math unit32 angles, circles and tangents
TRANSCRIPT
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Unit 32Angles, Circles and Tangents
Presentation 1 Compass BearingsPresentation 2 Angles and Circles: ResultsPresentation 3 Angles and Circles: ExamplesPresentation 4 Angles and Circles: ExamplesPresentation 5 Angles and Circles: More ResultsPresentation 6 Angles and Circles: More ExamplesPresentation 7 Circles and Tangents: ResultsPresentation 8 Circles and Tangents: Examples
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Unit 3232.1 Compass Bearings
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Notes1.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°
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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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Unit 3232.2 Angles and Circles: Results
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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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Unit 3232.3 Angles and Circles:
Examples
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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.?
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ExampleFind the angles marked with letters in the diagram if O is the centre of the circle
SolutionAs 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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Unit 3232.4 Angles and Circles:
Examples
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SolutionIn triangle OAB, OA is a radiusand AB a tangent, so the anglebetween them = 90°HenceIn triangle OAC, OA and OC are both radii of the circle.Hence OAC is an isosceles triangle, and b = c.
ExampleFind the angles a, b and c, if AB is a tangent and O is the centre of the circle.
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Unit 3232.5 Angles and Circles: More
Results
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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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Unit 3232.6 Angles and Circles: More
Examples
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SolutionOpposite angles in a cyclic quadrilateral add up to 180°So
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ExampleFind the angles marked in the diagrams. O is the centre of the circle.
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SolutionConsider arc BD. The angle subtended at O = 2 x aSo
also
ExampleFind the angles marked in the diagrams. O is the centre of the circle.
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Unit 3232.7 Circles and Tangents:
Results
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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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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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Unit 3232.8 Circles and Tangents:
Examples
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Example 1Find the angles x and y in the diagram.
SolutionFrom 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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ExampleFind the unknown lengths in the diagram
Solution
Since AT is a tangent
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ThusAs AC and BD are intersecting chords
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