Properties of the Circle


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1 9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference of a circle Concentric circles: Circles that have the same centre Concyclic points: Points that lie on the circumference of the same circle Cyclic quadrilateral: A cyclic quadrilateral is a figure whose four vertices are concyclic points. The four vertices lie on the circumference of a circle Radius: A radius is the distance from the centre of a circle out to the circumference (radii is plural, meaning more than one radius) Subtend: Form an angle at some point (usually the centre or circumference of a circle) Tangent: A straight line external to a curve or circle that just touches the curve or circle at a single point
2 Chapter 9 Properties of the Circle 499 INTRODUCTION IN CHAPTER 4, YOU STUDIED the geometry of angles, triangles, quadrilaterals and other polygons. This chapter shows you some properties of the circle. DID YOU KNOW? A rainbow is the shape of an arc of a circle. If you could see the whole rainbow, it would form a circle. Research the rainbow on the Internet and find out more about its shape and other properties. Parts of a Circle An arc is a part of the circumference. A tangent touches the circle at one point.
3 500 Maths In Focus Mathematics Extension 1 Preliminary Course Concentric circles are circles that have the same centre. Equal circles have the same radius. DID YOU KNOW? Leonardo da Vinci ( ) was a great artist, scientist and inventor. He studied geometry, and many of his model drawings show this influence. His drawings included designs for flying machines, springdriven automobiles, bridges and weapons. Leonardo s designs were revolutionary, and the scientists of his time did not have the knowledge needed to make the models work. Arcs, Angles and Chords Equal arcs subtend equal angles at the centre of the circle.
4 Chapter 9 Properties of the Circle 501 Proof Let two equal arcs have lengths l 1 and l 2, and subtend angles of a and b at the centre of the circle. Using l = ri, l = ra and l = rb 1 2 But l = l 1 2 ra = rb a = b The converse is also true: You will study the formula l = ri in Chapter 5 of the HSC Course book. If two arcs subtend equal angles at the centre of the circle, then the arcs are equal. EXAMPLE AB is a diameter of the circle with centre O. Arc CB = arc BD. Prove + AOC = + AOD. Solution Since arc CB = arc BD, + COB = + DOB Let + COB = + DOB = x Then + AOC = 180c  + COB ( + AOB is a straight angle) = 180c  x Also + AOD = 180c  + DOB (similarly) = 180c  x + AOC = + AOD
5 502 Maths In Focus Mathematics Extension 1 Preliminary Course Equal chords subtend equal angles at the centre of the circle. Proof OA = OC (equal radii) OB = OD (similarly) AB = CD (given) by SSS, DOAB / D OCD + AOB = + COD (corresponding + s in congruent Ds ) The converse in also true: Equal angles subtended at the centre of the circle cut off equal chords. EXAMPLE AD is a diameter of the circle with centre O, where AB = CD. Prove that BC < AD.
6 Chapter 9 Properties of the Circle 503 Solution Since AB = CD, + AOB = + COD Let + AOB = + COD = x Then+ BOC = 180c ( x + x) ( + AOD is a straight+ ) = 180c 2x OB = OC ( equal radii) D OBC is isosceles with + OBC = + OCB + OBC + + OCB + 180c 2x = 180c ( + sum of DOBC) + OBC + + OCB = 2x + OBC = + OCB = x + OBC = + AOB But these are equal alternate angles BC < AD The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. These figures show that this property can look quite different in different situations. Proof Join AO and produce to D. Let+ BAO = xand+ CAO = y + BAC = x + y OA = OB ( equal radii)
7 504 Maths In Focus Mathematics Extension 1 Preliminary Course + OBA = x (base+ s of isosceles D) = + BAO OA = OC ( equal radii) + OCA = y (base+ s of isosceles D) = + CAO + BOD = x + x (exterior+ of DOBA) = 2x + COD = y + y (exterior+ of DOCA) = 2y + BOC = + BOD + + COD = 2x + 2y = 2( x + y) = 2+ BAC EXAMPLES 1. Find the values of x and y. Solution x = 40 ( + at centre is twice the+ at the circumference) Reflex+ BOC = 360c80c ( + of revolution) = 280c y = 140 ( + at centre is twice the+ at the circumference) 2. Prove + BOC is twice the size of + OCA.
8 Chapter 9 Properties of the Circle 505 Solution Let + OAC = x + BOC = 2 + OAC ( + at centre is twice the + at the circumference) = 2x OA = OC ^equal radiih + OCA = + OAC = x (base + s of isosceles D) + BOC = 2+ OCA Angles in the same segment of a circle are equal. Proof Join A and D to centre O + AOD = 2 + ABD ( + at centre is twice the + at the circumference) + AOD = 2+ ACD + ABD = + ACD
9 506 Maths In Focus Mathematics Extension 1 Preliminary Course EXAMPLE Prove D ABC and D DEC are similar. Hence find the value of y correct to 1 decimal place. Solution + ABC = + DEC ( + s in same segment) + BCA = + ECD ( vertically opposite + s) D ABC DDEC ED EC = BA BC y 65. = y = 17. # # 65. y = 12. = 92. cm correct to 1 decimal place. The angle in a semicircle is a right angle. Proof
10 Chapter 9 Properties of the Circle AOB = 180c (straight+) + AOB = 2+ ACB ( + at centre is twice the+ at the circumference) + ACB = 90c EXAMPLE AB is a diameter of the circle below. If BC = 2 m and AB = 9 m, find the exact length of AC. Solution + ACB = 90c AB = AC + BC = AC = AC = AC AC = 77 m 9.1 Exercise s 1. Find values of all pronumerals ( O is the centre of each circle). (b) (a)
11 508 Maths In Focus Mathematics Extension 1 Preliminary Course (c) 3. Find values of all pronumerals ( O is the centre of each circle). (a) (d) (b) (e) (c) (f) (d) 2. The circumference of a circle is 16r cm. Find the length of the arc that subtends an angle of 40c at the centre of the circle. (e)
12 Chapter 9 Properties of the Circle 509 (f) 4. Find values of all pronumerals ( O is the centre of each circle). (a) (g) (b) (h) (c) (i) (d) (j)
13 510 Maths In Focus Mathematics Extension 1 Preliminary Course (e) (j) (f) 5. (a) Prove D ABC DDEC. (b) Hence find the value of x correct to 1 decimal place. (g) 6. Find x and y, giving reasons. (h) 7. Find x and y, giving reasons. (i)
14 Chapter 9 Properties of the Circle Evaluate x, giving reasons for each step in your calculation. 11. Find x, giving reasons for each step in your calculations. 9. Prove D STV and D WUV are similar. Hence find x. 12. The circle below has centre O. D O y C A x 38c 10. AB = 6cm and BC = 3cm. O is the centre of the circle. Show that the radius of the circle is 3 5 cm. 2 B (a) Evaluate x and y. (b) Show that AD = BC. 13. Show that AD < BC in the circle below. A D 33c 114c B C
15 512 Maths In Focus Mathematics Extension 1 Preliminary Course 14. A circle has centre O and radius r as shown. A 15. The circle below has centre O and + DAB = i. A i O B O B C C D (a) Show that triangles AOB and ABC are similar. (b) Show that BC = 2 r. Show that + DAB and + BCD are supplementary. Chord Properties A perpendicular line from the centre of a circle to a chord bisects the chord. Proof
16 Chapter 9 Properties of the Circle ADO = + BDO = 90 c (given) OA = OB (equal radii) OD is common by RHS DOAD / DOBD AD = BD (corresponding sides in congruent Ds) So OD bisects AB The converse is also true: A line from the centre of a circle that bisects a chord is perpendicular to the chord. EXAMPLES 1. Line OC is perpendicular to chord AB. If the radius of the circle is 6 cm and the chord is 11 cm long, find the length of OC, correct to 1 decimal place. Solution AB = 11 AC = 5.5 ( OC bisects AB) Also OA = 6 (radius given) OA = AC + OC = OC = OC 5.75 = OC OC = 575. = 2.4 cm 2 2 CONTINUED
17 514 Maths In Focus Mathematics Extension 1 Preliminary Course 2. Given XY = YZ and + OXY = 54 c, find +XOY. Solution + OYX = 90 c ( OY bisects XZ) + XOY = 180 c (90c+ 54 c) ( + sum of DOXY) = 36c Equal chords are equidistant from the centre of the circle. Proof Let CD = AB + OEB = + OFD = 90c OB = OD AB = CD (given) (equal radii) (given)
18 Chapter 9 Properties of the Circle BE = AB 2 ( OE bisects AB) 1 DF = CD 2 ( OF bisects CD) BE = DF by RHS DOEB / DOFD OE = OF (corresponding sides in congruent Ds) The converse is also true: Chords that are equidistant from the centre are equal. Class Exercise Prove that chords that are equidistant from the centre are equal. EXAMPLE In the circle below, with centre O, OE = 35 mm, DE = 56 mm and OC = 21 mm. Show that AB = DE. Solution EF = 28 mm ( OF bisects DE) OE = EF + OF = 28 + OF = OF = OF OF = 441 = 21 mm OF = OC ( chords equal when equidistant from the centre) So AB = DE.
19 516 Maths In Focus Mathematics Extension 1 Preliminary Course The products of intercepts of intersecting chords are equal. AE: EB = DE: EC Proof + AED = + CEB ( vertically opposite + s) + DAE = + ECB ( + s in same segment) + ADE = + EBC (similarly) DAED < ;DCEB AE DE = EC EB AE: EB = DE: EC EXAMPLE Given chord AB = 12.3 cm, EB = 2.7 cm and DE = 10.6 cm, find the length of EC, correct to 1 decimal place.
20 Chapter 9 Properties of the Circle 517 Solution AE = AB  EB = = 96. AE: EB = DE: EC 96. # 27. = 106. # EC 96. # 27. EC = = 24cm. 9.2 Exercises 1. Find the values of all pronumerals ( O is the centre of each circle). (d) (a) (e) (b) (f) (c)
21 518 Maths In Focus Mathematics Extension 1 Preliminary Course (g) 3. A circle with radius 89 mm has a chord drawn 52 mm from the centre. How long, to the nearest millimetre, is the chord? (h) 4. O and P are the centres of intersecting circles with radii 20 cm and 8 cm respectively. If AB = 10 cm, find the distance OP, correct to 1 decimal place. (i) 5. Show AB = CD. (j) 6. AC = 20 cm and AD = 26 cm. Find OB, correct to 1 decimal place. 2. Find the exact radius of a circle with a chord that is 8 cm long and 5 cm from the centre.
22 Chapter 9 Properties of the Circle Evaluate x and y, correct to 1 decimal place. 9. A circle with centre O has radius r and chord AB = x. D A O C 8. Find the values of all pronumerals r + 4r 2  x Show that CD =. 2 A B E C D B (a) Prove that triangles ABC and CDE are similar. (b) Show that AC : CD = BC : CE. Concyclic Points Concyclic points are points that lie on the circumference of a circle.
23 520 Maths In Focus Mathematics Extension 1 Preliminary Course Any 3 noncollinear points are concyclic. They lie on a unique circle, with centre at the point of intersection of the perpendicular bisectors of the intervals joining these points. Four or more noncollinear points may not necessarily lie on a circle. Cyclic quadrilaterals A cyclic quadrilateral is a figure whose 4 vertices are concyclic points. The opposite angles in a cyclic quadrilateral are supplementary.
24 Chapter 9 Properties of the Circle 521 Proof Join B and D to O. Obtuse + DOB = 2+ A Reflex + DOB = 2+ C Obtuse + DOB + reflex + DOB = 360c 2+ A + 2+ C = 360c + A + + C = 180c ( + at centre is doubl e +at circumference) ( + of revolution) Similarly, it can be proven that + B + + D = 180c by joining A and C to O. The converse is also true: If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. The property of opposite angles being supplementary in a cyclic quadrilateral can also be used to prove the following property: The exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle.
25 522 Maths In Focus Mathematics Extension 1 Preliminary Course Proof Let + A = x Then + BCD = 180c  x ( opposite + s supplementary in cyclic quadrilateral) + BCD + + BCE = 180c ( + DCE straight angle) + BCE = 180c ( 180c  x) = 180c  180c + x = x + A = + BCE EXAMPLE Evaluate a, b and c. Solution OB = OD + ODB = 20c ( equal radii) ( base + s of isosceles D equal) a + 20c+ 20c = 180c ( + sum of D) a = 180c 40c = 140c b = 70c c = 180c 70c ( + at centre double + at circumference) ( opposite + s in cyclic quadrilateral) = 110c
26 Chapter 9 Properties of the Circle Exercises 1. Find the values of all pronumerals. (e) (a) (f) (b) (g) (c) (h) (d) (i)
27 524 Maths In Focus Mathematics Extension 1 Preliminary Course (j) 118c xc (e) O yc xc 46c 111c 2. Find the values of all pronumerals. (f) (a) (g) (b) (h) (c) (i) (d) (j)
28 Chapter 9 Properties of the Circle Show that ABCD is a cyclic quadrilateral. (c) B i C (a) A D B 58c 58c C D i (b) B C A E D A Tangent Properties The tangent to a circle is perpendicular to the radius drawn from the point of contact. The perpendicular distance is the shortest distance any other distance would be greater than the radius. The converse is also true: The line perpendicular to the radius at the point where it meets the circle is a tangent to the circle at that point.
29 526 Maths In Focus Mathematics Extension 1 Preliminary Course Here is another property of tangents to a circle: Tangents to a circle from an exterior point are equal. Proof Join OB + A = + C = 90c (tangent = radius) OB is common OA = OC (equal radii) by RHS, DOAB / DOCB AB = CB (corresponding sides in congruent Ds) EXAMPLE A circle with centre O and radius 10 cm has a tangent AB drawn to it where OB = 26 cm. Find the length of AB.
30 Chapter 9 Properties of the Circle 527 Solution OA = 10 ( radius) + OAB = 90c (tangent = radius) OB = OA + AB = 10 + AB = AB = AB AB = 576 = 24 cm When two circles touch, the line through their centres passes through their point of contact. Proof You could also prove this result for when circles touch internally. AB is a tangent to circle with centre O + OCB = 90c ( tangent = radius) AB is a tangent to circle with centre P + PCB = 90c + OCB + + PCB = 90c+ 90c = 180c (similarly) OCP is a straight line.
31 528 Maths In Focus Mathematics Extension 1 Preliminary Course EXAMPLE Two circles touch at A and the larger circle has centre O. Prove (a) T ABC and T AOD are similar (b) CB< DO (c) BC = 2DO. Solution (a) OA is a diameter of small circle + ODA = 90c ( + in semicircle) Since AB is a diameter of the larger circle, + BCA = 90c (similarly) + BCA = + ODA + A is common DABC ; D AOD (b) + BCA = + ODA [from (a)] These are equal corresponding angles. CB< DO (c) AB = 2 OA ( OA radius) AB OA AB OA BC = 2 AC BC = = = 2 AD DO = 2DO (line through centres passes through point of contact) (by similar Ds) The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
32 Chapter 9 Properties of the Circle 529 Proof Draw in diameter CF and join EF. Let+ ECB = x. + FCB = 90c ( tangent = radius) + FCE = 90c  x + FEC = 90c ( + in semicircle) + EFC = 180c ( 90c+ 90c x) ( angle sum of D) = x + EFC = + EDC ( angles in same segment) + EDC = + ECB EXAMPLE Evaluate x and y. CONTINUED
33 530 Maths In Focus Mathematics Extension 1 Preliminary Course Solution + ABC = 180c ( 95c+ 50c) ( + sum of D) = 35c x = 35c y = 35c ( + s in alternate segment) ( y and+ ABCin same segment) The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point. PQ 2 = QR: QS where PQ is a tangent to the circle. Proof The third pair of angles is equal by angle sum of a triangle. + QPR = +PSR ( angles in alternate segments) + Q is common PQR; ;DSPQ PQ PR QR = = QS SP PQ PQ QR = QS PQ 2 PQ = QR: QS
34 Chapter 9 Properties of the Circle 531 EXAMPLE AB is a tangent to the circle and CD = 1.3 cm, BC = 1.7 cm. Find the length of AB, correct to 1 decimal place. Solution BD = = 3 2 AB = BC: BD = 17. # 3 = 51. AB = 51. = 2. 3 cm correct to 1 decimal place. 9.4 Exercises 1. Find the values of all pronumerals. (b) (a)
35 532 Maths In Focus Mathematics Extension 1 Preliminary Course (c) (h) (d) y (i) x O 38c (e) (j) (f) (g) 2. Find the values of all pronumerals (all external lines are tangents to the circles). (a)
36 Chapter 9 Properties of the Circle 533 (b) (g) (c) (h) (i) (d) 63c xc O yc (j) (e) 9 cm 15 cm 13 cm O y P (f)
37 534 Maths In Focus Mathematics Extension 1 Preliminary Course 3. Find the values of all pronumerals, giving reasons for each step of your working ( O is the centre of circle, AB is a tangent). 6. Find the values of all pronumerals ( O is the centre of each circle; all external lines are tangents). (a) 4. AB, BC and AC are tangents, with AB = 24 cm, BC = 27 cm and BM = 15 cm. Find the length of AC. (b) (c) 5. AB = 6.5 m, AC = 3.9 m and BC = 5.2 m. Prove A lies on a diameter of the circle, given BC is a tangent to the circle. (d)
38 Chapter 9 Properties of the Circle 535 (e) (i) (f) (j) (g) 7. Find the values of all pronumerals. (a) (h) (b)
39 536 Maths In Focus Mathematics Extension 1 Preliminary Course (c) (h) (d) (i) (e) (j) (f) 8. Find AB, given AD = 49m,. BC = 51. mandcd = 78. m. (g)
40 Chapter 9 Properties of the Circle 537 Test Yourself 9 1. O is the centre of the circle. Evaluate i. 5. Evaluate x, y and z, giving reasons for each step of your working. 2. Evaluate y to 1 decimal place. 6. O is the centre of the larger circle. Find the value of x. 3. AB is a tangent to the circle. Find the value of x to 1 decimal place. 7. AB is a tangent to the circle. Evaluate ab, and c. 4. O is the centre of the circle. Find the length of tangents x and y. 8. O is the centre of the circle, and AB is a tangent. Evaluate a, b, c and d, giving reasons for each step of your working.
41 538 Maths In Focus Mathematics Extension 1 Preliminary Course 9. Find the length of the radius of the circle. O is the centre. 13. Calculate the length of AB to 3 significant figures, given that A and B are the centres of the circles. 10. Find the length of tangent AB. 14. Find the value of x to 1 decimal place. 11. Evaluate a and b. 15. Find the length of AB. 12. O is the centre of the circle. Find the value of a and b. 16. Evaluate a and b.
42 Chapter 9 Properties of the Circle Evaluate x and y, giving reasons for your working. 19. Prove that D BCD is similar to D ABC. 18. Evaluate xy, and z. 20. O is the centre of the circle. (a) Prove that D OAC and D OBC are congruent. (b) Show that OC bisects AB. Challenge Exercise 9 1. Find the length of the radius, to the nearest centimetre, if AC = 10 cm and BD = 3 cm. 2. In the circle below with centre O, OD = DC. Prove + AOE = 3+ DCB.
43 540 Maths In Focus Mathematics Extension 1 Preliminary Course 3. Two circles meet at points A and B. A tangent to each circle is drawn from A to meet the circles at D and C. Prove + DAC = 90c. 5. The triangles below have + BDE = 2+ BAD and + CDE = 2+ CAD. Prove that a circle can be drawn through A, B and C with centre D. 4. Three equal circles touch each other, as in the figure. (a) Prove that the triangle with vertices the centres of the circles is equilateral. (b) Find the perimeter of the curved figure DEF in terms of the radius r of the circles. (c) Find the exact area of the shaded region. 6. Two chords AB and CD intersect at 90c Prove, for obtuse + AOD, + AOD + + COB = 180c where O is the centre of the circle. 7. Prove that any kite ABCD with + ADC = + ABC = 90c is a cyclic quadrilateral with diameter AC.
44 Chapter 9 Properties of the Circle A large circle with radius R is surrounded by 7 smaller circles with radius r. A circle is drawn through the centres of the 3r smaller circles. If R =, find the shaded 2 area in terms of r. 9. Prove that if an interval subtends equal angles at two points on the same side of it, then the endpoints of the interval and the two points are concyclic. 10. Prove that if both pairs of opposite angles in a quadrilateral are supplementary, then the quadrilateral is cyclic.
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