Introduction...iv. Glossary of Statistical Terms Calculator Instructions...231

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1 CONTENTS Introduction...iv. Coordinate Geometr of the Line.... Geometr Theorems.... Constructions.... Transformation Geometr...6. Trigonometr I Trigonometr II: Real-World Applications Perimeter Area Nets and Volume Fundamental Principles of Counting Probabilit Statistics I: Statistical Investigations...8. Statistics II: Central Tendenc and Spread of Data...9. Statistics III: Representing Data...06 Glossar of Statistical Terms...9 Calculator Instructions... Please note: The philosoph of Project Maths is that topics can overlap so ou ma encounter Paper material on Paper and vice versa. The Eam questions marked b the smbol in this book are selected from the following:. SEC Eam papers. Sample eam papers. Original and sourced eam-tpe questions

2 /8/ 8:8 PM Page Coordinate Geometr of the Line To know where to find the coordinate geometr formulae in the booklet of formulae and tables To learn how to appl these formulae to procedural and in-contet eamination questions To gain the abilit with practise to recall and select the appropriate technique required b the eam questions Coordinating the plane and plotting points Coordinates are used to describe the position of a point on a plane (flat surface). Two lines are drawn at right angles to each other. The horizontal line is called the -ais. The vertical line is called the -ais. The two aes meet at a point called the origin. B( ) 0 The plane is called the Cartesian (kar-tee-zi-an) plane. Ever point on the plane has two coordinates an -coordinate and a -coordinate. A( ) The coordinates are enclosed in brackets. The -coordinate is alwas written first then a comma followed b the -coordinate. On the diagram the coordinates of the point A are ( ). This is usuall written as A( ). In a couple ( ) the order is important. The first number is alwas across left or right and the second number is alwas up or down. The graph above shows the point A( ) is different to the point B( ).

3 /8/ :8 PM Page LESS STRESS MORE SUCCESS Eample Write down the coordinates of the points P Q R S and T S Q P 6 T 6 R P ( 0) Q ( ) R ( ) S ( ) T ( ) An archaeologist has discovered various items at a site. The site is laid out in a grid and the position of each item is shown on the grid. The items found are a brooch (B) a plate (P) a ring (R) a statue (S) and a tile (T). (a) Write down the co-ordinates of the position of each item. B( 7 ) P( ) R( ) S( ) T( ) (b) Each square of the grid represents m. Find the total area of the grid. (c) Which of the items is nearest to the tile (T)? (d) Find the distance between the brooch (B) and the statue (S) B T R S P (a) P (7 ) R (6 ) S ( ) T (9 ) (b) Count the grids: 0 up b 0 across m (c) B observation the ring (R) is nearest to the tile (T ). (d) B( 7) to S( ) 6 m Count each bo as one unit.

4 /8/ :8 PM Page COORDINATE GEOMETRY OF THE LINE Midpoint of a line segment If ( ) and ( ) are two points their midpoint is given b the formula: Midpoint a b ( ) Midpoint + + ( ) (See booklet of formulae and tables page 8) When using coordinate geometr formulae alwas allocate one point to be ( ) and the other to be ( ) before ou use the formula. Eample Noah is positioned at (8 ) and a bus stop is positiond at ( 0 ). There is a traffic light eactl half wa between Noah and the bus stop. Find the coordinates of the traffic light. b Let ( ) (8 ) and ( ) ( 0 ) Coordinates of the traffic light a b a b ( 8) Midpoint (halfwa) formula a P Q and R are the midpoints of the sides of the triangle ABC. (i) Find the coordinates of P Q and R. (ii) The number of parallelograms in the diagram is (a) 0 (b) (c) (d) Tick the correct answer.

5 /8/ :8 PM Page LESS STRESS MORE SUCCESS A ( ) Q R C (9 ) P B ( ) (i) Use the midpoint formula a Midpoint of [AB] b three times. Midpoint of [AC] Midpoint of [BC] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (9 ) ( ) (9 ) b 0 R a b R (0 ) Q a 9 b Q a b Q (7 ) P a R a 9 b P a b P ( ) (ii) (d) The shaded triangle in the diagram forms half of three different parallelograms. A ( ) Q R C (9 ) P B ( )

6 /8/ :8 PM Page COORDINATE GEOMETRY OF THE LINE Translations In mathematics movement in a straight line is called a translation. Under a translation ever point is moved the same distance in the same direction. A translation is one of several tpes of transformations on our course. See Chapter for more on transformations. Eample Describe the translation that maps the points: (i) G to H (ii) R to D G H D R (i) G : H is described b units to the right and units down. b. This translation can be written as a (ii) R : D is described b 7 units to the left and units up. 7 b. This translation can be written as a Eample A ( ) and B ( ) are two points. : Find the image of the point ( ) under the translation AB : Under the translation AB: ( ) : ( ) Rule: Add to subtract from this can be written as a b.

7 6 /8/ :8 PM Page 6 LESS STRESS MORE SUCCESS Method Mathematical Method (Appl the rule directl) ( ) : ( ) ( 0) Method Graphical Method Plot the point ( ) and split the move into two parts: We sa the image of ( ) is ( 0). ( ) ( 0) Horizontal move: units to the right (add to ) Vertical move: units down (subtract from ) The image of ( ) is ( 0). In some questions we will be given the midpoint and one end point of a line segment and be asked to find the other end point. To find the other end point use the following method:. Draw a rough diagram.. Find the translation that maps (moves) the given end point to the midpoint.. Appl the same translation to the midpoint to find the other end point. Eample If K ( ) is the midpoint of [PQ] and P ( ) find the coordinates of Q.. Rough diagram P( ) K( ) Q(??)

8 /8/ :8 PM Page 7 COORDINATE GEOMETRY OF THE LINE :. Translation from P to K PK. Rule: add to subtract from. This can be written as a b.. Appl this translation to K: K( ) : ( ) (6 7) The coordinates of Q are (6 7). Distance between two points The given diagram shows the points A( ) and B( ). ƒ BC ƒ ƒ AC ƒ and B( ) A ( ) C 0 Using the theorem of Pthagoras: ƒ AB ƒ ƒ AC ƒ ƒ BC ƒ ƒ AB ƒ ( ) ( ) AB ( ) ( ) The distance between A( ) and B( ) is ƒ AB ƒ (see booklet of formulae and tables page 8). ( ) ( ) 7

9 8 /8/ :8 PM Page 8 LESS STRESS MORE SUCCESS Eample Carol is positioned at the point ( ) and Siobhan is positioned at the point ( ). Find the distance between them. ( ) Let ( ) ( ) and ( ) ( ) Distance between Carol and Siobhan: ( ) ( ) ( ) ( ) ( ) () ( 6) 6 At this stage all numbers are alwas positive. 0 0 (using a calculator) ABCD is a rectangle with A( ) and B( 9). Given ƒ BC ƒ ƒ AB ƒ calculate the area of ABCD. C B Let ( ) ( ) and ( ) ( 9) ƒ AB ƒ ( ) ( ) ( ) (9 ) ( 6) (8) ƒ BC ƒ ƒ AB ƒ (0) Area rectangle ABCD (length)(width) (0)() 0 square units D A

10 /8/ :8 PM Page 9 COORDINATE GEOMETRY OF THE LINE (a) Write down the coordinates of the point A and the point B on the diagram. (b) Use the distance formula to find ƒ AB ƒ. (c) Write down the distance from O to A and the distance from O to B. A( ) (d) Use the theorem of Pthagoras to find the length of the hpotenuse of the triangle OAB. B( ) O (a) A (0 ) B ( 0) (b) Let ( ) ⴝ (0 ) and ( ) ( 0) ( ) ( ) ( 0) (0 ) () ( ) ƒ AB ƒ 6 9 (c) ƒ OA ƒ ƒ OB ƒ (d) The theorem of Pthagoras A (Hp) (Opp) (Adj) ƒ AB ƒ () () ƒ AB ƒ 9 6 ƒ AB ƒ Hp Answer to part (b) Answer to part (d). B 9

11 0 /8/ :8 PM Page 0 LESS STRESS MORE SUCCESS Given the points on the diagram: B C E ( 0) ( ) ( 6 0) F ( ) E B (i) Find: (a) (b) (c) (d) ƒ BE ƒ ƒ CF ƒ ƒ EC ƒ ƒ BF ƒ C F (ii) Hence prove the triangle BCE is congruent to the triangle BCF. (i) B observation from the diagram we can sa: For parts (a) and (b) count the width of each bo as one unit. (a) ƒ BE ƒ 8 and (b) ƒ CF ƒ 8 We use ( ) ( ) for both ƒ EC ƒ and ƒ BF ƒ. (c) ƒ EC ƒ ( ) ( 6 0) (d) ( ) ( ) ƒ EC ƒ ( ) ( ) ( ( 6)) ( 0) ( 6) ( ) () 6 6 ƒ BF ƒ ( ) ( 0) ( ) ( 0) () ( ) ƒ BF ƒ ƒ EC ƒ ƒ BF ƒ and ƒ BE ƒ ƒ CF ƒ. These two pieces of information will be ver useful in answering part (ii) of this question.

12 /8/ :8 PM Page COORDINATE GEOMETRY OF THE LINE (ii) Consider BCE and BCF ƒ BC ƒ ƒ BC ƒ same ƒ EC ƒ ƒ BF ƒ 0 E B ƒ EB ƒ ƒ CF ƒ 8 Hence (b SSS) BCE is congruent (identical) to BCF. C F The four cases for congruent triangles are covered in Chapter Geometr. Part (ii) above is an ecellent eample of an eam question linking two different topics on our course. In this case we see coordinate geometr of the line linked with geometr theorems. In a recent eam a similar question on congruence was asked but it was worth ver few marks. For not answering this part candidates lost mark out of a total of 7 marks awarded for the question. Remember: Do not become disheartened continue to do our best for ever part of ever question and ou will do well. Henr travels in a swarm from Zone A to Zone B. The swarms movement from Zone A to Zone B can be modelled b the translation that maps (0 0) ( 708 0) (i) If Henrs starting position in the swarm in Zone A is (00 98) find his position when the swarm move to Zone B. (ii) Henrs best friend Harriet is also part of the swarm. Her position in Zone B is ( 7 99) find her starting position. (iii) Find the distance in cm between Henr and Harriet as the travel in the swarm when each unit represents one millimetre.

13 /8/ :8 PM Page LESS STRESS MORE SUCCESS Zone B (i) (0 0) ( 708 0) 708 on the -component 0 on the -component Zone A Henr: (00 98) ( ) ( 70 0) Zone B (ii) Harriet: (??) ( 7 99) Working backwards from Zone B to Zone A 708 on the -component 0 on the -component Zone A Harriet s Zone A position ( ) (997 0) (iii) The distance between Henr and Harriet in the swarm is alwas the same. Hence we can find the distance from: Zone A. (00 98) to (997 0) or Zone B. ( 70 0) to ( 7 99) Here we find the distance from (00 98) to (997 0). Let ( ) (00 98) and ( ) (997 0) ( ) ( ) (997 00) ( 0 ( 98)) ( 8) ( 6) Distance units Since each unit is mm then 0 units is 0 mm cm As an eercise ou could verif the distance from ( 7 99) to ( 70 0) is also 0 units cm. Slope of a line All mathematical graphs are read from left to right. The measure of the steepness of a line is called the slope. The vertical distance up or down is called the rise. The horizontal distance across is called the run. The slope of a line is defined as: Slope Rise Run Rise Run The rise can also be negative and in this case it is often called the fall. If the rise is zero then the slope is also zero.