JUNIOR CERTIFICATE HIGHER LEVEL

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1 JUNIR CERTIFICTE HIGHER LEVEL ctive Maths 2 Strands 1 5 Supplementary Material for 2016 exam and onwards m = y 2 - y 1 x 2 - x 1 πr 2 Michael Keating, Derek Mulvany and James Loughlin Special dvisors: liver Murphy, Colin Townsend and Jim McElroy

2 Contents Chapter 2 Number Systems 2.2 Commutative, ssociative and Distributive Properties... 1 Commutative Property...1 ssociative Property...1 Distributive Property...1 Chapter 18 Geometry Revision of Geometry Rotations...3 nswers... 5 dditional copies of this booklet are available to download from folens.ie Editor: Priscilla Connor Designer: Liz White Layout: Compuscript Illustrations: Compuscript ISBN: Michael Keating, Derek Mulvany, Colin Townsend and Jim McElroy, 2014 Folens Publishers, Hibernian Industrial Estate, Greenhills Road, Tallaght, Dublin 24, Ireland ll rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher. The publisher reserves the right to change, without notice, at any time, the specification of this product, whether by change of materials, colours, bindings, format, text revision or any other characteristic.

3 2 2.2 CMMUTTIVE, SSCITIVE ND DIST RIBUTIVE PRPERTIES The commutative, associative and distributive properties also apply to the real numbers. Commutative Property n operation is commutative if a change in the order of the numbers does not change the result. In ctive Maths 1, we saw that multiplication and addition for rational numbers is commutative. Multiplication and addition of real numbers is also commutative. For example: = and (1 + 5 )(3 + 2 ) = (3 + 2 )(1 + 5 ) a + b = b + a, where a, b R FRMUL and a b = b a, where a, b R Number SySTEMS ssociative Property n operation is associative if a change in grouping does not change the result. ddition and multiplication are associative operations on the real numbers. For example: (1 + 2 ) = 1 + ( ) and (1 2 ) 1 2 = 1 ( ) FRMUL (a + b) + c = a + (b + c), where a, b, c R and (a b) c = a (b c), where a, b, c R Distributive Property The distributive property for the real numbers says that multiplication distributes over addition and subtraction. For example: 1 ( ) = and 2 ( ) = FRMUL a(b + c) = ab + ac, where a, b, c R 1

4 2 Number SySTEMS Worked Example 2.9 Use the commutative, associative or distributive properties to show that each of the following expressions are rational: (i) 3 ( ) (ii) ( 5 )( ) (iii) ( ) 3 7 Solution (i) 3 ( ) = (Multiplication distributes over addition) = = = 9 is rational (ii) ( 5 )( ) = ( 5 ) 3 + ( 5 ) 5 (Distributive property) = 3 ( 5 ) + 5 ( 5 ) (Commutative property) = = 2 is rational (iii) ( ) 3 7 = ( ) 3 7 (Commutative property) = 6 + ( ) (ssociative property) = = 6 is rational Exercise Use the commutative, associative or distributive properties to show that each of the following expressions are rational. (i) 2 ( ) (ii) ( 11 2 )( ) (iii) ( )

5 18 Rotations nother type of transformation is a rotation. The amount the shape rotates is called the angle of rotation. This is given either as an angle or as a fraction of a complete turn, for example, 270 or 3_ 4 turn. The direction of rotation is given as clockwise (negative) or anti-clockwise (positive). The fixed point about which the object is rotated is called the point (centre) of rotation. Therefore, when describing the rotation of an object, we should include, if possible: (i) The centre of rotation (ii) The angle of rotation (iii) The direction of rotation (positive or negative) Point of rotation ngle of rotation Image bject Image bject bject Geometry I P 60 Image Every point on this object has been rotated through an angle of 60 about the point P. Positive (anti-clockwise) rotation of 90 about the point. This is denoted as R 90. Negative (clockwise) rotation of 90 about the point. This is denoted as R 90. Worked Example y Z Y X x Describe the rotation that takes triangle XYZ to: (i) Triangle (ii) Triangle B (iii) Triangle C Solution (i) Rotation of 90 about the origin (ii) Rotation of 180 about the origin (iii) Rotation of 90 about the origin 3 2 B C

6 18 Exercise 18.1 Extra Questions 9. In each case, identify the rotation that maps: y D C B x (i) onto B (ii) C onto B (iii) onto D (iv) onto C Make sure in your answer to include: The centre of rotation The angle of rotation Whether the rotation is positive or negative Geometry I 12. In each question below, three images labelled, B and C are the images of the object under a transformation. The transformations could be a translation, an axial symmetry, a central symmetry or a rotation. For each image, state which transformation is used, and in the case of a rotation, state the angle and direction. (i) bject B C (ii) bject B C (iii) bject B C 4

7 13. The diagram shows a rectangle on the co-ordinate plane (see diagram on page 347). 18 (a) Copy this diagram and draw the image of rectangle under the following transformations: (i) xial symmetry in the x-axis (ii) Central symmetry in the point (0,0) (iii) xial symmetry in the y-axis (iv) positive rotation of 90 about the origin (b) Hence, write down the co-ordinates of the images of the vertices of under each of the transformations. Transformation Co-ordinates of vertices xial symmetry in the x-axis (, ), (, ), (, ), (, ) Central symmetry in the point (0,0) (, ), (, ), (, ), (, ) xial symmetry in the y-axis (, ), (, ), (, ), (, ) Positive rotation of 90 about the origin (, ), (, ), (, ), (, ) 13. Consider the following shapes. Shape D Shape B Shape C C B Shape D Shape E Geometry I How many times can you rotate each shape about the point so that the image fits exactly over the object? nswers Exercise (i) onto B; positive 90 rotation about the origin (ii) C onto B; negative 90 rotation about the origin (iii) onto D; negative 90 rotation about the origin (iv) onto C; positive/negative 180 rotation about the origin 12. (i) : xial symmetry, B: Translation, C: Central symmetry (ii) : xial symmetry, B: Central symmetry, C: Positive rotation of 90 (iii) : Positive/negative rotation of 180, B: Central symmetry, C: xial symmetry 13. (b) (i) ( 1, 1), ( 4, 1), ( 4, 3), ( 1, 3) (ii) (1, 1), (4, 1), (4, 3), (4, 3), (1, 3) (iii) (1,1), (4,1), (4,3), (1,3) (iv) ( 1, 1), ( 3, 1), ( 3, 4), ( 1, 4) 13. Shape : 4, Shape B: 2, Shape C: 3, Shape D: 5, Shape E: 1 5

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