Learning Resources
Lesson
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.
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
If you have not already done so, go to the Overview for Unit 04 to see what materials you need to do the Investigations and other work as described in your text book. Record them and then be sure to have them with you every time you begin a lesson in this unit.
Now get into groups, open your text to page 206 and try to complete Step A and Step B. Some help and clues to doing the Investigation are provided, but use them only if you are having difficulty. If you do have difficulty, you might go here before contacting your teacher for help.
Part 1
Step C:
There are two ways you can proceed for this step. One is to complete it as described in your text on page 207. If you use this method and have trouble you can go here for some clues and directions.
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
There are two ways you can proceed for Steps D to I. One is to complete it as described in your text on page 207. If you use this method and have trouble you can go here for some clues and directions.
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
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". The two first statements about the chords of a circle can thus be written as:
Two chords of a circle are congruent if 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.
Investigation 1: Chords (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
- Complete the Investigation Questions 1, 2, & 4 - 8 on pages 208 & 209 in your text.
- Do the CYU Questions 9 - 11 on page 209.
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.
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.