Circle Properties Boardworks Ltd 2002 Circle Properties In
Circle Properties © Boardworks Ltd 2002
Circle Properties In this unit you will learn about: ü The names of the parts of a circle ü Inscribed regular polygons ü The geometry of tangents and chords © Boardworks Ltd 2002
Circle Definitions X • A circle has the special property that all the points on the outside are the same distance from the centre. • The radius is the distance from the centre to the edge. X X s u i d X ra X X X X © Boardworks Ltd 2002
Circle Definitions • A line joining two points on the circle is called a chord. • If the line also passes through the centre, it is called the diameter. chord diam X eter © Boardworks Ltd 2002
Circle Definitions • The distance round the circle is the circumference. • A part of the circumference is an arc. • A ‘pie slice’ of the circle is called a sector arc © Boardworks Ltd 2002
Chords and Tangents Imagine a line moving towards a circle. At first, the line just touches the circle. This is called a tangent. Then, the line crosses the circle and now has two points where it intersects. It is now a chord. X © Boardworks Ltd 2002
Chords and Tangents The chord splits the circle into two segments – a major segment and a minor segment. Minor segment X Major segment © Boardworks Ltd 2002
Circle parts F © Boardworks Ltd 2002
Useful formulae ü Area of circle = r 2 ü Circumference = d ü Area of sector = q x r 2 = q x d 360 ü Length of arc 360 © Boardworks Ltd 2002
Circle Properties © Boardworks Ltd 2001
Circle Property Proofs and Definitions 2 Isosceles triangles + perp. bisectors Angle at center and at circumference 3 Angle in a semi circle 4 Angles in the same segment Opposite angles in a cyclic quadrilateral Right angle between a tangent and radius Tangential lines the same length Alternate segment theorem 1 5 6 7 8 Proof Definition Proof Definition © Boardworks Ltd 2002
Measure the angles What do you notice? © Boardworks Ltd 2002
Triangles in circles © Boardworks Ltd 2002
Circle Property 1 Triangles formed using two radii will form an isosceles triangle. The perpendicular bisector of a chord passes through the centre of the circle. Remember to spot isosceles triangles and perpendicular bisectors in circle diagrams. © Boardworks Ltd 2002
2 Circle Property Example x = 200 Property 1 x 1400 x © Boardworks Ltd 2002
3 Circle Property Example x = 540 Property 1 x x 630 © Boardworks Ltd 2002
Circle Property 2 The angle subtended by x an arc at the centre of a circle, is twice the angle subtended at the x 2 x circumference. The angle at the centre is half the angle on the circumference. © Boardworks Ltd 2002
Circle Property 2 a a 2 b x 2 a b b Using Property 1 we can label the angles at the centre. Full Version of Property © Boardworks Ltd 2002
4 Circle Property Example x = 420 Property 2 x x 840 © Boardworks Ltd 2002
5 Circle Property Example x = 700 Property 350 2 x x y = 550 Property y 1 © Boardworks Ltd 2002
7 Circle Property Example x = 2600 Property 2 x x 1300 © Boardworks Ltd 2002
9 Circle Property Example x = 860 Property 2 430 x x © Boardworks Ltd 2002
Circle Property 3 Any angle subtended on the circumference of a semi- circle will be a right angle. 1800 x The angle in a semi-circle is a right angle. © Boardworks Ltd 2002
Circle Property 3 If PQ is a diameter then the angle at the centre is 1800. Using Property 1. . . The angle on the circumference is half of 1800 = 900. a P 1800 x Q The angle in a semi-circle is a right angle. © Boardworks Ltd 2002
14 Circle Property Example x = 420 Property 3 x 480 y = 750 x 150 y Property 3 © Boardworks Ltd 2002
Circle Property 4 a a b a and b are both “subtended” by the same chord Bitesize © Boardworks Ltd 2002
Circle Property 4 a a b x x 2 a 2 a If a is half the angle at the centre then so is b. So b = a. Full Version of Property © Boardworks Ltd 2002
16 Circle Property Example x = 350 Property 4 350 x x y y = 700 Property 2 © Boardworks Ltd 2002
17 Circle Property Example x = 680 Property 4 y x y = 1360 Property x 680 2 © Boardworks Ltd 2002
18 Circle Property Example x = 430 Property 430 4 x x y = 470 Property 860 y 4+1 © Boardworks Ltd 2002
19 Circle Property Example x = 120 y Property 340 4 x y = 340 x 120 Property 4 © Boardworks Ltd 2002
20 Circle Property Example x = 700 Property 4 y 640 x y = 640 700 x Property 4 © Boardworks Ltd 2002
Circle Property 5 Opposite angles in a cyclic quadrilateral add up to 1800. a (A cyclic quadrilateral is a x 4 sided shape with all four points on the circumference of a circle. ) b Opposite angles in a cyclic quadrilateral add to 1800. © Boardworks Ltd 2002
25 Circle Property Example x = 1600 Property 2 800 y = 1000 x x y Property 5 © Boardworks Ltd 2002
26 Circle Property Example x = 730 Property 2 x x y = 1070 1460 y Property 5 © Boardworks Ltd 2002
27 Circle Property Example x = 830 Property 2 x y = 970 x 1660 y Property 5 © Boardworks Ltd 2002
28 Circle Property Example x = 950 Property 2 x y = 850 x 1900 y Property 5 © Boardworks Ltd 2002
29 Circle Property Example x = 500 Property 5 x x y y = 1000 Property 2 1300 © Boardworks Ltd 2002
30 Circle Property Example x = 700 Property 5 x y = 2200 Property 2 1100 © Boardworks Ltd 2002
Exercise 31. 1 • Page 333 © Boardworks Ltd 2001
Circle Property 6 The tangent to a circle Q OPQ = 900 P is perpendicular to the radius drawn at the point of contact. x O A tangent to a circle is at right angles to its radius. © Boardworks Ltd 2002
33 Circle Property Example x = 400 Property x 6 500 y = 250 Property x y 2 © Boardworks Ltd 2002
Circle Property 7 Q Two tangents drawn to QP = QR P a circle from the same point are equal in length. QP = QR R x O The tangents drawn from a point to a circle are equal in length. © Boardworks Ltd 2002
36 Circle Property Example x = 1240 y Property 2 x x y = 560 Property 620 6+7 © Boardworks Ltd 2002
Circle Property 8 The angle between a tangent and a chord drawn at a point of contact is x y x z y equal to any angle in the alternate segment. The angle between a tangent and a chord is equal to the angle in the alternate segment. © Boardworks Ltd 2002
Circle Property 8 The angle between a tangent and a chord drawn at a point of contact is x y x z y equal to any angle in the alternate segment. The angle between a tangent and a chord is equal to the angle in the alternate segment. © Boardworks Ltd 2002
Circle Property 8 This property is called the alternate segment theorem and states that the b x a angle a is equal to the angle b The angle between a tangent and a chord is equal to the angle in the alternate segment. Full Version of Property © Boardworks Ltd 2002
41 Circle Property Example x = 700 Property 8 x z y = 600 Property Triangle y 500 700 z = 1200 Property 8 © Boardworks Ltd 2002
42 Circle Property Example x = 1200 Property 1 y = 600 Property 2 z = 600 y x x 300 z Property 8 © Boardworks Ltd 2002
44 Circle Property Example x = 500 Property 8 x y = 1000 x Property y 2 y = 400 z 500 Property 1 © Boardworks Ltd 2002
Calculating the size of unknown angles Calculate the size of the labelled angles in the following diagram: a = 29° (angles at the base of an isosceles triangle) c b = 180° – 2 × 29° = 122° (angles in a triangle) O b d 29° 41° a c = 122° ÷ 2 = 61° (angle at the centre is twice angle on the circumference) d = 180° – (29° + 41° + 61°) = 20° (angles in a triangle) © Boardworks Ltd 2002
The tangent and the radius © Boardworks Ltd 2002
Two tangents from a point © Boardworks Ltd 2002
The alternate segment theorem © Boardworks Ltd 2002
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