CfE Higher Mathematics Assessment Practice 1: The straight line

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1 SCHOLAR Study Guide CfE Higher Mathematics Assessment Practice 1: The straight line Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy A Watson Heriot-Watt University Edinburgh EH14 4AS, United Kingdom.

2 First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Assessment Practice: CfE Higher Mathematics 1. CfE Higher Mathematics Course Code: C747 76

3 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders.

5 1 Topic 7 The straight line Contents 7.1 Learning points Assessment practice

6 2 TOPIC 7. THE STRAIGHT LINE Learning objective You should be able to: interpret gradients of pairs of lines; show that points are collinear; determine equations of parallel lines; determine equations of perpendicular lines; use the gradient to find an angle; use the properties of medians; use the properties of altitudes; use the properties of perpendicular bisectors; find the centroid, orthocentre and circumcentre of a triangle; show that lines are concurrent and find the coordinates of the centroid, orthocentre and circumcentre of a triangle. By the end of this topic, you should have identified your strengths and areas for further revision.

7 TOPIC 7. THE STRAIGHT LINE Learning points Read through the learning points before you attempt the assessments and go back to the course materials if you need to revise further. The straight line The distance formula for two points is (x 2 x 1 ) 2 + (y 2 y 1 ) 2. The formula for the gradient, m, of a straight line requires two points (x 1,y 1 ) and (x 2,y 2 ). The formula is m = y 2 y 1 x 2 x 1 A horizontal straight line has gradient 0. The equation of a horizontal straight line with y-intercept (0,c) is y = c. A vertical straight line has gradient undefined. The equation of a vertical straight line with x-intercept (k,0) is x = k. The equation of a straight line with gradient m and y-intercept (0,c) is y = mx + c. The equation of a straight line with gradient m and passes through the point (a,b) is y b = m(x a). The general form of the equation of a straight line is Ax + By + C = 0. To be able to identify the gradient or y-intercept: The equation of a straight line must be in the form y = mx + c. If the equation is not in that form it must be re-arranged. Parallel lines have the same gradient and distinct y-intercepts. Collinearity means that a set of points lie on a single straight line: Find the gradients of the lines joining pairs of points. Equal gradients indicate parallel lines. A common point determines collinearity m = tan θ where θ is the angle made by the line and the positive direction of the x-axis. Straight lines with gradients m 1 and m 2 are perpendicular if and only if m 1 m 2 = 1 The midpoint formula is ( x1 + x 2 2., ) y 1 + y 2 2 A median of a triangle is a line from a vertex to the midpoint on the opposite side. The medians of a triangle are concurrent and the point of intersection is called the centroid. If the points M(a, b), N(c, d) and P(e, f ) are the vertices of a triangle. ( ) The point of intersection of the medians has coordinates a + c + e b + d + f 3, 3.

8 4 TOPIC 7. THE STRAIGHT LINE An altitude of a triangle is a line from a vertex, which is perpendicular to the opposite side. The altitudes of a triangle are concurrent and the point of intersection is called the orthocentre. The perpendicular bisector of a line or a side in a triangle is the line which is perpendicular through the midpoint of the side. The perpendicular bisectors of the sides of a triangle are concurrent and the point of intersection is called the circumcentre.

9 TOPIC 7. THE STRAIGHT LINE Assessment practice Make sure that you have read through the learning points and completed some revision before attempting these questions. Assessment practice: The straight line Go online SQA Past Paper: 2001 Paper 1 Q1: What is the equation of the straight line which is parallel to the line with equation 2x + 3y = 5 and passes through the point (2,-1)? SQA Past Paper: 2003 Paper 1 Q2: Find the equation of the line which passes through the point (-1,3) and is perpendicular to the line with equation 4x + y 1 = 0 in the form ax + by + c = 0. (3 marks) SQA Objective Question Q3: The line joining (-2,-3) and (6,k) has gradient 2 3. What is the value of k? SQA Past Paper: 2009 Paper 1 Q4: The line GH makes an angle of π 6 radians with the y-axis, as shown in the diagram. What is the gradient of GH?

10 6 TOPIC 7. THE STRAIGHT LINE SQA Objective Question Q5: What is the distance between the points (-2,5,3) and (4,-1,1) as an exact value in its simplest form? SQA Objective Question Q6: The points A(0,8), B(-4,0) and C(p,-4) are collinear. What is the value of p? SQA Objective Question Q7: A straight line passes through the points G(-2,5), M(4,-3) and H. M is the midpoint of GH. What are the coordinates of H? (1 mark)... Q8: Triangle OP Q has vertices at O(0,0), P (2,6) and Q(-6,2). If OS is an altitude, what are the coordinates of S? (4 marks) SQA Objective Question Q9: The triangle with vertices A(8,6), B(9,-5) and C(-2,y) has centroid with coordinates (a, 1). What are the values of a and y?

11 TOPIC 7. THE STRAIGHT LINE 7 SQA Past Paper: 2004 Paper 1 Q10: The diagram shows the line OA with equation x 2y = 0. The angle between A and the positive direction of the x-axis is a. What is the value of a? (3 marks)... Q11: The second diagram shows lines OA and OB. The angle between these two lines is 30. What is the gradient of OB correct to 1 decimal place? (1 mark) The triangle ABC has vertices A(2,2), B(12,2) and C(8,6). Q12: What is the equation of l 1, the perpendicular bisector of AB? (1 mark)... Q13: What is the equation of l 2, the perpendicular bisector of AC, in the form ax + by + c = 0? (4 marks)... Q14: What are the coordinates of the point of intersection of the lines l 1 and l 2? (1 mark)... Q15: What is the equation of the circle passing through A, B and C?

12 8 TOPIC 7. THE STRAIGHT LINE SQA Past Paper: 2002 Paper 1 The triangle ABC has vertices A(-1,6), B(-3,-2) and C(5,2). Q16: What is the equation of the line p, the median from C of the triangle ABC?... Q17: What is the equation of the line q, the altitude from A?... Q18: What are the coordinates of the point of intersection of the lines p and q? (1 mark) SQA Past Paper: 2004 Paper 1 Q19: The point A has coordinates (7,4). The straight lines with equations x +3y +1=0and 2x +5y = 0 intersect at B. What is the gradient of AB? (3 marks)... Q20: How many of the these lines is AB perpendicular to? a) both b) one c) neither (5 marks)

13 ANSWERS: UNIT 2 TOPIC 1 9 Answers to questions and activities Topic 1: The straight line Assessment practice: The straight line (page 5) Q1: Hints: Change the equation to the form y = mx + c to identify m. What is the gradient of the line with equation 2x +3y = 5? 2 3 What is the gradient of the parallel line? 2 3 Use the gradient and the point in y b = m(x a) to find the equation. Answer: 2x +3y 1 = 0 Q2: What is the gradient of the line with equation 4x + y 1 = 0?-4 What is the gradient of the line perpendicular to the line with equation 4x + y 1 = 0? 1 /4 Answer: x +4y 13 = 0 Q3: Hints: k ( 3) 6 ( 2) = 2 3 make k the subject of the formula. Answer: 7 3 Q4: What is π 6 radians in degrees? 30 What size is the angle HGO in degrees? 60 Find the gradient of GH using the property m = tan θ. Answer: 3 Q5: Hints: Distance = = (4 ( 2)) 2 + ( 1 5) 2 + (1 3) ( 6) 2 + ( 2) 2 = 76

14 10 ANSWERS: UNIT 2 TOPIC 1 Answer: 2 19 Q6: Hints: 4 0 = 2 p ( 4) 4 = 2(p + 4) 4 = 2p + 8 What is the gradient of AB? 2 What is the gradient of BC? 2 Find an expression for the gradient of BC to find the value of p. Answer: -6 Q7: To get from G to M, how many units right do you go? 6 To get from G to M, how many units down do you go? 8 Answer: (10,-11) Q8: What is the gradient of PQ? 1 /2 What is the equation of PQ? y = 1 2 x + 5 What is the gradient of the altitude OS? -2 What is the equation of OS? y = 2x Find the point of intersection of PQ and OS. Answer: (-2,4) Q9: a = 5 and y = 2 Q10: What is the gradient of the line OA? 1 /2 Use this value to find the value of a. Answer: 26 6

15 ANSWERS: UNIT 2 TOPIC 1 11 Q11: What is the angle between OB and the positive direction of the x-axis? 56 6 Use this answer to find the gradient and remember to round your answer. Answer: 1 5 Q12: What is the mid point of AB? (7,2) What is the gradient of AB? 0 What is the gradient of the perpendicular bisector? Undefined Is the perpendicular bisector horizontal or vertical? Vertical Find the equation through the midpoint. Answer: x = 7 Q13: What is the mid point of AC? (5,4) What is the gradient of AC? 2 3 What is the gradient of the perpendicular bisector? 3 2 Use y b = m(x a) to find the equation through the midpoint (a,b). Answer: 3x +2y 23 = 0 Q14: Hints: Use simultaneous equations with the equations from your answers to the previous two questions. Answer: (7,1) Q15: The point of intersection of the perpendicular bisectors of a triangle is called the circumcentre.

16 12 ANSWERS: UNIT 2 TOPIC 1 What are the coordinates of the centre of the circle passing through A, B and C? (7,1) What length is the radius? 26 Express the equation of the circle in the form (x a) 2 +(x b) 2 = r 2. Answer: (x 7) 2 +(x 1) 2 = 26 Q16: What are the coordinates of R, the midpoint of AB? (-2,2) What is the gradient of CR? Undefined Look at the coordinates of C and R then identify the equation of the line p. Answer: y = 2 Q17: What is the gradient of BC? 1 /2 What is the gradient of the perpendicular bisector of BC? -2 Now find the equation of the line q. Answer: y = 2x + 4 Q18: Use simultaneous equations with your answers to the previous questions to this question. Answer: (1,2) Q19: Using simultaneous equations what are the coordinates of B? (5,-2) Answer: Q20: What is the gradient of the line x + 3y + 1 = 0? 1 3 What is the gradient of the line 2x +5y = 0? 2 5 What is the gradient of any line perpendicular to AB? 1 3 Answer: b) one

National 5 Mathematics Assessment Practice Topic 2: Volumes, arcs, sectors, angles and similar shapes

SCHOLAR Study Guide National 5 Mathematics Assessment Practice Topic 2: Volumes, arcs, sectors, angles and similar shapes Authored by: Margaret Ferguson Heriot-Watt University Edinburgh EH14 4AS, United