Learning Resources
Lesson
In this lesson you will use the technique of paper folding ( i.e. transformational geometry) to discover how perpendicular bisectors of chords and the centre of a circle are related.
If you have classmates at your site, it would be better to work together in groups of 2 or 3 to complete this investigation, as it will save you time and you can discuss and help each other. Also, be sure to record the answers to all the investigation steps in your own notebook.
There are two ways of completing this investigation. One is to use the method of paper-folding described in your text, the other is to use Geometer's Sketch Pad in the interactive window.
Text Book Method
To complete this investigation you will need the following materials:
- a ruler
- a set of compasses to draw a circle
- a pair of scissors
- a protractor
Part 1
In Step A, make sure you construct a reasonably large circle (12 to 18 cm) so that you can more easily find the conclusion the investigation is attempting to get you to reach. In fact, since you will need another similar circle later, you may as well make two now. Look at the diagram in the sidebar on page 210 to ensure that you draw the proper chord for Step B.
In Step C, if you have done things correctly, you will have folded the circle in half and the fold line is the diameter. Refer to the diagram in your text book's sidebar to confirm your labelling in Step D.
For Step E, you will noticed that M is the midpoint and that the angles are right angles.
Steps F & G requires you to repeat the above procedures to verify what you found.
If you can not make a conjecture for Step H based on the paper folding, try the alternate investigation suggested in the following part of the main lesson.
Part 2
Follow the instructions for Step I through Step L inclusive and make the requested measurements.
If you can not make a conjecture for Step M based on the paper folding, try the alternate investigation suggested in the following part of the main lesson.
In Step N, bring together your two conclusions form Part I and Part II to make one general statement - an "iff" statement.
Alternate Method
For Part 1 of the Investigation, an alternate to paper folding for discovering the relation could be to use Geometer's Sketch Pad. To use this alternate method simply click here.
For Part 2 of the Investigation, a similar alternate to the paper folding using Geometer's Sketch Pad is provided. To use this alternate method simply click here.
Do not proceed past this point unless you have done Investigation 2, either by the method presented in your text or by using the interactive windows with Geometer's Sketch Pad.
Summary
The key ideas of Investigation 2 are contained in Steps H, M, and N of the investigation. They are summarized below for your convenience. If you have completed the investigations as directed you should have discovered these important properties of chords in a circle.
There are several ways of stating this very important property. For example, you could use or state it in any of the following forms:
1. The perpendicular bisector of a chord of a circle always passes through the centre of the circle.
The above statement applied to figure on the right means:
If FE is the perpendicular bisector of CD (that is 
and Ð FGC and ÐFGD are right angles)
Then, FE must pass through A the centre of the circle.
2. If a line passes through the centre of a circle and is perpendicular to a chord, then it will bisect the chord.
The above statement applied to figure on the right means
If FE passes through centre A and FE is perpendicular to CD
Then,
3. If a line passes through the centre of a circle and bisects a chord, then it is perpendicular to the chord.
The above statement applied to figure on the right means
If FE passes through centre A and
Then, FE is perpendicular to CD.
These statements can be summarized as follows:
"Lines perpendicular to chords of a circle are perpendicular bisectors iff they intersect at the centre of the circle."
You should now be ready to do some work using the concepts developed in Investigation 2, so continue on to the Activities section below and complete what is assigned.
Activity
Investigation Questions p.211. Complete 12, 13, 14 and 15
Check Your Understanding p.212. Complete 16, 17, 18, 19, 20 and 21
When you have completed these questions, ask your on-site teacher to get the solutions for you from the Teacher's Resource Binder and check them against your answers. After you do this, if there is something you had trouble with and still do not understand, contact your on-line teacher for help.
After you do the assigned activities, continue on to the Test Yourself section for a quick quiz on this lesson.
Test Yourself
- You are given three non-collinear (not in a straight line) points A, B, and C. Construct a circle that passes through all three.
The solution is presented as a Viewlet that will run automatically unless you pause it by clicking on the pause button which is the one marked II. After you pause, you can step through the frames by clicking on the advance button marked >>.
Click here for suggested solutions.