# 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled?

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1 1.06 Circle Connections Plan The first two pages of this document show a suggested sequence of teaching to emphasise the connections between synthetic geometry, co-ordinate geometry (which connects algebra and synthetic geometry) and trigonometry, using the circle as the unifying theme. Further connections can be made to complex numbers e.g. multiplication by, modulus of a complex number and polar form of a complex number but these are not dealt with specifically in this document. The remainder of the document (pages 15 to 25) looks at each part of the sequence in more detail. Part A 1. Terms and definitions associated with the circle 2. Transformations of the circle 3. Lengths and areas associated with the circle including such questions as: (i) What happens to the circumference if the radius length is doubled? (ii) What happens to the area if the radius length is doubled? 4. Angle measures in the circle: degrees and radians Part B This part connects synthetic and co-ordinate geometry. However, the students just go as far as finding the centre and radius length of the circle in Part B, and do not derive the equation of the circle until Part C. The equation of the circle, which may appear new to students at this stage, is left until later, in order to emphasise the connections to geometry first. It is a very small step to take, from knowing the centre and radius of a circle, to finding its equation. Finding the centre and radius, given various clues, is where most of the thinking occurs. Deriving the equation of the circle is yet another example of where students can themselves construct new knowledge from prior knowledge. The theorems and constructions associated with the circle, are listed below and later on (pages 15 to 25) more detail is given on connecting them to the coordinated plane. The intention is that students will move seamlessly between the uncoordinated and the coordinated plane, verifying geometric results using algebraic methods. For example, students could construct a tangent to a circle at a point on an uncoordinated plane. This could be followed by constructing a tangent to a circle with a given centre, at a given point of contact on the coordinated plane. Using algebraic methods, the equation of the tangent is found and plotted, thus verifying the construction of the tangent. Note: It is envisaged that students would have a list of the theorems and constructions available to them to facilitate making connections. 13

2 (1) Theorem 19, Corollary 5 Theorem 19: The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum to 180, (and converse). (2a) Theorem 20, Construction 19 and Corollary 6 Theorem 20: (i) Each tangent is perpendicular to the radius that goes to the point of contact. (ii) If P lies on the circle, and a line through is perpendicular to the radius to, then is a tangent to. Construction 19: Tangent to a given circle at a given point on the circle Corollary 6: If two circles share a common tangent line at one point, then the two centres and that point are collinear. (2b) Theorem 20, Construction 1 and Construction 17 Theorem 20: (i) Each tangent is perpendicular to the radius that goes to the point of contact. (ii) If lies on the circle, and a line through is perpendicular to the radius to, then is a tangent to. Construction 1: Bisector of a given angle, using only compass and straight-edge. Construction 17: Incentre and incircle of a given triangle, using only straight-edge and compass. (3) Constructions 2 & 16, Corollaries 3 & 4 and Theorem 21 Construction 2: Perpendicular bisector of a segment, using only compass and straight-edge. Construction 16: Circumcentre and circumcircle of a given triangle, using only straight edge and compass. Corollary 3: Each angle in a semi-circle is a right angle. Corollary 4: If the angle standing on a chord at some point of the circle is a right angle, then is a diameter. Theorem 21: (i) The perpendicular from the centre to a chord bisects the chord. (ii) The perpendicular bisector of a chord passes through the centre. Part C 1. The equation of the circle derived by students using prior knowledge 2. Two equivalent forms of the circle equation Part D 1. The unit circle and the trigonometric functions for all angles 2. The graphs of and 14

5 Part B (1) Theorem 19 and corollaries 2 and 5 (LCHL and JCHL) Theorem 19: The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. Corollary 2: All angles at points of a circle, standing on the same arc, are equal. Corollary 5: If is a cyclic quadrilateral, then opposite angles sum to 180. Activity for students: Constructing, investigating through measurement, use of dynamic geometry software, proof) Assessment & Problem Solving: 2012 LCHL Q6 (b) Paper 2 (Cyclic quadrilateral) Assessment & Problem Solving: 2008 LCHL Q1 (c) (ii) Paper 2 below shows an application of Theorem 19. Given that at this point in the sequence, students have not yet dealt with the equation of a circle, this question can be modified to ask just for the centre and radius of the circle for part (i) The connection to Theorem 19 is in part (ii). 17

6 (2a) Theorem 20 & Construction 19 and Corollary 6 A discussion of construction 19 for the tangent to a circle at a point, carried out (i) on an un-coordinated plane and (ii) on a co-ordinated plane, where the centre of the circle and the point of contact are given. The equation of the tangent can be found using algebraic methods. This tangent line can be plotted, thus verifying the construction. Theorem 20: (i) Each tangent is perpendicular to the radius that goes to the point of contact. (ii) If lies on the circle, and a line through is perpendicular to the radius to, then is a tangent to s. Construction 19: Tangent to a given circle at a given point on it. Extension: Link the construction of the tangent to the construction of the perpendicular bisector of a line segment. Construct the tangent to a circle centre (1,2) at a point (3,4) on the circle on the coordinate plane and find the equation of the tangent Assessment & Problem Solving: 2012 LCHL Q3 Paper 2 (Various solution methods are possible) Connection:. Assessment & Problem Solving: UL 2012 problem solving questions: 18 The area referred to is the internal area of the triangular frame containing the snooker balls.

7 Corollary 6: If two circles share a common tangent line at one point, then the two centres and that point are collinear. What is the relationship between and when the circles touch externally/internally? On the coordinate plane: Prove that two circles touch externally/internally given both centres and both radii. Prove that two circles do not intersect given both centres and both radii. If we know the point in common and both centres, find the equation of the common tangent. If we know the centres, and the ratio of the lengths of the radii (if they touch externally), find the point of contact and the equation of the common tangent. Given the centre and the equation of the tangent to a circle, find the radius of the circle. (Perp. Dis) Assessment & Problem Solving: Given the centres and radii of two circles touching externally, find the point of contact of the two circles. Connection: Proportional reasoning (e.g Q4 LCHL Paper 2: Find the coordinates of the point of contact knowing the centres of the circles and lengths of the two radii without using a formula.) Assessment & Problem Solving: Questions showing the use of Corollary 6 on the coordinate plane LCHL Q9 (a) (ii) 2013 LCHL Q4: Find the point of contact and the equation of the tangent common to both circles: 2012 LCHL Q2: Prove that circles are touching given centres and radii, verify the point of contact, find the equation of the common tangent 2012 LCHL Sample Paper Q4: Using the perpendicular distance from the centre of the circle to the tangent being equal to the radius 2010 LCHL Q4: Find centre and radius given -axis, as tangents and the line through the centre of the circle. 19

8 (2b) Construction 17 (incircle) using Theorem 20 and Construction 1 Theorem 20: (i) Each tangent is perpendicular to the radius that goes to the point of contact. (ii) If lies on the circle, and a line through is perpendicular to the radius to, then is a tangent to. Construction 1: Bisector of a given angle, using only compass and a straight-edge Construction 17: Incentre and incircle of a given triangle using only straight-edge and compass Draw a sketch of the incircle of a triangle; the circle must touch the three sides of the triangle. circle? What is the relationship between the sides of the triangle and the three radii of the incircle at the points of contact? Describe the distances between the centre of the incircle and the sides of the triangle at the point of contact given the definition of a circle. What construction will allow you to find the centre of the incircle? What information does an angle bisector give about the points on the angle bisector? Construct the bisectors of all the angles in a triangle. What do you notice about all the angle bisectors? Construct the incircle of the triangle. Construct the incircle of various triangles. Assessment & Problem Solving: LCHL 2012 Q6B Paper 2 (incorporates Theorem 20, corollary 5, Theorem 19, Properties of an angle bisector) This question (or the construction of an incircle) could be transferred to a coordinate plane. 20

9 (3) Construction 2, Construction 16, corollaries 3 and 4 and Theorem 21 How many circles can be drawn through one point? How many circles can be drawn through two points? How are you constrained when given two points? Connection: Students should see that since the two points given have to be equidistant from one point i.e. the centre of the circle, that the construction of the perpendicular bisector of a line segment will give them the locus of all points equidistant from two points. How many circles can be drawn through three non-collinear points? Can you build on the previous outcome? Carry out the construction. Could you have drawn more than one circle through those three points? How many points define a unique circle? Join up all three points to form a triangle. Draw the perpendicular bisector of the third side of the triangle constructed. What do you notice? (The 3 perpendicular bisectors pass through one point i.e. they are concurrent. New vocabulary: Concurrent) Using the list of constructions, which construction have you carried out? (From the above activity the students have carried out Construction 16: Circumcentre and circumcircle of a given triangle using only compass and straight-edge. Ask students to label the circumcircle, circumcentre and circumradius on their constructions.) The sides of the triangle are chords of the circumcircle. What do you notice about the relationship between the chords of this circle, the perpendicular bisectors of the chords and the centre of the circle? Using the list of theorems, can you identify a theorem associated with your construction? (Students now see an instance of Theorem 21: (1) The perpendicular from the centre of the circle to a chord bisects the chord. (2) The perpendicular bisector of a chord passes through the centre. ) Construct the circumcircle for different types of triangles. Carry out the constructions with given points on the coordinate plane and verify the coordinates of the circumcentre and the length of the circumradius using algebra. Examples of triangles with integer coordinates: (1) Triangle with vertices H(-1,-3), I(5,9) and J(11,1) Acute (2) Triangle with vertices D(1,6), E(10,3) and F(2,3) Obtuse (3) Triangle with vertices A(1,3), B(7,5) and C(9,-1) Right angled (Different groups of students present their work (possibly using a visualizer) and patterns are identified. Verify Corollaries 3 & 4 on the coordinate plane using slopes and or/distances with triangle ABC Corollary 3: Each angle in a semicircle is a right angle. Corollary 4: If the angle standing on a chord [BC] at some point of the circle is a right angle, then [BC] is a diameter of the circle. Given 2 points and on a circle and the equation of a line through the centre of the circle, find the centre and radius of the circle. Assessment & Problem Solving: LCHL 2013 Q9 (d) (Arbelos and cyclic quadrilaterals) Assessment & Problem Solving: Workshop 6 questions 21

10 Part C Equation of a circle Students should understand that the derivation of the formula for the equation of a circle is based on If students have been exposed to deriving the formula for the distance between two points, they will see that We could start with a circle with centre as shown first below, or we could start with a circle with centre not and show that the equation of a circle with centre is a special case of the equation of a circle with centre. Note: To derive the equation of the circle, one could begin with a circle with centre passing through a specific point and then generalise to find the equation of a circle with centre and passing through the point. The following sequence (i) to (iv) however, begins with a circle passing through a centre which is not and in part (iii) we derive the equation of a circle with centre and passing through a point. We then show that the equation of the circle with centre passing through a point is a special case where =. Both approaches help students make sense of the equation of the circle formula. Part (i): Circle with centre and containing the point (starting with a specific circle) Construct a line segment from to and a line segment from to. Complete the right triangle using a line segment from to as the hypotenuse, of length of the triangle Mark in the lengths of the other two sides of the right triangle. use it to find the radius length Find the distance from to using the distance formula and compare with the result using Part (ii): Using a general point on any circle with centre (1, 2) Construct a circle, centre passing through a point A( ). Construct line segments from to and from to. Construct a right triangle using a line segment from to as the hypotenuse of the triangle of length. Mark in the lengths of the other two sides of the right triangle in terms of and. for this triangle and use it to find the radius length Find the distance from to using the distance formula and compare with the equation for 22

11 Part (iii): A circle with any centre and any point on its circumference Construct a circle, centre passing through a point. Construct line segments from to and from to. Construct a right triangle using a line segment from to as the hypotenuse of length of the triangle Mark in the lengths of the other two sides of the right triangle in terms of. Write down Pythag and use it to find the radius length Find the distance from to using the distance formula and compare with the result using Is this equation true for all circles? (Go back to the definition of the circle.) Part (iv): Showing that the equation of circle with centre and radius length is a special case of the equation of a circle with centre and radius length. Given the equation of the circle centre and radius length? Follow up:, what would be the equation of a circle with Use the equation of the circle formula to get the equations of circles with different centres and different radii Write down circle equations and then find their centres and radii Use two ways of checking if a point is on, inside, or outside the circle. Compare both ways. Given a circle with a radius which is the largest number of a Pythagorean triple, use the particular Pythagorean triple to find 12 points on the circle with integer values. 23

13 Part D Unit circle and the trigonometric functions Construct a circle with centre and radius 1 on the coordinate plane. This is called the unit circle as it has centre and radius equal to one unit. Name 4 points with integer coordinates on the unit circle. Choose any point with coordinates ( ) on the unit circle, in the first quadrant. Draw in the radius from to ( ). Draw a line segment from ( ) to. Draw a line segment from to. This completes the right angled triangle with vertices, and ( ). Alternatively use the distance formula to find the distance from to. Rewrite the equation without square roots. Mark in the angle Write down the relationship between intersects the unit circle. ( in standard position in the triangle drawn in the first quadrant. and the -coordinate of where the terminal ray of the angle Write down the relationship between and the -coordinate of where the terminal ray of the angle intersects the unit circle. The fact that is true for all values of in the unit circle enables us to extend our use of the trigonometric ratios (learned in Junior Certificate and restricted to angles between and ) to trigonometric functions which model real life phenomena that occur in cycles. Triangles occur in circles. Circles link to cycles. Trigonometric functions can be used to model real life phenomena that occur in cycles. Prove that. Using the real-life analogy with movement of a point on a Ferris wheel, describe how and vary as a point moves from around the wheel back to. We are looking at co-variation here. What happens to the sine (or cosine) of the angle as we rotate around the coordinate grid? The sine (or cosine) value of the angle is a function of the angle of rotation. Sometimes we refer to these trigonometric functions as circular functions because they can be defined using the unit circle. Plot graphs of and and compare. Plot graphs of ). Relate and back to the circle. 25

14 Notes on Circle Connections: 26

15 1.07 Theorems and Corollaries 1. Vertically opposite angles are equal in measure. 2. In an isosceles triangle, the angles opposite the equal sides are equal, (and converse). 3. If a transversal makes equal alternate angles on two lines, then the lines are parallel, (and converse). 4. The angles in any triangle add to 5. Two lines are parallel if and only if, for any transversal, corresponding angles are equal. 6. Each exterior angle of a triangle is equal to the sum of the interior opposite angles. 7. The angle opposite the greater of two sides is greater than the angle opposite the lesser side, (and converse). 8. Two sides of a triangle are together greater than the third. 9. In a parallelogram, opposite sides are equal, and opposite angles are equal, (and converses). Corollary 1: A diagonal divides a parallelogram into two congruent triangles. 10. The diagonals of a parallelogram bisect one another. 11. If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal. 12. Let be a triangle. If a line is parallel to and cuts in the ratio, then it also cuts in the same ratio, (and converse). 13. If two triangles and are similar, then their sides are proportional, in order, (and converse). 14. In a right-angled triangle the square of the hypotenuse is the sum of the squares of the other two sides. 15. If the square of one side of a triangle is the sum of the squares of the other two, then the angle opposite the first side is a right angle. 16. For a triangle, base times height does not depend on choice of base. 17. A diagonal of a parallelogram bisects the area. 18. The area of a parallelogram is the base by the height. 19. The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc. Corollary 2: All angles at points of a circle, standing on the same arc, are equal (and converse). Corollary 3: Each angle in a semi-circle is a right angle. Corollary 4: If the angle standing on a chord at some point of the circle is a right angle, then is a diameter. Corollary 5: If is a cyclic quadrilateral, then opposite angles sum to,(and converse). 20. (i) Each tangent is perpendicular to the radius that goes to the point of contact. (ii) If lies on the circle, and a line through is perpendicular to the radius to, then is a tangent to. Corollary 6: If two circles share a common tangent line at one point, then the two centres and that point are collinear. 21. (i) The perpendicular from the centre to a chord bisects the chord. (ii) The perpendicular bisector of a chord passes through the centre. 27

16 1.08 Constructions 1. Bisector of a given angle, using only compass and straight edge. 2. Perpendicular bisector of a segment, using only compass and straight edge. 3. Line perpendicular to a given line, passing through a given point not on. 4. Line perpendicular to a given line, passing through a given point on. 5. Line parallel to given line, through given point. 6. Division of a segment into equal segments, without measuring it. 7. Division of a segment into any number of equal segments, without measuring it. 8. Line segment of given length on a given ray. 9. Angle of given number of degrees with a given ray as one arm. 10. Triangle, given lengths of three sides. 11. Triangle, given data. 12. Triangle, given data. 13. Right-angled triangle, given the length of the hypotenuse and one other side. 14. Right-angled triangle, given one side and one of the acute angles (several cases). 15. Rectangle, given side lengths. 16. Circumcentre and circumcircle of a given triangle, using only straight edge and compass. 17. Incentre and incircle of a given triangle, using only straight-edge and compass. 18. Angle of without using a protractor or set square. 19. Tangent to a given circle at a given point on the circle. 20. Parallelogram, given the length of the sides and the measure of the angles. 21. Centroid of a triangle. 22. Orthocentre of a triangle. 28

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