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Unit 32
Angles, Circles and Tangents
Presentation 1 Compass Bearings
Presentation 2 Angles and Circles: Results
Presentati...
Unit 32
32.1 Compass Bearings
Notes
1.Bearings are written as three-figure numbers.
2.They are measured clockwise from North.
The bearing of
A from O is...
What is the bearing of
(a) Kingston from Montego Bay 116°
(b) Montego Bay from Kingston 296°
(c) Port Antonio from Kingsto...
Unit 32
32.2 Angles and Circles: Results
A chord is a line joining any two points
on the circle.
The perpendicular bisector is a second
line that cuts the first li...
Unit 32
32.3 Angles and Circles:
Examples
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 li...
Example
Find the angles marked with letters in the
diagram if O is the centre of the circle
Solution
As both the triangles...
Unit 32
32.4 Angles and Circles:
Examples
Solution
In triangle OAB, OA is a radius
and AB a tangent, so the angle
between them = 90°
HenceIn triangle OAC, OA and OC...
Unit 32
32.5 Angles and Circles: More
Results
The angle subtended by an arc, PQ, at the
centre is twice the angle subtended on the
perimeter.
Angles subtended at the ci...
Unit 32
32.6 Angles and Circles: More
Examples
Solution
Opposite angles in a cyclic quadrilateral add up to 180°
So
and
Example
Find the angles marked in the
diagrams. O...
Solution
Consider arc BD. The angle subtended at O = 2 x a
So
also
Example
Find the angles marked in the
diagrams. O is th...
Unit 32
32.7 Circles and Tangents:
Results
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 tangent...
For any two intersecting
chords, as shown,
The angle between a tangent and
a chord equals an angle on the
circumference su...
Unit 32
32.8 Circles and Tangents:
Examples
Example 1
Find the angles x and y in the diagram.
Solution
From the alternate angle segment theorem, x = 62°
Since TA and ...
Example
Find the unknown
lengths in the diagram
Solution
Since AT is a tangent
So
Thus
As AC and BD are intersecting chord...
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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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