Download presentation

Presentation is loading. Please wait.

Published byAlexandra Gilmore Modified over 8 years ago

1
Unit 32 Angles, Circles and Tangents Presentation 1Compass Bearings Presentation 2Angles and Circles: Results Presentation 3Angles and Circles: Examples Presentation 4Angles and Circles: Examples Presentation 5Angles and Circles: More Results Presentation 6Angles and Circles: More Examples Presentation 7Circles and Tangents: Results Presentation 8Circles 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 Bay116° (b) Montego Bay from Kingston296° (c) Port Antonio from Kingston060° (d) Spanish Town from Kingston270° (e) Kingston from Negril102° (f) Ocho Rios from Treasure Beach045° ? ? ? ? ? ?

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

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° Hence In 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
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° ? ? ? ? ? ?

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. ? ? ? ? ? ? ?

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
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 ? ?

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
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 ? ? ? ? ? ? ?

22
Example Find the unknown lengths in the diagram Solution Since AT is a tangent So Thus As AC and BD are intersecting chords ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Similar presentations

© 2024 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google