Lesson 7: Parallel Chords

Introduction

Student Audience:

This lesson is designed for implementation in a high school Euclidean Geometry class.  This lesson is designed to be completed in 1 one-hour class period.

Objectives:

á      Students will construct a set of parallel chords within a circle

á      Students will investigate the location of midpoints of parallel chords

á      Students will prove that the midpoints of parallel chords are collinear

á      Students will create and investigate an activity extension of their choice, forming and proving their conjectures

Mathematical Concepts:

á      Chords of a circle

á      Parallel segments

á      Midpoint of a segment

á      Colinearity

á      Right triangle congruence

á      Line perpendicular to a segment and passing through a particular point

á      Supplementary angles

á      Definition of a right angle

Lesson Synopsis:

Students will begin the lesson by constructing a circle and a set of parallel chords within the circle.  Students will then investigate the midpoints of these chords, making conjectures about their location and colinearity.  Students will then formally prove the conjecture that the midpoints of parallel chords are collinear and lie on a diameter of the circle.  Finally, students with instructor guidance, students will examine an extension of this investigation.  They will form conjectures, develop a proof of their hypotheses, and provide a short demonstration to the class.

Lesson

Materials:

á      Sample

Implementation:

The instructor will introduce the lesson as an investigation into the properties of parallel chords.  After using GeometerŐs Sketchpad to construct a circle with containing a set of parallel chords, the students will explore properties of these chords.  In particular, the instructor should guide students to investigate the midpoints of the chords.  After students have developed an hypothesis about the midpoints of parallel chords, the instructor will guide them to construct a proof of their conjecture.  Student may work individually or in small groups to compete this proof.  A sample proof is shown below:

The final component of the lesson is a student-selected extension of this activity.  In particular, students should investigate a related but different geometry scenario, looking for pattern or inconsistencies and offering examples, counterexamples, and proofs as necessary.  Below is a menu of possible extensions.

á      Instead of constructing the midpoints of the chords, divide the segments into a different proportion, such as 1:4 or into three congruent segments.  Are the points associated with your new division collinear?  Do they lie on a circle? An ellipse? Something else?

á      Investigate ŇchordsÓ of polygons.  Image that a ŇchordÓ of a polygon is a segment connecting two sides of the polygons.  Are the midpoints of parallel chords collinear?  Sometimes? Always? Does your finding depend on the type of polygon you use?  Does your finding depend on whether the sides being connecting are adjacent or not?

á      Construct chords of an ellipse?  Is the same property true for the midpoints of parallel chords of an ellipse?

Students should utilize the dynamic features of GeometerŐs Sketchpad software to aid in their investigations and assist them in generating examples and counterexamples.