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Master Angles and Circles: Comprehensive Guide

Learn about bearings, angles, and circles with examples and results, including tangents and isosceles triangles within a circle. Understand the alternate segment theorem and intersecting chords in circle geometry.

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Master Angles and Circles: Comprehensive Guide

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  1. Unit 32Angles, Circles and Tangents

  2. Unit 32 32.1 Compass Bearings

  3. Notes • Bearings are written as three-figure numbers. • 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 • Kingston from Montego Bay 116° • Montego Bay from Kingston 296° • Port Antonio from Kingston 060° • Spanish Town from Kingston 270° • Kingston from Negril 102° • Ocho Rios from Treasure Beach 045° ? ? ? ? ? ?

  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. Example Find the angles a, b and c, if AB is a tangent and O is the centre of the circle. 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. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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

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

  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, • PT = RT • Triangles TPO and TRO are congruent ? ?

  19. 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 ? For any two intersecting chords, as shown, ?

  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 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

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