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Congruent Tangents Discovery Activity

Congruent Tangents Discovery Activity
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This activity uses Geometer’s Sketchpad 5 and is intended to introduce students to tangent lines and using congruent tangents to find unknown measurements.  The student will construct, from a given circle O, another circle and various rays and segments to create two congruent tangents to circle O. Instructions are given simply but clearly in Comic Sans because that font is easiest to read, even for students with dyslexia.  After the construction, students are presented with three open-ended discovery questions:

"Do you think that the rays you created are actually tangents to circle O?  How could you prove that they either are or are not tangents?"

"Using the line tool, construct segments between points O and A and points O and B (A and B are the points of tangency on circle O).  What can be said about the newly-created triangles OAP and OBP?"

"Construct segments between points P and A and P and B (point P is the point of intersection of the tangents).  If segment AP has length of 3x-7 and segment BP has length 2x+2, what is the value of x? What about if segment AP has length x+3 and segment BP has length y-4?"

The goal of these questions is for students to puzzle out and discover on their own how two such tangent lines can be used to apply concepts of triangle congruence and solving equations with both one variable and two.  This activity can be used as a review of previous concepts and an introduction of new ones. It also can be used to introduce and/or review the concept and format of geometric proofs.

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