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Math unit32 angles, circles and tangents

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Angles, circles and Tangents

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Math unit32 angles, circles and tangents

  1. 1. Unit 32 Angles, Circles and Tangents Presentation 1 Compass Bearings Presentation 2 Angles and Circles: Results Presentation 3 Angles and Circles: Examples Presentation 4 Angles and Circles: Examples Presentation 5 Angles and Circles: More Results Presentation 6 Angles and Circles: More Examples Presentation 7 Circles and Tangents: Results Presentation 8 Circles and Tangents: Examples
  2. 2. Unit 32 32.1 Compass Bearings
  3. 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. 4. What is the bearing of (a) Kingston from Montego Bay 116° (b) Montego Bay from Kingston 296° (c) Port Antonio from Kingston 060° (d) Spanish Town from Kingston 270° (e) Kingston from Negril 102° (f) Ocho Rios from Treasure Beach 045° ? ? ? ? ? ?
  5. 5. Unit 32 32.2 Angles and Circles: Results
  6. 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. 7. Unit 32 32.3 Angles and Circles: Examples
  8. 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. 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. 10. Unit 32 32.4 Angles and Circles: Examples
  11. 11. Solution In triangle OAB, OA is a radius and AB a tangent, so the angle between them = 90° HenceIn 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. 12. Unit 32 32.5 Angles and Circles: More Results
  13. 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. 14. Unit 32 32.6 Angles and Circles: More Examples
  15. 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. 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. 17. Unit 32 32.7 Circles and Tangents: Results
  18. 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. 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. 20. Unit 32 32.8 Circles and Tangents: Examples
  21. 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. 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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