Up and Down. - Circle Theorems 2: The Converse of the Chord Theorem -

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1 - Circle Theorems 2: The Converse of the Chord Theorem - Revision Label the circle diagram showing: the circumference the centre a diameter a chord a radius State the Chord Theorem. Checkpoint An Example of a Converse Theorem Many geometry theorems are equally true if you write them in reverse. For example, You should have learned that if two lines are parallel, then a pair of corresponding angles α will be equal. It is also true that if the pair of corresponding angles are equal, then the lines must be parallel. We say that this is the converse of the first statement. When you do more advanced mathematics you will be shown examples of theorems for which the converse statement is not true. You will have to be very careful about the direction of your logic when you use words like therefore, ifthen, because and implies. Look very carefully at the bits before and after these words and make sure that you have them the right way around. Copyright 2007, Hartley Hyde Page 1 of 9

2 Restating the Chord Theorem Using Symbols Most theorems can be written in the form If something is true then something else is true. The Chord Theorem can be written If BD = CD Then AD BC Where the symbol means perpendicular to. There are several different ways to write this same idea. Some older text books use the symbol for therefore like this BD = CD AD BC More recent text books use the symbol for implies like this BD = CD AD BC Other text books try to use every-day words like so. The Converse of the Chord Theorem To write the Converse of a Theorem we just switch the two Something bits. The Converse of the Chord Theorem is written If AD BC Then BD = CD Or it could be written AD BC BD = CD or using the symbol for implies like this AD BC BD = CD Copyright 2007, Hartley Hyde Page 2 of 9

3 Stating the Converse of the Chord Theorem in Words The straight line drawn from the centre of a circle, perpendicular to a chord, bisects the chord. Deductive Proof Given: To Prove: Construction: Proof: AD BC BD = CD Join the radii AB and AC In s ADB and ADC m ADB = m ADC = 90 AB = AC AD is common (AD BC) (radii) s ADB and ADC are congruent (RHS) in particular, BD = CD QED Notes on this Deductive Proof On the previous page we wrote the theorem in symbols as If AD BC Then BD = CD This matches the Given and To Prove sections of the Deductive Proof. The QED at the bottom of each proof stands for the Latin phrase quod erat demonstrandum from Euclid s Greek περ δει δειξαι. Both can be literally translated as "which was to be demonstrated". English school boys preferred the translation Quite enough done. My son uses w 5 which stands for Which was what was wanted. Purpose To demonstrate the Converse of the Chord Theorem from Book III of Euclid s Elements Using a ClassPad to demonstrate the Converse Theorem Switch on your ClassPad o and tap on the Geometry icon G. If you have already saved a file called ChordSt, open it now and jump to the starline (shown as **********) on the next page. Copyright 2007, Hartley Hyde Page 3 of 9

4 Building the ChordSt file From the Draw Menu tap on Line Segment. Tap once near the centre of your screen and then tap toward the left edge. The first point will be automatically labelled A and the second B. From the Draw Menu tap on Circle. Tap once on the point A and once on the point B. Your screen should look like the first screen: From the Draw Menu tap on Point. Tap on the circle at a point away from B. The new point will be labelled C. Choose Select from the View Menu or you can tap on the first tool in the Tool Bar which is the Selection Arrow. Tap on C and your screen should look like the second screen above. Now try to move the point C away from the circle. The circle should move with the point: this shows that you have correctly attached the point to the circle. From the Draw Menu tap on Line Segment. Join AC and BC by tapping on the end points. At this point Save your work as ChordSt. Checkpoint *************************************** The file ChordSt gives you a starting point for many investigations of Circle Theorems. By building a library of such starting points you can save yourself much time. This applies to other ClassPad Applications such as Spreadsheets and Statistics Investigations. From this point we can proceed to demonstrate the Converse of the Chord Theorem using two different methods. Copyright 2007, Hartley Hyde Page 4 of 9

5 Demonstrating the Converse Theorem (Method 1) From the left-hand side of the Tool Bar choose the Selection Arrow. Tap on the point A and on the line segment BC. From the Draw Menu choose Construct and then tap on Perpendicular. This will draw a line through the point A, perpendicular to the chord BC. Using the Selection Arrow, highlight the new perpendicular line and the chord BC. From the Draw Menu choose Construct and then tap on Intersection. This will create the point D where the lines intersect. Using the Selection Arrow, highlight the line AD and the circle. From the Draw Menu choose Construct and then tap on Intersection. This will create the points E and F where the line AD cuts the circle. Using the Selection Arrow, highlight the line AD. From the Edit Menu choose Properties and then tap on Hide. To demonstrate the theorem, we need to measure BD and CD and as far as ClassPad is concerned, it only knows about BC. We therefore have to hide BC and draw BD and CD before we can make those measurements. Using the Selection Arrow, highlight the chord BC. From the Edit Menu choose Properties and then tap on Hide. From the Draw Menu choose Construct and then tap on Line Segment. Now join BD and CD separately. Although this looks the same as what we had before, BD and CD are now separate line segments and we can find their individual lengths. Copyright 2007, Hartley Hyde Page 5 of 9

6 Use the Selection arrow to highlight the line segment BD. From the Draw Menu choose Measurement and tap Length. The Length of BD will appear at the top of your screen. Click in free space to de-select BD. Now follow the same process to find the length of CD. Save your work as Chord1. Always save before you play. Notice that when you move point B, the two lengths remain the same. Join AD. Select AD and BC and from the Draw Menu tap on Attach Angle. Because you defined the line AD as a perpendicular, you should not be surprised to find that the angle is a right angle. Building an Animation Join AF. Use the Selection Arrow to choose the point D and the radius AF. From the Edit Menu choose Animate and then tap on Add Animation. From the Edit Menu choose Animate and then tap on Go (to and fro). As the point D moves up and down along the radius AF you will notice that the length of BD and CD remain the same, whatever the length of AD. From the View Menu you can select Animate UI and this will display an Animation Bar that will help you choose different types of animation much as you can control a DVD player. Checkpoint Copyright 2007, Hartley Hyde Page 6 of 9

7 Demonstrating the Converse Theorem (Method 2) Open the file ChordSt. From the Draw Menu tap on Point. Tap on a point D somewhere along BC but not near the midpoint. From the Draw Menu tap on Line Segment and join AD. Use the Selection Arrow to highlight AD and BC. At the extreme right-hand end of the Tool Bar is a tick R and an arrow u. Tap on the arrow and follow the Tool Bar around the corner to the Measurement Bar. The ClassPad is well ahead of us. Because we have selected two intersecting line segments, it knows that we must be looking for the measurement of the angle of intersection. In the screen dump this is about 69. Now here is the really clever bit. Tap on the angle in the measurement bar and the number is highlighted. Now simply overtype this number with the number 90 and then tap the tick. As soon as you tap the tick, the whole construction springs into a new position so that AD and BC are perpendicular. This is called Adding a Constraint. Wherever you try to move the points around, the angle ADC is fixed at 90. Even some of the most expensive computer geometry packages can t do that! If you ever need to remove a constraint you have to go to the Edit Menu and tap on Clear Constraints. However, be warned: this will remove all constraints, even the places where you have attached a point to a circle will be disconnected. It is usually safer to start again. That s why it is a good idea to keep backing up your work. Yes! Even on your ClassPad. Copyright 2007, Hartley Hyde Page 7 of 9

8 Measuring Lengths We still have the same problem that we met when following Method 1. To demonstrate the theorem, we need to measure BD and CD and as far as ClassPad is concerned, it only knows about BC. We therefore have to hide BC and draw BD and CD before we can make those measurements. Using the Selection Arrow, highlight the chord BC. From the Edit Menu choose Properties and then tap on Hide. From the Draw Menu choose Construct and then tap on Line Segment. Now join BD and CD. Although this looks the same as what we had before, BD and CD are now separate line segments and we can find their individual lengths. Use the Selection arrow to highlight the line segment BD. From the Draw Menu choose Measurement and tap Length. The Length of BD will appear at the top of your screen. Click in free space to de-select BD. Now follow the same process to find the length of CD. Save your work as Chord2. Always save before you play. Notice that when you move point B, the two lengths remain the same. Summary What do each of these models demonstrate? Checkpoint Copyright 2007, Hartley Hyde Page 8 of 9

9 Checkpoints Checkpoint 1 Parts of Circle Chord Theorem The straight line joining the centre of a circle to the midpoint of a chord (which is not a diameter) is perpendicular to the chord. Checkpoint 2 Students will only reach this Checkpoint if the forgot to save the file ChordSt when they did Circle Theorems 1: Chord Theorem (Round and Round). All that is needed is to check that they have a triangle as described and that they have saved it as ChordSt. Checkpoint 3 Just to check that Method 1 works. If you have a slower class, you may skip Method 1 and go straight to Method 2. Checkpoint 4 Check that Method 2 works. Summary Their experiments have not rejected the hypothesis that: If AD BC Then BD = CD Copyright 2007, Hartley Hyde Page 9 of 9