over an equation in the Algebra margin will bold the circle that the equation corresponds to. By right clicking and selecting Equation, you can toggle between the different forms of the equation.) Next, drag the equations of the circles onto the screen from the Algebra margin. Verify that the circles resulting from the two equations are the same. Finally, paste a screenshot of your results in the space below.

d. The equation of a circle in general form is Ax2 + By2 + Cx + Dy + E = 0, where A = B ≠ 0. How do the coefficients C and D change as the center of the circle crosses over the x- and y-axes? How do the coefficients C and D change if the radius of the circle changes? Experiment with different circles using the general form of the equation in GeoGebra, if you wish.

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.