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##### Circle Properties

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**Circle Properties**• In this unit you will learn about: • The names of the parts of a circle • Inscribed regular polygons • The geometry of tangents and chords**Circle Definitions**radius • The radius is the distance from the centre to the edge. X X X • A circle has the special property that all the points on the outside are the same distance from the centre. X X X X X X X X**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 X diameter**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 sector arc**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**Chords and Tangents**The chord splits the circle into two segments – a major segment and a minor segment. Minor segment Major segment X**Useful formulae**• Area of circle = p r2 • Circumference = d • Area of sector = q x p r2 360 • Length of arc = q x p d 360**Circle Property Proofs and Definitions**Isosceles triangles + perp. bisectors 1 Proof Definition Angle at center and at circumference 2 Proof Definition Angle in a semi circle 3 Proof Definition Angles in the same segment 4 Proof Definition Opposite angles in a cyclic quadrilateral 5 Proof Definition Right angle between a tangent and radius 6 Proof Definition Tangential lines the same length 7 Proof Definition Alternate segment theorem 8 Proof Definition**Measure the angles**What do you notice?**Circle Property 1**Triangles formed using two radii will form an isosceles triangle. The perpendicular bisector of a chord passes through the the centre of the circle. Remember to spot isosceles triangles and perpendicular bisectors in circle diagrams.**x**Circle Property Example 2 x = 200 Property 1 1400 x**x**Circle Property Example 3 x = 540 Property 1 x 630**Circle Property 2**The angle subtended by an arc at the centre of a circle, is twice the angle subtended at the circumference. x x 2x The angle at the centre is half the angle on the circumference.**Circle Property 2**a a 2b x 2a b b Using Property 1 we can label the angles at the centre. Full Version of Property**x**Circle Property Example 4 x = 420 Property 2 x 840**x**Circle Property Example 5 x = 700 Property 2 350 x y = 550 y Property 1**x**Circle Property Example 7 x = 2600 Property 2 x 1300**x**Circle Property Example 9 x = 860 Property 2 430 x**Circle Property 3**Any angle subtended on the circumference of a semi- circle will be a right angle angle. x 1800 The angle in a semi-circle is a right angle.**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 x 1800 Q The angle in a semi-circle is a right angle.**x**Circle Property Example 14 x = 420 Property 3 x 150 480 y y = 750 Property 3**Circle Property 4**a a b a and b are both “subtended” by the same chord Bitesize**Circle Property 4**a a b x x 2a 2a If a is half the angle at the centre then so is b. So b = a. Full Version of Property**x**Circle Property Example 16 x = 350 x Property 4 350 y y = 700 Property 2**x**Circle Property Example 17 x = 680 Property 4 y y = 1360 680 x Property 2**x**Circle Property Example 18 x = 430 x Property 4 430 860 y = 470 y Property 4 + 1**x**Circle Property Example 19 x = 120 y Property 4 340 x 120 y = 340 Property 4**x**Circle Property Example 20 x = 700 Property 4 y 640 700 y = 640 x Property 4**Circle Property 5**Opposite angles in a cyclic quadrilateral add up to 1800. (A cyclic quadrilateral is a 4 sided shape with all four points on the circumference of a circle.) a x b Opposite angles in a cyclic quadrilateral add to 1800.**x**Circle Property Example 25 x = 1600 Property 2 x 800 y y = 1000 Property 5**x**Circle Property Example 26 x = 730 Property 2 x 1460 y y = 1070 Property 5**x**Circle Property Example 27 x = 830 Property 2 1660 x y y = 970 Property 5**x**Circle Property Example 28 x = 950 Property 2 1900 x y y = 850 Property 5**x**Circle Property Example 29 x = 500 x Property 5 y y = 1000 1300 Property 2**x**Circle Property Example 30 x = 700 x Property 5 y y = 2200 1100 Property 2**Exercise 31.1**Page 333**Circle Property 6**Q The tangent to a circle is perpendicular to the radius drawn at the point of contact. OPQ = 900 P x O A tangent to a circle is at right angles to its radius.**Circle Property Example**33 x = 400 x Property 6 500 x y = 250 y Property 2**Circle Property 7**Q Two tangents drawn to a circle from the same point are equal in length. QP = QR QP = QR P x O R The tangents drawn from a point to a circle are equal in length.**x**Circle Property Example 36 x = 1240 y Property 2 x y = 560 620 Property 6 + 7**Circle Property 8**The angle between a tangent and a chord drawn at a point of contact is equal to any angle in the alternate segment. y x x z y The angle between a tangent and a chord is equal to the angle in the alternate segment.**Circle Property 8**The angle between a tangent and a chord drawn at a point of contact is equal to any angle in the alternate segment. y x x z y The angle between a tangent and a chord is equal to the angle in the alternate segment.**Circle Property 8**This property is called the alternate segment theorem and states that the angle a is equal to the angle b b x a The angle between a tangent and a chord is equal to the angle in the alternate segment. Full Version of Property**Circle Property Example**41 x = 700 Property 8 x z y = 600 y 500 Property Triangle 700 z = 1200 Property 8**Circle Property Example**42 x = 1200 Property 1 y x y = 600 x 300 Property 2 z z = 600 Property 8**Circle Property Example**44 x = 500 Property 8 x x y = 1000 y Property 2 z 500 y = 400 Property 1