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

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1 CONTENTS Introduction...iv. Coordinate Geometr of the Line.... Geometr Theorems...8. Constructions...7. Transformation Geometr Trigonometr I Trigonometr II: Real Life Applications Perimeter, Area, Nets and Volume Fundamental Principles of Counting Probabilit Statistics I: Statistical Investigations...9. Statistics II: Central Tendenc and Spread of Data Statistics III: Representing Data... Glossar of Statistical Terms... 8 Calculator Instructions... 0 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. SEC Eam papers. Sample eam papers. Original and sourced eam-tpe questions in this book are selected from the following:

2 /9/ 7:8 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 practice, 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. The plane is called the Cartesian (kar-tee-zi-an) plane. Ever point on the plane has two coordinates, an -coordinate and a -coordinate. 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(, ). B(, ) A(, ) 0 In a couple (, ) the order is important The graph above shows that the point A(, ) is different to the point B(, )

3 /9/ 7:8 PM Page LESS STRESS MORE SUCCESS The four quadrants The intersecting -ais and -ais divide the plane into four regions called quadrants. These are numbered st, nd, rd and th, as shown on the right. nd quadrant + Eample (, ) st quadrant + + Eample (, ) 0 rd quadrant Eample (, ) th quadrant + Eample (, ) 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 (, ) Translation 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.

4 /9/ 7:8 PM Page COORDINATE GEOMETRY OF THE LINE Eample Describe the translation that maps the points R H (i) G to H (ii) E to Q E (iii) R to D Q D G (i) G : H is described b units to the right and units up. This can be written as a b. (ii) E : Q is described b units to the left and 0 units up (or down). This can be written as a (iii) R : D is described b units to the right and units down. This can be written as a b. A more comprehensive 0 treatment of translations can be found in Chapter on transformation geometr. b. Midpoint of a line segment If (, ) and (, ) are two points, their midpoint is given b the formula: Midpoint a, b (See booklet of formulae and tables page 8) (, ) Midpoint +, + (, ) Before using coordinate geometr formulae, alwas allocate one point to be (, ) and the other to be (, ).

5 /9/ 7:8 PM Page LESS STRESS MORE SUCCESS Eample A(8, ) and B( 0, ) are two points. Find the midpoint of [AB]., b Let (, ) (8, ) and (, ) ( 0, ) Midpoint formula a Midpoint a 8 0 6, b a, b (, 8) In some questions, we will be given the midpoint and one end point of a line segment. We will 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(?,?) :. Translation from P to K, PK. Rule: add to, subtract from. b. This can be written as a. Appl this translation to K: K(, ) : (, ) (6, 7) The coordinates of Q are (6, 7).

6 /9/ 7:8 PM Page COORDINATE GEOMETRY OF THE LINE Distance between two points The given diagram shows the points A(, ) and B(, ). ƒ BC ƒ B(, ) ƒ AC ƒ and Using the theorem of Pthagoras: ƒ AB ƒ ƒ AC ƒ ƒ BC ƒ ( ) ( ) ƒ AB ƒ ( ) ( ) A(, ) 0 The distance (length) between A(, ) and B(, ) is ƒ AB ƒ (See booklet of formulae and tables page 8). ( ) ( ) Eample Find the distance between the points A(, ) and B(, ). Let (, ) (, ) and (, ) (, ) ( ) ( ) ( ) ƒ AB ƒ ( ) ( 6) C At this stage the numbers are alwas positive.

7 6 /9/ 7:8 PM Page 6 LESS STRESS MORE SUCCESS 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 ( ) ( ) ( 6) ( ) (9 ) (8) D A ƒ BC ƒ ƒ AB ƒ (0) Area rectangle ABCD (length)(breadth) (0)() 0 square units Henr the bee travels in a swarm from Zone A to Zone B. The swarm s movement from zone A to zone B can be modelled b the translation that maps (0, 0) : ( 708, 0). (i) If Henr s starting position in the swarm in Zone A is (00, 98) find his position when the swarm moves to Zone B. (ii) Henr s best friend, Harriet, is also part of the swarm. If 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 is one mm.

8 /9/ 7:8 PM Page 7 COORDINATE GEOMETRY OF THE LINE Zone B (i) (0, 0) : ( 708, 0) 708 on the -component 0 on the -component Zone A Henr: (00, 98) : (00 708, 98 0) ( 70, 0) (ii) Harriet: (?,?) : ( 7, 99) Working backwards from Zone B to Zone A 708 on the -component 0 on the -component. Zone B Zone A Harriet s Zone A position ( 7 708, 99 0) (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) Distance ( ) ( ) (997 00) ( 0 ( 98)) ( 8) ( 6) units Since each unit is mm Then 0 units is 0 mm cm As an eercise, ou could verif that the distance from ( 7, 99) to ( 70, 0) is also 0 units cm. When geese fl in formation, the form an inverted v-shape. 7

9 8 /9/ 7:8 PM Page 8 LESS STRESS MORE SUCCESS (i) If the lines of geese can be represented b the equations 0 and 6 0, find the coordinates of the leading goose. After hour, the leading goose has flown to a point (7, 67). (ii) Assuming the geese flew in a straight line and taking each unit to represent km, find the distance travelled b the geese to the nearest km. (iii) Hence, find the average fling speed in m/s. (i) Solving the linear equations in two variables: Put into or 6 () 6 8 (Add) 7 8 Solving linear equations is a skill ou must know. Another eample appears later in this chapter. The solution is and or (, ) (ii) Use distance formula ( ) ( ) Let (, ) (, ) and (, ) (7, 67) Distance (7 ) (67 ),089,096,8 7 # 0069 Distance to nearest km 7 km 7,000 Distance 0 m>sec (iii) Speed Time The eam ma contain in-contet questions at an stage. Be prepared to emplo techniques learned elsewhere, as in the above question where Distance. This would seem to have no link to coordinate geometr. Speed Time Slope of a line The slope of the line AB is defined as the vertical change rise or run horizontal change. The slope of AB 0 B A 0 Horizontal change Vertical change

10 /9/ 7:8 PM Page 9 9 COORDINATE GEOMETRY OF THE LINE In the diagram on the right, the slope of AB is found b getting the vertical change horizontal change B(, ) A(, ) 0 Slope m (See booklet of formulae and tables page 8) Positive and negative slopes As we move from left to right the slope is positive if the line is rising and the slope is negative if the line is falling Negative slope + Positive slope Eample Write down the slopes of the following lines in the diagram. (i) GR (ii) BR (iii) HJ (iv) GA (v) AB (vi) BG H J A R G B

11 0 /9/ 7:8 PM Page 0 LESS STRESS MORE SUCCESS Use rise (b counting the boes) in each case to find run 8 8 Slope BR 0 Slope HJ 0 8 Slope GA Slope AB Slope BG 9 (i) Slope GR (ii) (iii) (iv) (v) (vi) Line going up Q positive slope Line going down Q negative slope Horizontal line Q slope-zero Line going up Q positive slope Line going down Q negative slope Line going down Q negative slope An accountant plots the straight line value of a computer over a three-ear period on the given graph. Justif our answer. (0, 000) Value of computer in (i) Find the slope of the line. (ii) Hence write down the average rate of change in the value of the computer (, 900) Number of ears of use