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Workbook Spring Year 9 Workbook Spring Year 9 2 Copyright © Mathematics Mastery 2018-19 Contents Unit 8: Construction .................................................................................................................................................. 4 8.1: Constructing triangles (review of Year 8)........................................................................................... 4 8.2: Constructing perpendicular bisectors and angle bisectors .......................................................... 7 8.3: Mixed questions (constructing triangles, perpendicular bisectors and angle bisectors) .................................................................................................................................................................................... 10 8.4: Mixed problems ........................................................................................................................................... 12 Unit 9: Congruence .................................................................................................................................................. 16 9.1: Congruent figures ....................................................................................................................................... 16 9.2: Congruence conditions for triangles ................................................................................................... 20 9.3: Mixed problems ........................................................................................................................................... 25 Unit 10: Pythagoras’ Theorem ............................................................................................................................ 30 10.1: Pythagoras’ Theorem.............................................................................................................................. 30 10.2: Calculating the missing length of the hypotenuse ....................................................................... 32 10.3: Calculating the lengths of other sides .............................................................................................. 35 10.4: Mixed problems ........................................................................................................................................ 39 Unit 11: Angles in polygons .................................................................................................................................. 44 11.1: Sum of interior angles in polygons .................................................................................................... 44 11.2: Sum of exterior angles of polygons ................................................................................................... 47 11.3: Mixed questions ........................................................................................................................................ 51 Unit 12 Linear equations and inequalities ..................................................................................................... 56 12.1: Writing Linear equations and inequalities ..................................................................................... 56 12.2: Solving equations and inequalities with the unknown on only one side ........................... 58 Concept corner: Properties of equality ...................................................................................................... 62 12.3: Solving equations with the unknown on both sides of the equality ..................................... 63 Unit 13: Graphical solutions ................................................................................................................................ 70 13.1: Simultaneous equations ........................................................................................................................ 70 13.2: Quadratic graphs ...................................................................................................................................... 80 13.3: Further quadratic graphs ...................................................................................................................... 85 13.4: Exponential, reciprocal and piecewise linear graphs ................................................................ 90 Copyright © Mathematics Mastery 2018-19 3 Unit 8: Construction 8.1: Constructing triangles (review of Year 8) Concept corner In order to construct a triangle, you need to have some information: All 3 sides: Side Side Side (SSS) Two side lengths and the angle between them: Side Angle Side (SAS) Two angles and the length of the side between them: Angle Side Angle (ASA) Note: AAS does not give a unique triangle. For example: B B 56° 72° 72° 56° A 7 cm C A 7 cm C Throughout this section, make sure you show all your construction lines. 1. Draw the following triangles accurately and measure the sides and angles not given in the diagram. a) 4 Copyright © Mathematics Mastery 2018-19 b) 2. Draw ∆CED, if ED = 6 cm, EC = 4 cm and ∠퐶퐸퐷 = 40°. Measure the sides and angles not given in the diagram. Copyright © Mathematics Mastery 2018-19 5 3. Draw an isosceles triangle that has two sides of length 7 cm and an angle between them of 40°. a) Measure the length of a base and the corresponding height of the triangle to the nearest millimetre. base = …………………………. cm height = ……………………….. cm b) What is the area of the triangle? 6 Copyright © Mathematics Mastery 2018-19 8.2: Constructing perpendicular bisectors and angle bisectors Concept corner Angle bisector Perpendicular bisector Throughout this section, make sure you show all your construction lines. 1. Construct the perpendicular bisector of the line segment AB. Copyright © Mathematics Mastery 2018-19 7 2. Measure the size of DEF ……………………° Construct the angle bisector of DEF. 3. Construct the line which passes through M and is perpendicular to the line segment PQ. 8 Copyright © Mathematics Mastery 2018-19 4. Construct a right angle at Y using only a pencil, a pair of compasses and a ruler. 5. In the space below, construct an equilateral triangle of side length 6 cm using only a pencil, a pair of compasses and a ruler. Copyright © Mathematics Mastery 2018-19 9 8.3: Mixed questions (constructing triangles, perpendicular bisectors and angle bisectors) Concept corner When constructing geometrical figures, use a sharp pencil to draw points and lines clearly. All construction lines must be clearly shown. Do not erase the construction lines you have drawn. Throughout this section, make sure you show all your construction lines. 1. Construct triangle 푃푄푅 where PQ = 4 cm, PR = 4.5 cm and QR = 4.9 cm. Measure and write down the size of ∠푃푅푄. Construct the perpendicular bisector of PR. 2. Construct triangle XYZ where XY = 4.4 cm, XZ =4.6 cm and YZ =5.2 cm. Measure and write down the size of ∠푌푋푍. Construct the angle bisector of ∠푋푌푍. 10 Copyright © Mathematics Mastery 2018-19 3. In the space below, construct triangle PQR with PQ = 4.8 cm, QR = 6.5 cm and ∠PQR=90°using only a pencil, a pair of compasses and a ruler. Measure and write down the size of ∠PQR giving your answer correct to one decimal place. 4. Construct △XYZ with XY = 9.4 cm, XZ = 8.8 cm and ∠Y= 60° using only a pencil, a pair of compasses and a ruler. Measure and write down the length of YZ. Copyright © Mathematics Mastery 2018-19 11 8.4: Mixed problems Throughout this section, make sure you show all your construction lines. 1. Explain what is meant by, ‘equidistant from two points’. 2. Here is a plan of some land. There will be a fence that is always the same distance from tree A as from tree B, going all the way from one road to the other road, Use compasses and a ruler to accurately show, on the plan, where the fence will be. You must leave in your construction lines. 12 Copyright © Mathematics Mastery 2018-19 3. a) Use a ruler and a pair of compasses to draw a triangle that has these side lengths: 5 cm, 5 cm, 8 cm b) Fin says it is possible to draw a triangle with these side lengths: 5cm, 5cm, 12cm Is she correct? Yes No Explain how you know. …………………………………………………………………………………………………………………………… 4. Construct the angle bisector of angle ABC. Copyright © Mathematics Mastery 2018-19 13 5. In the space below, construct and label the following angles using only a pencil, a pair of compasses and a ruler. a) Construct an angle of 30˚. b) Construct an angle of 45˚. 14 Copyright © Mathematics Mastery 2018-19 c) Construct an angle of 330˚. d) Construct an angle of 345˚. Copyright © Mathematics Mastery 2018-19 15 Unit 9: Congruence 9.1: Congruent figures 1. In the diagram below, which triangles can be moved to overlap ∆ABC completely? d b c a Concept corner Congruent shapes are identical but can have different orientations. For example, these five shapes are all congruent Properties of congruence Quadrilateral 퐴퐵퐶퐷 and quadrilateral 퐴’퐵’퐶’퐷’ are congruent. This can be written as 퐴퐵퐶퐷 ≡ 퐴’퐵’퐶’퐷’, where the symbol ‘≡’ means ‘congruent’. The angle marked in the diagram below can be written as 퐴퐵̂퐶 or ∠퐴퐵퐶. Complete the statements below: A and A’ are corresponding v………………… AB and A’B’ are corresponding s…………….. ∠퐴퐵퐶 and ∠퐴′퐵′퐶′ are corresponding a…………….. 16 Copyright © Mathematics Mastery 2018-19 2. The two pentagons below are congruent. a) Name two pairs of corresponding vertices ……………………………………………………….…………………………………………………………………….. b) Name two pairs of corresponding sides ……………………………………………………….…………………………………………………………………….. c) Name two pairs of corresponding angles ……………………………………………………….…………………………………………………………………….
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