Center Home -> Content Areas Home -> Math Home -> Project Activities -> Sketchpad Activities ->

Determining the Uniqueness of a Circle

Activity Description Activity GuideResources


Part 1:
  • It is a common notion that through any two given points, exactly one line can be constructed. Do you think there is a certain number of arbitrary points, such that there exists a unique circle through them? How many points do you think are needed?
  • Given a single point, can you construct a unique circle that passes through this point? Discuss and hypothesize an answer with your partner.
  • Construct a single point in your sketch window. Construct a circle through the given point. Is the circle you constructed unique? That is, can you construct a different circle through the same initial point? Try constructing several different circles that pass through your original point. Take turns describing your group's constructions with the group next to you. What conclusions can you make? Can you construct a circle through any given point?
Part 2:
  • Can you construct a unique circle that passes through any two given points? Discuss and hypothesize an answer with your partner.
  • Construct two points in your sketch window. Explore ways to construct a circle that passes through both of your points. Try constructing several different circles that pass through your original points. Take turns describing your group's constructions with the group next to you. How many circles could you construct? What conclusions can you make from your constructions? Can you construct a circle through any two given points? Further conjecture about the centers of all of the circles that you have constructed.
To construct a circle through two given points:
  • Select two points and connect them by a segment.
  • Construct the perpendicular bisector of this segment.
  • Construct a point on the perpendicular bisector.
  • Using this point as the center of your circle, construct a circle that passes through one of the two initial points.
  • In a New Sketch, construct a circle through two given points. Verify both of the initial points are on the constructed circle. How can you be sure? (Instructor Note: A Sketchpad file illustrating this construction has been saved as part2.gsp.)

  • Move the center point of the circle. What do you notice about the circle in relation to the initial points? Make a conjecture about the distance from each initial point and the center point of the circles. Formally prove or disprove your findings.
  • Based on your observations and formal proof, what can you conclude about the number of circles that can be constructed through any two points? Where must the centers of these circles lie? Prove the conjecture that you make about the centers of the circles.
Part 3:
  • Can you construct a unique circle that passes through any three given points? Discuss and hypothesize an answer with your partner.
  • Construct three points in your sketch window. Explore ways to construct a circle that passes through your initial points. Try constructing several circles that pass through your original points. Take turns describing your group’s constructions with the group next to you. How many circles could you construct? What conclusions can you make from your constructions? Can you construct a circle through any three given points? (Instructor Note: A Sketchpad file illustrating this construction has been saved as part3.gsp.)
To construct a circle through three given points:

  • Construct three points

  • Choose one pair of points and construct a line segment between them.

  • Choose a second pair of points and construct a line segment between them.

  • Construct the perpendicular bisectors of these two segments.

  • Construct the intersection of these two bisectors.

  • Using this point of intersection as the center of your circle, construct a circle that passes through one of the initial three points.

  • Does the circle pass through all of the initial points? Verify all of the initial points are on the constructed circle. How can you be sure?

  • Manipulate one of the three points that you initially constructed? What do you notice about the other two initial points in relation to the circle? What do you notice about the three lengths that you calculated?

  • Based on your observations, do you believe that a circle can be constructed through any three given points? Do you believe that this circle is unique? Prove your assertion.

Extensions:

  • Conduct a similar investigation using four initial points. Can you construct a unique circle through any four points? Hypothesize if you can construct a unique circle through any five points. In general, can you construct a circle through an arbitrary number of points?

  • Consider points (a1, b1), (a2, b2), and (a3, b3). Determine the coordinates of the center of the unique circle that these three points determine. Express the coordinates of this center point in terms of a1, b1, a2, b2, a3, and b3.
 


Back to Project Activities | Back to Math Homepage

Send questions or comments here.
Last modified on June 18, 2002.