Lesson 9: Intersecting Chords

 

 

Introduction

 

Student Audience:

            This lesson is designed to be a component of a high school Euclidean Geometry course.  Completion of the activities in this lesson is anticipated to require 1 one-hour class periods.

 

Objectives:

Š      Students will create a dynamic sketch of intersecting chords within a circle

Š      Students will analyze their dynamic sketch, looking for a relationship between the segments created when two chords intersect

Š      Students will create and analyze special cases of intersecting chords to aid in their discovery of a relationship between the segments of intersecting chords

Š      Students will articulate a conjecture based on their observations

Š      Students will utilize formal proof to confirm their conjecture

 

Mathematical Concepts:

Š      Chords of a circle

Š      Properties of quadrilaterals

Š      Inscribed angles

Š      Similar triangles

Š      Methods of proof

Š      Median of a triangle

Š      Proportions

 

 

Lesson Synopsis:

            Using Geometer’s Sketchpad, students will create a dynamic sketch of intersecting chords.  Analyzing this sketch, students will formulate a conjecture of the relationship between the segments created by two intersecting chords.  Finally students will construct a formal proof to support their conjecture. 

   

Lesson

 

Materials:

Š      Geometer’s Sketchpad software

 

Implementation:

            The instructor will commence the lesson by posing the question of what picture would occur when two chords are drawn in a circle.  Students will likely respond by elaborating the following two cases, (1) the chords intersect and (2) the chords do not intersect.  The instructor will present a sketch (either hand-drawn or dynamic) of the two situations, asking students to compare and contrast them.  In particular, the instructor should aim to initiate an observation that four segments result from two chords intersecting.

            After students have made this observation, the instructor will challenge students to discover if any relationship exists between these four segments.  Students should work individually or in small groups to construct an appropriate dynamic sketch using Geometer’s Sketchpad software. A sample sketch can be found here.  Each student will then analyze their sketch, observing lengths of segments for numerous cases in an effort to determine a general relationship.  During this student exploration, the instructor will circulate the classroom addressing concerns and facilitating investigations by incorporating facilitator questions as appropriate.  

            After students have formed conjectures that the product of the segments of the chords is equal, this instructor will guide students to create a proof.  A sample proof is shown below. 

Given: A circle with intersecting chords AB and CD.

 

 

 

By connecting points A, C, B, and D we form quadrilateral ABCD.  Notice that angles ACD and ABD both subtend chords AD. Consequently, angles ACD and ABD must be congruent. 

 

Similarly, angles CAB and CDB both subtend chord CB and are therefore congruent.  

 

 

 

Examine triangles AEC and DEB.  These triangles have two pairs of congruent angles.  Therefore triangle AEC is similar to triangle DEB.  Therefore the following proportion is true:

                                    AE    =     DE

                                    CE            BE

Therefore (AE)(BE) = (CE)(DE). 

 

 

 

 

            When all students have succeeded in developing a conjecture, the instructor will reassemble the class to discuss possible extensions of this problem.  In particular, the instructor may lead students in an exploration of intersecting “chords” of a triangle, that is what relationships, if any, occur when segments joining two sides of a triangle intersect.  A sample sketch can be found here. Students should investigate whether the relationship seen for the circle is always, sometimes, or never true for the triangle.  While the instructor may lead this investigation extension as a whole-class activity, students may also work individually or in small groups to gather evidence and form conjectures.  

 

Facilitator Questions:

Š      What properties of the segments might be useful to examine (e.g. length and slope)

Š      Are there any unique or degenerate cases that might be helpful to examine?

Š      Can you generate a situation in which a segment of one chord is nearly the same length as a segment of the second chord? 

Š      One technique for proof is to draw in “missing” lines and segments.  Are there any “missing” segments in your sketch?

Š      What shape(s) results from drawing the missing segments?

Š      Each chord cuts the circle into two pieces.  What are these pieces called? 

Š      How are arcs measured? 

Š      How might angles and arc lengths be useful?

Š      Do any segments always appear congruent?

Š      Do any angles always appear congruent?

Š      How can one be certain that two inscribed angles will be congruent?

Š      Can any similar triangle be found in your sketch?

Š      How might similar triangles help prove the product conjecture?

Š      What might a “chord” of a triangle look like? 

Š      Can ‘chords’ of a triangle ever intersect?  When would this happen?

Š      When will chords of a triangle fail to intersect?

Š      When the chords of a triangle intersect, how many segments are formed?

Š      Is the relationship between the segments of intersecting chords of a circle true for intersecting ‘chords’ of a triangle?  When?

 


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