Investigating Circles from Three Points

Investigative Circle Activity Using Three Points
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Activity Challenge: Locate the center and radius of a circle given only three points on the circle, using slopes and equations.

Note to Teacher: This activity works best if developed in a general form as a teacher led activity and then ask students to use the findings independently (or in groups) with actual numeric points. Unlike using a system of equations with the general formula for a circle to solve this problem, this approach uses numerous geometric concepts.

General Form Demonstration:

The three points on the circle are labeled

Draw a line through points A and B, labeled r.

Draw a line through points B and C, labeled t.

Write the equations of these two lines:

line r:	line t
slope:
equation:

Geometrically, we know that the center of the circle will lie on lines that pass through the midpoints of chords and and are perpendicular to each chord. Lines that are perpendicular have negative reciprocal slopes. We will call these new lines rp and tp to indicate their perpendicular connection to lines r and t.

Write equations for lines the perpendicular lines passing through the midpoints:

On line r, segment :	On line t, segment :
midpoint:

These two lines will intersect at the center of the circle. ()

Solve for x:

Once we have the formula for finding x, we can just substitute into one of the line equations for the perpendiculars to find the y value of the center. Whew!!
The radius can be found using the distance formula with the center and any of the points.

Apply the formula: Find the center and radius of a circle which passes through the points (5,5), (6,-2), and (2,-4).

Use the new found formula for the x-coordinate of the center of the circle: