Segment Lengths in Circles

Objectives:  Students are to discover three theorems concerning how to find the lengths of segments in circles.

Lesson:

Have students discover the following theorems:

 Theorem 1:  If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Theorem 2:  If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment. Theorem 3:  If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

Here are the interactive GSP sketches for the above theorems:  Theorem 1, Theorem 2, Theorem 3.

Conclusion
:  Have the students discuss the conclusions they formed based on the activity.  Review the theorems that the students should have discovered during the activity.

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