Learning Resources
Lesson
The introduction to this lesson sets up the problem that will be used throughout the unit to develop the ideas of circles - the building of a sports and activity complex. Read it carefully to get a feel for the problem.
As you begin the lesson a number of definitions are given on the side bar of your text; read and recall these from earlier mathematics courses. It is a good idea to start a notebook of these terms and write a proper definition for each with a diagram to illustrate as you progress through the unit. The work cannot be done if you do not understand the language.
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. However, be sure to record the answer to all the investigation steps in your own notebook.
Make note of the two-fold purpose for this investigation and how it will tie in with the central problem.
Part 1
Steps A and B
These should be complete as they are described on page 206 in your text. To do this you will need the following materials:
- a ruler longer than 18 cm
- a set of compasses to draw a circle
- a pair of scissors
- graph paper
Step A is very easy to complete; for Step B, be sure to make good fold lines that can be easily measured. The fold lines will be diameters of the circle, 18 cm in length, and their intersection point will be the centre of the circle. If you have been careful with the folding, the relationship between the lines will be obvious - use the proper geometric terms to describe this relationship. See the diagram in the sidebar in your text.
Step C
For Step C, there are two ways you can proceed
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One is to complete it as described in your text on page 207. To draw the parallel chords, simply align the edge of your ruler with the previous chord. Alternately, lay the circle on top of a piece of graph paper and use the grid lines as guides.
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An alternate method for Step C is to use Geometer's Sketch Pad to draw the diagram and record the values for the length of the segment and its distance from the centre as you move the chord to different positions. If you decide to use Geometer's Sketch Pad, be sure to move the chord to a least 10 different positions and record the values you obtain in the table for Step C. To use this alternate method simply click here.
Part 2
Step D & E
Refer to the diagrams in the margin on page 207 of your text to see what is intended for this step in the investigation. Be sure that the segment is on the OUTSIDE of the fold.
Step F
Be sure that PX falls along PC, otherwise the fold you make will not be perpendicular to the segments you drew and without this the investigation is meaningless.
Step J
The conjecture you should have arrived at is on the next page of the lesson.
An alternate method for Steps D to I is to use the interactive investigation with Geometer's Sketch Pad. To access that interactive investigation click here.
Have you answered Step J? Do not go on unless you have! In fact, do not proceed past this point unless you have done all of Investigation 1, either by the method presented in your text or by using the alternate method presented in the interactive windows.
Converses
In formal mathematical language what you should have discovered in Investigation 1 Step J is the following:
If two chords of a circle are equidistant from the centre of the circle, then they are congruent.
If you used the Alternate method to do the Investigation, you also discovered the following:
If two chords of a circle are congruent, then they are equidistant from the centre of the circle.
Two statements in which the subject of the "if" and "then" clause is interchanged are said to be converses of each other. In the above example, both the original statement and its converse are true. But that is not always the case. For example consider the following statement:
If a quadrilateral is a square, then it contains four right angles.
Its converse is:
If a quadrilateral contains four right angles, then it is a square.
Obviously the converse is not true, as the figure could be a rectangle with unequal sides.
When a statement and its converse are both true, we can write both statements as one using the mathematical/logical conjunction "iff" ( sometimes this is read as "if and only if"). The two first statements about the chords of a circle can thus be written as:
Two chords of a circle are congruent iff they are equidistant from the centre of the circle.
Another example of this "iff" conjunction is given in the example in the margin of page 209 in your text.
You should now be ready to do some work using the concepts developed in Investigation 1, so go to the top of the page and click the Activities Button and complete what is assigned.
Summary
- The diameter is the longest chord of a circle.
- The diameters of a circle intersect in the centre.
- The longer a chord the farther it is from the centre.
- In a circle, two chords that are equidistant from the centre are equal in length.
- In a circle, if two chords are equal in length, they are equidistant from the centre.
- Items 4 and 5 above are called converses of each other.
- Items 4 and 5 above can be written as one sentence using "iff".
Activity
Investigation Questions p.208 - 209. Complete 1 to 8 inclusive
Think About p.208
Check Your Understanding p.209. Complete 9, 10 and 11
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 below for a quick quiz on this lesson.
Test Yourself
Write converses of the following statements. If the statement and the converse are both true, write them using "iff". If they are not true, explain why.
- If x is divisible by 10, then x is divisible by 5.
- If a triangle is isosceles, then it has two equal sides.
- If the chords of two different circles are congruent, then they are equidistant from the centres of the circle.
Click here for suggested solutions.