Lesson 9: Chord Length and Distance from Center

 

Introduction

 

Student Audience:

            This lesson is designed for implementation in a high school Euclidean geometry class.  Completion of this lesson is anticipated to require 2 one-hour class periods.

 

Objectives:

Š      Students will investigate the properties of the chords of a circle

Š      Students will examine distance between a point and a segment

Š      Students will form conjectures about the relationship between chord length and distance from center

Š      Students will create a dynamic sketch which addresses the question of how a chord’s length relates to its distance from the circle’s center

Š      Students will collect data including chord length and distance from center

Š      Students will graph and analyze data to form conjectures about the precise relationship between chord length and distance from center

Š      Students will prove that the graph relating chord length and distance from center is a portion of an ellipse

Š      Students will evaluate the role that domain plays in the graph of their data and the graph of a representative equation (equation of the ellipse)

 

Mathematical Concepts:

Š      Chord of a circle

Š      Distance measured between a segment and a point

Š      Pythagorean theorem

Š      Graph of an ellipse

Š      General equation of an ellipse

 

 

Lesson Synopsis:

            Students will be investigating the question of how the length of a chord varies with its distance from the center of a circle.  Students will create a dynamic sketch of the situation using Geometer’s Sketchpad software.  Using this sketch, students will collect data of the length of a cords and their distance from the circle’s center.  Using a spreadsheet software, students will input and graph their data.  Students will make conjectures about the relationship between chord length and distance from a circle’s center.  A proof of this conjecture will be created through applying the Pythagorean theorem. 

 

Lesson

 

Materials:

Š      Geometer’s Sketchpad software or similar

Š      Microsoft Excel software or similar

Š      Student Data Organizer worksheet (optional)

 

Implementation:

            The instructor will begin the lesson with a brief discussion of chords.  In particular, the instructor should inquire whether all of the chords in a particular circle have the same length.  As students decide that the chords of a circle have varying length, the instructor should inquire which chords will be longest and which will be shortest, in essence framing the underlying question of how chord length varies with distance from the circle’s center.  After being presented this underlying question, students will use Geometer’s Sketchpad software (either individually or in small groups) to create a dynamic sketch in which the length of various chords of a circle can be examined.  A sample sketch can be found here.

            Using their dynamic sketch, students should collect data for chord length and distance from the circle’s center.  The instructor may encourage students to either individually organize their data or utilize the Data Organization worksheet provided.  Students will subsequently input their data into an electronic spreadsheet and create a scatterplot of the data.  Using this scatterplot, students will form conjectures about the relationship between distance and length.  A sample data set and scatterplot can be found here.  

            After students have reported their conjectures to the class, the instructor will direct the students to support their conjecture that the scatterplot is part of an ellipse, that is the students will be pushed to prove their conjecture.  The instructor will guide students to create a sketch of a general chord and the segment which measures its distance from the circle’s center.  Students should use their knowledge that distance is measured along a segment through the center that is perpendicular to the chord to create the following sketch: 

Finally, students will use this sketch to develop an equation relating chord length and distance.  One possible solution method is shown below. 

 

The second leg of the right triangle shown is half of the chord.  By the Pythagorean theorem the length of that leg (half of the chord) is √(r2 – x2).  Thus, the length of the chord is 2√(r2 – x2). 

 

If y represents the length of the chord then the following statements are true:

y = 2√(r2 – x2)

y/2 = √(r2 – x2)

y2/4 = r2 – x2

x2 – y2/4 = r2          which is a familiar equation for an ellipse. 

 

Following this proof component, the instructor will guide students to compare the graph of the elliptical equation to their scatterplots.  In particular, the instructor should guide students to notice that only a portion of the ellipse can represent data points since lengths are solely positive real numbers.

 

Facilitator Questions:

Throughout the lesson, the instructor my incorporate the following facilitator questions to guide student activity. 

 

Š      How is the distance from a point to a line measured?

Š      Does the relationship between distance and length appear linear? Quadratic? Something else?

Š      If your scatterplot were reflected over the x-axis and the result was then reflected over the y-axis, what shape would form?

Š      Are there multiple chords of the circle that have the same lengths?  How do you know?

Š      Are chords that have the same length the same distance from the center?  How do you know?

Š      What would your scatterplot look like if two chords were the same distance from the center but had different length?

Š      What would your scatterplot look like if two chords were the same length but at different distances from the center?

Š      What might a sketch that showed the relationship between a chord length and its distance from the center look like? 

Š      Does the equation you found fit every data point that you collected? 

Š      Does every point on the graph of this equation exist as a possible data point for our exploration?  Why?

 

Extension Activity:

Chords and elipses?  Is the same relationship true?
NAME __________________________________   DATE ________________________

 

 

Data Organization Form

 

 

Distance from Center

Length of Chord