Lesson 4: Tangents to a Circle
This lesson is designed for implementation in a high school Euclidean Geometry class. As written, this lesson is expected to be completed in 2 one-hour class periods.
1. Relationship between tangent and radius of the circle
2. Angle of a tangent and arc angles
3. Congruent tangents
4. Angle formed by a secant and a tangent
5. Angle formed by 2 secants through a common exterior point
6. Product of the lengths of the segments of secants through a common exterior point
Using a GeometerŐs Sketchpad script tool, students will investigate properties of the tangents of a circle. After students discover and articulate properties they will select one property and construct a proof to support their conjecture. Students proving the same property will then gather in groups to discuss and evaluate their method of proof. Each group will construct a brief presentation of their conjecture including one proof.
The instructor will begin the lesson by equipping individual students (or very small groups of students) at computers that has GeometerŐs Sketchpad software and the TangentScriptTool.gsp file loaded. After the instructor explains to students that they will be working with something called tangents to a circle, the students will launch into an investigation of the properties of the tangents. Using facilitator questions as appropriate, the instructor will direct students to explore such properties as length and angle. Students should utilize extreme or degenerate cases to help focus their investigation. Additionally, each student should keep a record of her findings, properties, and conjectures.
After a student has identified a significant number of tangent properties, the instructor will guide them to select one property and generate a proof of their conjecture. When all students have had an opportunity to complete a proof (possibly a homework assignment for the first day of this lesson), the instructor will group students who proved the same conjecture. During this group component of the lesson, students will compare, contrast, and gently critique proofs. Additionally, each group will be responsible for creating a presentation (either electronic or traditional) of their conjecture including 1 proof. The final component of the lesson will be a series of brief presentations from the groups.