- Circle Theorems 1: The Chord Theorem - A Historic Note The main ideas about plane geometry were developed by Greek scholars during the period between 600 and 300 B.C.E. Euclid established a school of mathematics in Alexandria, Egypt in about 300 B.C.E. He collected the main ideas about geometry and published them on thirteen papyrus scrolls called the Elements. The circle theorems were on scrolls Three and Four. The Elements became a text-book that has been studied by geometry students for the Euclid of Alexandria past 2300 years. Now it is your turn. But, you have a ClassPad to help you understand what it means. Purpose To demonstrate the Chord Theorem from Book III of Euclid s Elements Prior Learning Two geometric figures are said to be congruent if they have exactly the same size and shape. There are four sets of conditions under which two triangles can be congruent: SSS Two triangles are congruent if three sides of one are equal to three sides of the other. SAS Two triangles are congruent if two sides of one are equal to two sides of the other, and if the included angles are equal. AA c S Two triangles are congruent if two angles of one are equal to two angles of the other and if a side of one is equal to the corresponding side of the other. RHS Two right-angled triangles are congruent if the hypotenuse of one is equal to the hypotenuse of the other, and if a side of one is equal to a side of the other. Copyright 2007, Hartley Hyde Page 1 of 7
Definitions A circle is a plane figure bounded by a curve all points of which are at the same distance from a point called the centre of the circle. The curve is called the circumference of the circle. Any line segment drawn from the centre to a point on the circumference is called a radius (plural, radii). A line segment joining any two points of the circumference is called a chord. Any chord that passes through the centre is called a diameter. Many of these definitions are ambiguous. Some definitions of a circle include only the set of points making up the curve, but we will define a circle to include the whole figure, including the area contained by the curve. Each of the other underlined terms can mean either the set of points comprising the construct or they can refer to the length of the construct. 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. Copyright 2007, Hartley Hyde Page 2 of 7
Inductive Reasoning When we try something out and see a pattern we sometimes conclude that the pattern is always true. We call this a hypothesis. However, if we try something new that doesn t follow the pattern we have to reject our hypothesis. This is the type of thinking that scientists use when they experiment. It is called inductive reasoning. When we do geometry using a ClassPad we are doing experiments to test if an idea is true and so this type of investigation is also an inductive process. ClassPad Time Switch on your ClassPad o and tap on the Geometry icon G. To clear the screen, tap on the File menu and select New. You will be asked if you are sure and you then tap OK. 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 this: 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: the Selection Arrow G. 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. Copyright 2007, Hartley Hyde Page 3 of 7
Below the Menu Bar is the Tool Bar. Many of the Menu items you have been finding are also available from the Tool Bar. As you select each item you will see the different icons appear and as you learn to recognize them you may find it easier to get your drawing tools from the Tool Bar instead of the Draw Menu. However, the icons in the Tool Bar keep changing so it is much more precise if these notes explain what to do using items from the Menu Bar. The tool bar keeps changing icons so that you can see which tool is active. From the Draw Menu tap on Line Segment. Join AC and BC by tapping on the end points. Save your work as ChordSt. Use the Selection Arrow G and tap on BC. The black spots show that BC has been selected. From the Draw Menu tap on Construct and then tap on Midpoint. A point labelled D appears halfway between B and C. Use the Line Segment tool y to join AD. Use the Selection Arrow G to select BC and AD and from the Draw Menu choose to Attach Angle. You should now have a screen that looks like the second screen above. This shows that for this particular chord BC, the line from the centre of the circle to the midpoint of the chord is a right-angle just as the theorem claims. However, this may have just been a fluke. We can experiment by tapping on the point B to highlight it as shown in the third screen dump. Now tap and drag the point B to different positions around the circle. Notice that the angle between AD and BC is still 90 wherever you move B. Now try moving the point C. Copyright 2007, Hartley Hyde Page 4 of 7
Animation Tap the point B and tap the Circle. From the Edit Menu select Animate and then tap on Add Animation. Choose the Edit Menu again and try out each of the Go options. From the View Menu select Animation UI. This will give you a panel of controls for choosing various animation modes. Choose Animation UI to toggle the panel off. Notice that for every position that the figure is redrawn, the angle between AD and BC is always 90. Save your work as Chord1 You should now be able to write a hypothesis about the line from the centre of a circle to the midpoint of a chord. Write your hypothesis here. Checkpoint There are two points for which AD and BC are not perpendicular. If B and C coincide the whole structure collapses to just one radius. If B is directly opposite C, the chord BC becomes a diameter and again the whole structure collapses to a single line segment. You will find it very difficult to adjust your ClassPad accurately enough to see this. However, these examples show how careful we have to be when we form a hypothesis. Deductive Reasoning And now we are ready to look at an example of deductive reasoning. If we start with ideas you already understand such as congruent triangles, we can deduce the result that you have just discovered by experiment. Proof: In s ADB and ADC AB = AC (radii) AD is common BD = CD (D is midpoint of BC) s ADB and ADC are congruent (S.S.S.) Given: Figure as shown To Prove: m ADC = 90 Construction: Join AB and AC In particular m ADB = m ADC m ADB = m ADC = 90 (equal s on line) Q.E.D. Copyright 2007, Hartley Hyde Page 5 of 7
A Corollary of the Chord Theorem A corollary is an extra logical observation that is tacked onto the end of some theorems. In this case it is fairly obvious that: The Perpendicular bisector of a chord of a circle passes through the centre of a circle. Illustrating the Corollary You can now use your ClassPad to illustrate the Corollary to the Chord Theorem. Clear your screen. From the Draw Menu tap Circle. Tap near the centre of your screen and again near the bottom to get a circle. From the Draw Menu tap Line Segment and choose two points C and D which are on the circle but well clear of B. Use the Selection Arrow G to highlight the line segment CD as shown here. From the Draw Menu tap on Construct and then choose Perp. Bisector. Show your teacher what happens if you move points C or D. You may wish to set up an animation of the point C. Checkpoint Using the Corollary Imagine that you have a large circle drawn on a piece of cardboard. You may have drawn around the edge of a dinner plate. Explain how you could use the Corollary of the Chord theorem to find the centre of the circle. Checkpoint Use your ClassPad to show your teacher how your method works. Copyright 2007, Hartley Hyde Page 6 of 7
Checkpoints The Hypothesis Should amount to a restatement of the Chord Theorem. However, students are unlikely to have observed what happens when CD is a diameter. Checking the Corollary Students should be able to show you a screen like this. As they move either point C or D the perpendicular bisector should still pass through the centre. Attempts at animation should earn extra praise. Finding the centre Students may describe how they would draw two non-parallel chords, find the perpendicular bisector of each chord, and the perpendicular bisectors will intersect at the centre. Show students what goes wrong if the chords are parallel. Be alert for other ingenious schemes that may give equally satisfactory results. Copyright 2007, Hartley Hyde Page 7 of 7