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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. 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 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°
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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.
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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.
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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: ?
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Bottom Triangle: ? ?
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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°
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.
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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°?
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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.
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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.
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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?
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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
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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
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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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