National 5 Mathematics Assessment Practice 5: The Theorem of Pythagoras and vectors (Topics 16-17)

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1 SCHOLAR Study Guide National 5 Mathematics Assessment Practice 5: The Theorem of Pythagoras and vectors Topics Authored by: Margaret Ferguson 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: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75

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.

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5 1 Topic 7 The Theorem of Pythagoras and vectors Contents 7.1 Learning points Assessment practice

6 2 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS By the end of this topic, you should have identified your strengths and areas for further revision. Read through the learning points before you attempt the assessments and go back to the Course Materials unit if you need more help. Key point You should be able to: use the: theorem of Pythagoras in 3D shapes; converse of Pythagoras; find the distance between two points; add, subtract and scale 2D vectors using directed line segments; describe a vector journey; determine the coordinates of a point on a 3D object; add, subtract and scale vectors using components; calculate the magnitude of a vector.

7 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS Learning points The Theorem of Pythagoras The Converse of Pythagoras Check that the longest side squared is equal to the sum of the squares of the other two sides. If this is true then the triangle is right-angled otherwise the triangle is not. The Theorem of Pythagoras in 3D shapes A face diagonal is a diagonal on one of the faces of the 3D shape. A space diagonal goes through the space inside the 3D shape. Add a radius or diameter to a circle to form a right angled triangle. The distance between two points Given two points x 1,y 1 and x 2,y 2, the distance formula is Vectors A vector is a quantity which has both direction and magnitude. The magnitude of a vector is its size or length. A directed line segment from A to B is defined as AB. A vector or force can also be defined by a lower-case letter in bold. Vectors are equal if they have the same direction and magnitude. When a vector has its direction reversed it is negative e.g. a becomes -a. A vector can be multiplied by a scalar e.g. doubling a gives 2a. Vectors can be added by joining one to the end of another. Displacement is the shortest distance from A to B. A vector journey is a description of its displacement. The components of a vector describe the journey from A to B x x e.g. in 2D or y in 3D. y z Arithmetic can be performed on the components e.g. + = = x 2 x 1 2 +y 2 y 1 2 The magnitude is calculated from the components using the Theorem of Pythagoras x e.g. u =, u = x x 2 + y 2 and v = y, v = x 2 + y 2 + z 2 y z

8 4 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS 7.2 Assessment practice Make sure that you have read through the learning points or completed some revision before attempting these questions. Tailor your practice by choosing the most appropriate questions. The Theorem of Pythagoras: Questions 1 to 10 Vectors: Questions 11 to 32 Key point Questions 1, 6, 7 and 8 also assess your reasoning skills. Assessment practice: The Theorem of Pythagoras and vectors Go online Converse of Pythagoras Q1: 3,4,5 is a Pythagorean triple because = 5 2. This triangle is therefore right-angled by the converse of Pythagoras. Is 20, 99, 102 a Pythagorean triple? The television mast is 285 m high and is anchored 68 m from the base of the mast by a cable 293 m long. If the mast is vertical then the triangle is right-angled.

9 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS 5 Q2: Is the mast is vertical, Yes or No? Q3: Justify your answer. The Theorem of Pythagoras in 3D shapes PQRST is a rectangular based pyramid. Q4: What is the length of the face diagonal PR, leaving your answer as a surd? Q5: What is the vertical height of the pyramid UT, giving your answer correct to 3 significant figures? The shoe box below has length 35 cm, breadth 18 cm and height 13 cm. Q6: Calculate the length of the space diagonal in the shoe box, giving your answer correct to 1 decimal place.

10 6 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS The diagram below shows a pipe of diameter 48 cm with water flowing through it. The distance across the surface of the water is 28 cm. Q7: Calculate the height of the water in the pipe. Three identical dinner plates are placed together so that their edges touch. The diameter of each plate is 29 cm. A simplified diagram is shown below. Q8: Construct a triangle then calculate the distance x. Calculating distances using coordinates Q9: Calculate the distance between A17,2 and B8,-9. Q10: Calculate the distance between C-19,-1 and D20,29.

11 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS 7 Vectors in 2D Q11: Name the vector a as a directed line segment. Q12: Name the vector b as a directed line segment. Q13: Name the vector c as a directed line segment. Q14: Draw the resultant vector of 2a - b

12 8 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS Vector journeys In the diagram ABCD is a trapezium. Vectors u and v represent ADand BD. Q15: Express BA in terms of u and v. Q16: Express CD in terms of u and v. OABCDE is a triangular prism made from 2 congruent rectangles and 2 congruent isosceles triangles. Q17: State the coordinates of A. Q18: State the coordinates of B. Q19: State the coordinates of C. Q20: State the coordinates of E.

13 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS 9 Vector components in 2D Q21: Name the vector. Q22: State its components. Given a = 3 2, b = 1 3 and c = 4 4 State the components of the resultant vectors. Q23: a + b Q24: b - c Q25: 2a - b + 1 /2c Vector components in 3D Three forces have components a = 3 4, b = 1 2 and c = Q26: Calculate the components of a + b. Q27: Calculate the components 2b - c. Q28: Find the resultant force.

14 10 TOPIC 7. THE THEOREM OF PYTHAGORAS AND VECTORS Magnitude of vectors Two forces have components d = 5 2 and e = 1 4 Q29: Calculate d. Q30: Calculate 3d 2e. Two forces have components f = 1 3 and g = Q31: Calculate the magnitude of g. Q32: Calculate the magnitude of 2f + g

15 ANSWERS: UNIT 2 TOPIC 5 11 Answers to questions and activities Topic 5: The Theorem of Pythagoras and vectors Assessment practice: The Theorem of Pythagoras and vectors page 4 Q1: No The longest side squared is = = 400 and 99 2 = = is not equal to so 20, 99, 102 is not a Pythagorean triple. Q2: Yes Q3: The longest side squared is = = 4624 and = = Since = the triangle is right-angled by the Converse of Pythagoras and hence the mast is vertical. Q4: 296 = 2 74 Q5: Steps: What is the length of UR as a surd? 74 UR is the base of the right-angled triangle URT. Answer: 19 2 Q6: Steps: What is the length of the face diagonal on the base of the box, to 5 significant figures? A space diagonal goes from the bottom left front corner to the back right top corner. Use your answer with the height of the box to find the space diagonal. Answer: 41 4 Q7: Steps: Construct a right-angled triangle by adding a radius from one end of the chord to the centre of the circle then drop a line from the centre perpendicular to the chord. What is the radius of the pipe? 24 Use the right-angled triangle in your diagram to calculate its height. What is the height of your triangle? 19 5 Use this answer to calculate the height of the water.

16 12 ANSWERS: UNIT 2 TOPIC 5 Answer: 4 5 Q8: Steps: Join the centres of the circles to form an equilateral triangle. What is the height of the equilateral triangle? 25 1 Use this answer to help find the length x. Answer: 54 1 Q9: 14 2 Q10: 49 2 Q11: AB Q12: HG Q13: QP Q14: Q15: v - u Q16: Hints: Remember that a trapezium has one pair of parallel sides. CD is double BA. Use the distance formula. Answer: 2v - 2u

17 ANSWERS: UNIT 2 TOPIC 5 13 Q17: 8,0,0 Q18: 8,19,0 Q19: 0,19,0 Q20: 4,0,15 Q21: HG 7 Q22: 3 Q23: Q24: Q25: Q26: Q27: 2 3 Q28: Hints: The resultant force of a, b and c is the sum of the components of the three vectors. 6 Answer: 0 3 Q29: 29 or 5 4 Q30: 485 or 22 0 Q31: 77 or 8 8 Q32: 209 or 14 5