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Units_007-090 FV 2:23 PM Page 73 Angles Circle geometry 1 Key facts • The parts of a circle are shown in these diagrams. (a) angle at circumference on minor arc (b) segment chord eter diam umference circ 28 3 11/20/06 angle at centre radius tor sec tangent arc point of contact angle on major arc • There are four basic circle geometry facts to which others are linked. Fact 1 A tangent to a circle is perpendicular to the radius at the point of contact. (This can be proved by contradiction.) S C P T Deduction 1 The 2 tangents from a point to a x circle are equal. Triangles congruent (SSS). Fact 2 An angle at the centre of a circle is equal 2y 2x to twice any angle at the circumference which stands on the same arc. y Deduction 2 (a) All angles at the circumference which stand on the same arc are equal. (b) The angle in a semicircle is a right-angle. (c) Opposite angles of a cyclic quadrilateral are supplementary. (d) The exterior angle of a cyclic quadrilateral equals the interior opposite angle. (a) (b) (c) (d) a b d b c a d c a + c = 180° b + d = 180° e e=b continued 73 Units_007-090 FV 11/20/06 2:23 PM Page 74 Fact 3 The angle between a chord and tangent equals any angle in the alternate segment. x x Fact 4 A line from the centre of a circle perpendicular to the chord bisects the chord Worked example Example 1 Example 2 Given that TQ = TR calculate the sizes of the In this diagram C is the centre of the circle, PT is angles with letters. a tangent to the circle and M is the midpoint of Give reasons. chord PA. Name all the right-angles in the diagram. Give reasons. R Q c P B a b S d C 70° T A T M P Solution 1 a = 70° (Angle in the alternate segment.) Solution 2 b = 70° (Base angles of isosceles angle CPT = 90° triangle.) c = 110° d = 70° (Tangent is perpendicular to the radius.) (Opposite angles of cyclic angle PAB = 90° quadrilateral.) angle CMA = 90° (Angles in the same segment.) and angle CMP = 90° (CM bisects and so is (Angle in a semicircle.) perpendicular to chord AP.) 74 Units_007-090 FV 11/20/06 2:23 PM Definitions • Tangent a straight line which touches a circle at a point of Page 75 ! Examiner’s tips • • Do not assume that lines which cross inside a circle do so at the centre unless this is made clear in the question. • Remembering the angle in a semicircle is important but be careful not to assume angles at the circumference are 90° unless it is clear they stand contact on the circumference • Cyclic quadrilateral a quadrilateral with on a diameter. • Turning diagrams upside down may help in looking for related angles. • Add the sizes of angles to the diagram as you find them. all its vertices on the circumference Can you answer these questions? of a circle. 1. Find the angles marked with letters in these diagrams. Give reasons. (a) Did you know? • The word ‘tangent’ comes from the Latin word ‘tangere’, meaning ‘to touch’, which is also the root of ‘tangible’, meaning ‘can be touched’. (b) 70° 83° (c) (d) a b 78° C d 62° s x r f e C is the centre 2. A point P is 10 cm from a point C which is the centre of a circle radius 4 cm. Calculate the length of the tangent to the y x circle from P. (You will need to sketch a Look on the CD for more exam practice questions 58° diagram.) a 3. Prove fact 2 by using the isosceles b triangles in this diagram to find connections between angles a and x and between y and b. 4. c a d O is the centre of the circle and PT is a tangent O T b e P (a) Write down the value of a. Give a reason. (b) Find e, c and d in terms of b. Give reasons. 75