Lesson 4: Tangents to a Circle

Introduction

Student Audience:

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.

Objectives:

• Students will use a Geometer’s Sketchpad script tool to create the tangent lines to a circle
• Students will discover the relationship between the angle formed by a secant and tangent and the angle of the arcs of a circle
• Students will investigate properties of the tangents they have drawn, including the following:

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

Mathematical Concepts:

• Tangents of a circle
• Properties of circle tangents
• Secant
• Angle measure of an arc

Lesson Synopsis:

Using a Geometer’s 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.

Lesson

Materials:

• Geometer’s Sketchpad software
• TangentScriptTool.gsp file
• Presentation software (optional)

Implementation:

The instructor will begin the lesson by equipping individual students (or very small groups of students) at computers that has Geometer’s 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.

Facilitator Questions:

• Based on your observations, how would you describe a tangent?
• What appears the same in each of the figures you have created with your script tool?
• What changes as you vary the size of the circle or the location of the exterior point?
• How might one precisely and mathematically articulate the idea that a line touches the circle at only one point?
• How is the tangent line and the radius of the circle related?
• What is the largest possible measure of the angle between two tangents through a particular point? When does this occur?
• What is the smallest possible angles measure between two tangents through a particular point? When does this occur?
• What are some ways to increase the angle between the tangents?
• What are some ways to decrease the angle between the tangents?
• In trying to find a relationship between the angle between two tangents through an exterior point and the associated circle, what geometric entities might be useful (e.g. inscribed angles, central angles, arc angles)?
• What do you notice about the distance between the exterior point and the point of tangency for each tangent point?
• How is a secant like a tangent?
• How is a secant different from a tangent?
• Examine the measure of the angle between a secant and a tangent. How do you think that angle measure might relate to the circle?
• How many arcs are formed when one secant intersects a circle?
• How many arcs are formed when a tangent and secant through a particular exterior point intersect a circle?