1. Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples Presentation 4Angles and Circles: Examples Presentation 5Angles and Circles: More Results Presentation 6Angles and Circles: More Examples Presentation 7Circles and Tangents: Results Presentation 8Circles and Tangents: Examples

2. Unit 32 32.1 Compass Bearings

3. Notes 1.Bearings are written as three-figure numbers. 2.They are measured clockwise from North. The bearing of A from O is 040° The bearing of A from O is 210°

4. What is the bearing of (a) Kingston from Montego Bay116° (b) Montego Bay from Kingston296° (c) Port Antonio from Kingston060° (d) Spanish Town from Kingston270° (e) Kingston from Negril102° (f) Ocho Rios from Treasure Beach045° ? ? ? ? ? ?

5. Unit 32 32.2 Angles and Circles: Results

6. A chord is a line joining any two points on the circle. The perpendicular bisector is a second line that cuts the first line in half and is at right angles to it. The perpendicular bisector of a chord will always pass through the centre of a circle. ? ? When the ends of a chord are joined to centre of a circle, an isosceles triangle is formed, so the two base angles marked are equal. ?

7. Unit 32 32.3 Angles and Circles: Examples

8. When a triangle is drawn in a semi- circle as shown the angle on the perimeter is always a right angle. ? A tangent is a line that just touches a circle. A tangent is always perpendicular to the radius. ?

9. Example Find the angles marked with letters in the diagram if O is the centre of the circle Solution As both the triangles are in a semi- circles, angles a and b must each be 90° ? Top Triangle: ? ? ? ? Bottom Triangle: ? ? ? ?

10. Unit 32 32.4 Angles and Circles: Examples

11. Solution In triangle OAB, OA is a radius and AB a tangent, so the angle between them = 90° Hence In triangle OAC, OA and OC are both radii of the circle. Hence OAC is an isosceles triangle, and b = c. Example Find the angles a, b and c, if AB is a tangent and O is the centre of the circle. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

12. Unit 32 32.5 Angles and Circles: More Results

13. The angle subtended by an arc, PQ, at the centre is twice the angle subtended on the perimeter. Angles subtended at the circumference by a chord (on the same side of the chord) are equal: that is in the diagram a = b. In cyclic quadrilaterals (quadrilaterals where all; 4 vertices lie on a circle), opposite angles sum to 180°; that is a + c = 180° and b + d = 180° ? ? ? ? ? ?

14. Unit 32 32.6 Angles and Circles: More Examples

15. Solution Opposite angles in a cyclic quadrilateral add up to 180° So and Example Find the angles marked in the diagrams. O is the centre of the circle. ? ? ? ? ? ? ?

16. Solution Consider arc BD. The angle subtended at O = 2 x a So also Example Find the angles marked in the diagrams. O is the centre of the circle. ? ? ? ? ? ? ? ?

17. Unit 32 32.7 Circles and Tangents: Results

18. If two tangents are drawn from a point T to a circle with a centre O, and P and R are the points of contact of the tangents with the circle, then, using symmetry, (a) PT = RT (b) Triangles TPO and TRO are congruent ? ?

19. For any two intersecting chords, as shown, The angle between a tangent and a chord equals an angle on the circumference subtended by the same chord. e.g. a = b in the diagram. This is known by alternate segment theorem ? ?

20. Unit 32 32.8 Circles and Tangents: Examples

21. Example 1 Find the angles x and y in the diagram. Solution From the alternate angle segment theorem, x = 62° Since TA and TB are equal in length ∆TAB is isosceles and angle ABT = 62° Hence ? ? ? ? ? ? ?

22. Example Find the unknown lengths in the diagram Solution Since AT is a tangent So Thus As AC and BD are intersecting chords ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?