unit 32 angles, circles and tangents presentation 1compass bearings presentation 2angles and...
TRANSCRIPT
![Page 1: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/1.jpg)
Unit 32Angles, 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
![Page 2: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/2.jpg)
Unit 3232.1 Compass Bearings
![Page 3: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/3.jpg)
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°
![Page 4: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/4.jpg)
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°
?
?
?
?
?
?
![Page 5: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/5.jpg)
Unit 3232.2 Angles and Circles: Results
![Page 6: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/6.jpg)
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.
?
![Page 7: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/7.jpg)
Unit 3232.3 Angles and Circles:
Examples
![Page 8: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/8.jpg)
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.?
![Page 9: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/9.jpg)
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: ? ?
?
?
![Page 10: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/10.jpg)
Unit 3232.4 Angles and Circles:
Examples
![Page 11: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/11.jpg)
Solution
In 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.
Example
Find the angles a, b and c, if AB is a tangent and O is the centre of the circle.
?
? ?
?
?
?
?
?
?
??
??
?
?
?
![Page 12: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/12.jpg)
Unit 3232.5 Angles and Circles: More
Results
![Page 13: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/13.jpg)
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°?
??
??
?
![Page 14: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/14.jpg)
Unit 3232.6 Angles and Circles: More
Examples
![Page 15: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/15.jpg)
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.
?
???
???
![Page 16: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/16.jpg)
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.
?
??
?
?
??
?
![Page 17: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/17.jpg)
Unit 3232.7 Circles and Tangents:
Results
![Page 18: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/18.jpg)
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?
?
![Page 19: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/19.jpg)
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
?
?
![Page 20: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/20.jpg)
Unit 3232.8 Circles and Tangents:
Examples
![Page 21: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/21.jpg)
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
?
?
?
?
?
? ?
![Page 22: Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples](https://reader035.vdocument.in/reader035/viewer/2022062314/56649e505503460f94b477b4/html5/thumbnails/22.jpg)
Example
Find the unknown lengths in the diagram
Solution
Since AT is a tangent
So
ThusAs AC and BD are intersecting chords
?
?
?
?
? ?
? ?
? ? ?
?
?
?
?