Circle theorems CHAPTER 19

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1 352 CHTE 19 Circle theorems In this chapter you will learn how to: use correct vocabulary associated with circles use tangent properties to solve problems prove and use various theorems about angle properties inside a circle prove and use the alternate segment (intersecting tangent and chord) theorem prove and use the intersecting chords theorem. You will also be challenged to: investigate the nine-point circle theorem. Starter: Circle vocabulary Here are some words you will often encounter when working with circles: Centre adius Chord Diameter Circumference Tangent rc Sector Segment Match the correct words to the nine diagrams below: a) The French call a rainbow an arc-en-ciel (ciel sky). Do you think this is a good name? b) The English call a piece of an orange a segment. Do you think this is a good name?

2 19.1 Tangents, chords and circles Tangents, chords and circles In this section you will learn some theorems about circles, and then use them to solve problems. The theorems are concerned with tangents and chords. chord is a line segment joining two points on the circumference of a circle. tangent is a straight line that touches a circle only once. line segment drawn from the centre of a circle to the midpoint of a chord will intersect the chord at right angles. If M is the midpoint of, i.e. M and M are the same length M then these two angles will both be right angles. tangent and radius meet at right angles. adius The fact that the radius and the tangent meet at right angles is very obvious when one is vertical and the other is horizontal Tangent adius but is not quite so obvious when the situation is rotated, like this. Tangent

3 354 Chapter 19: Circle theorems The two eternal tangents to a circle are equal in length. Tangent Tangent These theorems may be used to help you determine the values of missing angles in circles. When you use them, remember to tell the eaminer which theorem(s) you have used. EMLE The diagram shows a circle, centre. T is a tangent to the circle. Find the value of. 48 T SLUTIN ngle T 90 (angle between the radius and tangent is 90 ). The angles in triangle T add up to 180, so:

4 19.1 Tangents, chords and circles 355 EMLE The diagram shows a circle, centre. is a chord across the circle. M is the midpoint of. Find the value of y. SLUTIN ngle M 90 (radius bisecting chord). The angles in triangle M add up to 180. y 62 M So: y y y y 28 Since a radius bisects a chord at right angles, there is often an opportunity to use ythagoras theorem. EMLE The diagram shows a radius T that bisects the chord at M. M 12 cm. The radius of the circle is 13 cm. Work out the length MT. M T SLUTIN First join : Now apply ythagoras theorem to triangle M: M M M 25 5 cm The distance T is a radius, that is, 13 cm. Thus MT cm T

5 356 Chapter 19: Circle theorems EECISE T is a tangent to the circle, centre. ngle T 29. y 29 T a) State, with a reason, the value of the angle marked. b) Work out the value of the angle marked y. 2 T is a tangent to the circle, centre. T 24 cm. 7 cm. 7 cm 24 cm T The line T intersects the circle at, as shown. Work out the length of T. 3 T and T are tangents to the circle, centre. ngle T is T a) Work out the size of angle. Give reasons. b) What type of quadrilateral is T? Eplain your reasoning.

6 19.1 Tangents, chords and circles T and T are tangents to the circle, centre. ngle is y T a) What type of triangle is triangle? b) Work out the value of. c) Work out the value of y. 5 The diagram shows a circle, centre. The radius of the circle is 5 cm. M is the midpoint of EF. M 3 cm. E M F Calculate the length of EF. 6 The diagram shows a circle, centre. 34 cm. M is the midpoint of. M 8 cm. M Work out the radius of the circle.

7 358 Chapter 19: Circle theorems 7 From a point T, two tangents T and T are drawn to a circle, centre. a) Make a sketch to show this information. The length T is measured, and found to be eactly the same as the length. b) What type of quadrilateral is T? 8 The diagram shows a circle, centre. and CD are chords. The radius T passes through the midpoints M and N of the chords. M 8 cm, NT 3 cm, 30 cm. 8 cm M C N 3 cm T D a) Eplain why angle M 90. b) Use ythagoras theorem to calculate the distance. Show your working. c) Calculate the distance MN. d) Calculate the length of the chord CD.

8 19.2 ngle properties inside a circle ngle properties inside a circle There are several important theorems about angles inside a circle. You will need to learn these, and use them to solve numerical problems. You may also be asked to prove why they are true. Consider two points, and say, on the circumference of a circle. The angle subtended by the arc at the centre is angle. ngle subtended by arc at The angle subtended by the arc at a point on the circumference is angle. ngle subtended by arc at There is a theorem in circle geometry which states that angle is eactly twice angle. This angle at the centre is twice as big as this one, at the circumference. 2 This result is quite easy to prove, and is the basic theorem upon which several other circle theorems are built.

9 360 Chapter 19: Circle theorems THEEM The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. F Make a diagram to show the arc, the centre, and the point on the circumference of the circle: From, draw a radius to, and produce it, which means etend it slightly: Triangle is isosceles, since both and are radii of the same circle. Therefore angles and are equal. These are marked on the diagram with a letter a: a a Likewise the triangle is isosceles, since both and are radii of the same circle. Therefore angles and are equal. These are marked on the diagram with a letter b: The angle at the circumference is angle a b. a a b b

10 19.2 ngle properties inside a circle 361 To obtain an epression for the angle at the centre, look at this magnified copy of the diagram: a b 180 2a 180 2b 2a 2b a Y b ngle 180 a a 180 2a So: angle Y 180 (180 2a) 2a Likewise: angle 180 b b 180 2b So: angle Y 180 (180 2b) 2b Thus the angle at the centre is: ngle 2a 2b 2(a b) ut angle a b, from above. Therefore angle 2 angle. Thus, the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. Two further theorems can be deduced immediately from this first one. You can quote the previous theorem to justify these proofs.

11 362 Chapter 19: Circle theorems THEEM ngles subtended by an arc in the same segment of a circle are equal. These three angles are all equal. F Join and so that they form an angle at the centre: If the angle at is, then the angle at the centre must be 2 2 and if the angle at the centre is 2, then the angle at must be. Thus angles and are equal. THEEM The angle subtended in a semicircle is a right angle. F Since is a diameter, is a straight line. If this line is a diameter Thus angle 180. Using the result that the angle at the circumference is half that at the centre: ngle then this angle will be a right angle.

12 19.2 ngle properties inside a circle 363 EMLE Find the values of the angles marked and y. Eplain your reasoning in both cases Diagram not to scale y SLUTIN ngle 44 (angles in the same segment are equal). 70 y 180 (angles in a triangle add up to 180 ) and 44, so: y y 180 y y 66 EMLE Find the values of the angles marked and y. Eplain your reasoning in both cases y SLUTIN y (angle at centre 2 angle at circumference) (angle in a semicircle is a right angle and angles in a triangle add up to 180 )

13 364 Chapter 19: Circle theorems EECISE 19.2 Find the missing angles in these diagrams, which are not drawn to scale. Eplain your reasoning in each case d 48 a f g 28 b 44 c e h 78 i j k 104 l m n p s 36 q r t u 75 v 2v

14 19.3 Further circle theorems Further circle theorems THEEM The angles subtended in opposite segments add up to 180. This angle plus this one add up to 180. F Denote the angles and as p and q respectively. p Then the angles at the centre are twice these, that is, 2p and 2q. ngles at point add up to 360, so: 2p 2q 360 2q 2p Thus 2(p q) 360 So p q 180 q The points,, and form a quadrilateral whose vertices lie around a circle; it is known as a cyclic quadrilateral. Thus the theorem may also be stated as: pposite angles of a cyclic quadrilateral add up to 180

15 366 Chapter 19: Circle theorems EMLE Find the angles and y. Diagram not to scale y SLUTIN For angle, we have (angles in opposite segments) Thus For angle y, two construction lines are needed: a y 96 Use angles in opposite segments, a Using the angle at centre is twice the angle at circumference: y

16 19.3 Further circle theorems 367 THEEM The angle between a tangent and chord is equal to the angle subtended in the opposite segment. (This is often called the alternate segment theorem.) This angle is equal to this one. F First, consider the special case of a tangent meeting a diameter: y Since the angle in a semicircle is 90, the other two angles in the triangle add up to 90. Hence y z 90. Since a radius and tangent meet at 90 : z 90 Hence z y z. From which it follows that y. z Now move around the circle to, say, so that it is no longer on the end of a diameter. The angle at is equal to the angle at, as they are angles in the same segment. Thus the theorem is proved. y y z

17 368 Chapter 19: Circle theorems EECISE 19.3 Find the missing angles in these diagrams, which are not drawn to scale. Eplain your reasoning in each case a b f d c e g h i j k l n m q s 87 p r t 52 v w 132 u

18 19.4 Intersecting chords Intersecting chords THEEM Consider a pair of chords and CD intersecting at a point inside a circle, as shown in this diagram: C D Then the lengths,, C and D are related by the following result: C D F Join C and D to complete two triangles C and D as below: C Then angles C and D are equal (angles in the same segment). ngles C and D are equal (angles in the same segment). ngles C and D are equal (vertically opposite). Thus the triangles C and D are mathematically similar, so triangle D is an enlargement of triangle C. The enlargement factor is C, but it is also D, and these must be equal, so C D Cross-multiplying, C D and the result is proved. D

19 370 Chapter 19: Circle theorems EMLE Find the missing length in the diagram below. 5 cm 8 cm 10 cm SLUTIN Using the result for intersecting chords we have So the missing length is 4 cm. S The same principle is valid even when the two chords cross over outside the circle, as in the net eample. EMLE and S are chords of a circle, and, when produced, intersect at. 10 cm 10 cm. S 6 cm. 9 cm. a) Write down the length S. b) Work out the length. c) Work out the length. 6 cm 9 cm SLUTIN a) S cm S b) S cm c) cm

20 19.4 Intersecting chords 371 EECISE Find the missing length, cm, in this diagram. 10 cm S 3 cm 9 cm cm 2 Find the missing length, y cm, in this diagram. y cm 12 cm S 5 cm 10 cm 3 Chords and DC, when produced, meet at point. 3 cm. 12 cm. C 10 cm. 3 cm 12 cm 10 cm cm C D a) Work out the length D. b) Hence work out the length DC, marked as cm. 4 In the diagram below, is a diameter of the circle. 2 cm. 6 cm. S 4 cm. cm. 6 cm 2 cm cm 4 cm S a) Work out the value of. b) Hence work out the radius of the circle.

21 372 Chapter 19: Circle theorems 5 Chords and S, when produced, meet at point W. 7 cm. W 8 cm. SW 10 cm. a) Work out the length W. b) Hence work out the length S, marked as cm. W 8 cm 7 cm 10 cm S cm EVIEW EECISE 19 1 circle of diameter 10 cm has a chord drawn inside it. The chord is 7 cm long. a) Make a sketch to show this information. b) Calculate the distance from the midpoint of the chord to the centre of the circle. Give your answer correct to 3 significant figures. 2 The diagram shows a circle, centre. T and T are tangents to the circle. ngle T a) Work out the size of angle T, marked. b) Is it possible to draw a circle that passes through the four points,, and T? Give reasons for your answer. 3 The diagram shows a circle, centre. T and T are tangents to the circle. ngle T 32. S y 32 T a) Work out the size of angle S, marked y. Hint: Draw in and. b) Is it possible to draw a circle that passes through the four points, S, and T? Give reasons for your answer.

22 eview eercise In the diagram,, and C are points on the circle, centre. ngle CE 63. FE is a tangent to the circle at point C. E Diagram not accurately drawn 63 C F a) Calculate the size of angle C. Give reasons for your answer. b) Calculate the size of angle C. Give reasons for your answer. [Edecel] 5,, and S are points on the circumference of a circle, centre. is a diameter of the circle. ngle S 56. S 56 Diagram not accurately drawn a) Find the size of angle. Give a reason for your answer. b) Find the size of angle. Give a reason for your answer. c) Find the size of angle. Give a reason for your answer. [Edecel]

23 374 Chapter 19: Circle theorems 6,, C and D are four points on the circumference of a circle. E and DCE are straight lines. ngle C 25. ngle EC 60. a) Find the size of angle DC. b) Find the size of angle D. 25 Diagram not accurately drawn 60 D C E ngle CD 65. en says that D is a diameter of the circle. c) Is en correct? You must eplain your answer. [Edecel] 7 The diagram shows a circle, centre. C is a diameter. ngle C 35. D is the point on C such that angle D is a right angle. Diagram not accurately drawn 35 D C a) Work out the size of angle C. Give reasons for your answer. b) Calculate the size of angle DC. c) Calculate the size of angle. [Edecel]

24 eview eercise ,, C and D are four points on the circumference of a circle. T is the tangent to the circle at. ngle DT 30. ngle DC 132. C Diagram not accurately drawn 132 D 30 T a) Calculate the size of angle C. Eplain your method. b) Calculate the size of angle CD. Eplain your method. c) Eplain why C cannot be a diameter of the circle. [Edecel] 9 oints, and C lie on the circumference of a circle with centre. D is the tangent to the circle at. CD is a straight line. C and intersect at E. Diagram not accurately drawn E C D ngle C 80. ngle CD 38. a) Calculate the size of angle C. b) Calculate the size of angle. c) Give a reason why it is not possible to draw a circle with diameter ED through the point. [Edecel]

25 376 Chapter 19: Circle theorems 10,, C and D are points on the circumference of a circle centre. tangent is drawn from E to touch the circle at C. ngle EC 36. E is a straight line. Diagram not accurately drawn E 36 D a) Calculate the size of angle C. Give reasons for your answer. b) Calculate the size of angle DC. Give reasons for your answer. [Edecel] 11, and are points on a circle. is the centre of the circle. T is the tangent to the circle at. ngle T 56. C Diagram not accurately drawn 56 T a) Find (i) the size of angle and (ii) the size of angle.,, C and D are points on a circle. C is a diameter of the circle. ngle CD 25 and angle CD 132. Diagram not accurately drawn C D b) Calculate (i) the size of angle C and (ii) the size of angle D. [Edecel]

26 Key points In the diagram below, and S are chords of the circle. 4 cm. 6 cm. 3 cm. S cm. 3 cm 4 cm 6 cm cm S Work out the value of. Key points asic circle properties line segment drawn from the centre of a circle to the midpoint of a chord will intersect the chord at right angles. tangent and radius meet at right angles. The two eternal tangents to a circle are equal in length. Tangent M adius Tangent Tangent

27 378 Chapter 19: Circle theorems Circle theorems The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference of the circle. ngles subtended by an arc in the same segment of a circle are equal. The angle subtended in a semicircle is a right angle. 2 The angles subtended in opposite segments add up to 180. y 180 The angle between a tangent and chord is equal to the angle subtended in the opposite segment. y Intersecting chords (internal) Intersecting chords (eternal) C C D D C D C D

28 Internet Challenge Internet Challenge 19 The nine-point circle theorem This diagram shows the nine-point circle. Here are the instructions to make it M C Start with any general triangle, whose vertices are, and C. Construct points 1, 2, 3. Can you see what rule is used to locate them? Construct the point M. Can you see how 1, 2, 3 are used to do this? Construct points 4, 5, 6. Can you see what rule is used to locate them? Construct points 7, 8, 9. Can you see what rule is used to locate them? Then it should be possible to draw a circle that passes through all nine of the points: 1, 2, 3, 4, 5, 6, 7, 8 and 9. Look at the diagram, and see if you can figure out how the various points are constructed. Use the internet to check that your deductions are correct. Try to make a nine-point circle of your own, using compass constructions. You might also try to do this using computer graphics software. Which mathematician is thought to have first made a nine-point circle? Can you find a proof that these nine points all lie on the same circle?

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