300 14 Position and transformation Think like a mathematician 3 Discuss the answers to these questions with a partner or in a small group. Zalika and Maha use different methods to find the midpoint of the line segment AB where A is (3, 4) and B is (11, 4). Zalika’s method Draw a diagram. x y A × B 2 1 0 4 midpoint at (7, 4) 3210 5 6 7 8 9 10 11 12 13 14 15 3 4 5 Maha’s method The y-coordinates of A and B are the same, so the y-coordinate of the midpoint is 4. To find the x-coordinate: 11 – 3=8, 8÷2=4, 3+4=7 The midpoint is at (7, 4) a Write the advantages and disadvantages of Zalika’s method. b Explain how Maha’s method works. c Write the advantages and disadvantages of Maha’s method. d Whose method do you prefer? Explain why. e Can you think of a better method? f Discuss your answers with other groups in your class. 2 Match each line segment with the correct midpoint. An example is done for you. Line segment AB and iii. i (1, −2) ii (−1, −6) iii (2, 3) iv (−5, 4) v (−3 1 2 , −3) vi (2 1 2 , 5) vii (−3, −1) viii (5, −2 1 2 ) x y I A C P M N H L K G Q FE D B J 2 1 0 –1 –4–5–6 –1–2–3 4 5 6321 –2 –3 –4 –5 –6 3 4 5 6 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
301 14.2 The midpoint of a line segment 4 Work out the midpoint of the line segment joining each pair of points. Write whether A, B or C is the correct answer. Use your preferred method. a (7, 1) and (7, 7) A (7, 6) B (7, 3) C (7, 4) b (4, 2) and (10, 2) A (7, 2) B (6, 2) C (5, 2) c (4, 11) and (4, 2) A (4, 9) B (4, 6 1 2 ) C (4, 4 1 2 ) d (8, 15) and (15, 15) A (111 2 , 15) B (7, 15) C (12 1 2 , 15) Worked example 14.2b You can calculate the midpoint of a line segment by finding the means of the x-coordinates and the y-coordinates of the end points. The diagram shows the line segment PQ. Calculate the coordinates of the midpoint of PQ. Answer 10 4 2 6 2 3 + − = = Add the x-coordinates of P and Q and divide the result by 2. 4 6 2 2 2 1 + − − = = − Add the y-coordinates of P and Q and divide the result by 2. The midpoint of PQ is (3,−1). The mean of a and b is a b + 2 x y P (10, 4) Q (–4, –6) 4 2 0 –2 –8 –2–4–6 8 10642 –4 –6 6 5 Copy and complete the workings to calculate the midpoint of the line segment joining each pair of points. a (2, 3) and (6, 7) 2 6 2 3 7 2 8 2 10 2 4 + + ( , , ) ( = =) ( ) , b (8, 0) and (12, 6) 8 12 2 0 6 2 20 2 6 2 + + ( , , ) ( = =) ( ) , c (5, 2) and (8, 10) 5 8 2 2 10 2 13 2 2 1 2 6 + + ( ) ( ) , , = = , d (0, 4) and (7, 11) 0 7 2 4 11 2 2 2 + + ( ) , , = = , We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
302 14 Position and transformation 6 a E is the point (6, 0), F is the point (14, 8) and G is the point (3, 15). Work out the midpoint of the line segments i EF ii EG iii FG b Draw a coordinate grid. Plot the points E, F and G. Check your answers to part a by finding the midpoints on your diagram. Think like a mathematician 7 Discuss the answers to these questions with a partner or in a small group. Shen and Hassan calculate the midpoint of the line joining the points (−5, −8) and (−1, 9). This is what they write. Shen x-coordinate: –5+–1 2 =–4 2 =–2 y-coordinate: –8+9 2 =–1 2 =– 1 2 Midpoint is at –2, – 1 2 Hassan x-coordinate: –1+–5 2 =–6 2 =–3 y-coordinate: 9+–8 2 = 1 2 Midpoint is at –3, 1 2 a Who, out of Shen and Hassan, has the correct midpoint? Explain the mistake the other student has made. b Look again at their methods. Shen added the x and y coordinates of (−5, −8) to (−1, 9). Hassan added the x and y coordinates of (−1, 9) to (−5, −8). Does it matter in which order you add the x and y coordinates? Explain your answer. c Discuss your answers with other groups in your class. 8 Calculate the midpoint of the line segment between a (5, −2) and (2, −6) b (−4, 5) and (3, 0) c (−7, 5) and (−10, 10). 9 A parallelogram has vertices at P (2, 5), Q (−2, 3), R (2, −1) and S (6, 1). The diagonals are PR and QS. Show that the diagonals have the same midpoint. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
303 14.2 The midpoint of a line segment 10 Calculate the coordinates of the midpoint of each side of this triangle. x y D F E 20 10 0 –10 –4 –1–2–3 4321 –20 –30 –40 30 40 11 A quadrilateral has vertices at (−2, 1), (0, 4), (5, 2) and (1, −1). Do the diagonals have the same midpoint? Justify your answer. Think like a mathematician 12 The midpoint of a line segment is (4, 1). One end of the line segment is (2, 5). a Work out the coordinates of the other end of the line segment. b Compare the method you used to answer part a with a partner’s method. Did you both use the same method? Did you use different methods? c Discuss the methods you used with other learners in the class. Which do you think is the best method to use to answer this type of question? Explain why. 13 The midpoint of a line segment is (7, 2). One end of the line segment is (−1, 6). Work out the coordinates of the other end of the line segment. 14 Here are six cards showing the coordinates of the points A to F. A (2, 0) B (−3, −2) C (−7, 5) D (1, 4) E (5, −3) F (−4, 2) Three line segments are made using the six cards. The midpoint of all three line segments is (−1, 1). What are the three line segments? Show how you worked out your answers. Summary checklist I can work out the coordinates of the midpoint of a line segment. I can work out the coordinates of the end of a line segment when I know the coordinates of the other end and the midpoint. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
304 14 Position and transformation 14.3 Translating 2D shapes In this section you will ... • translate shapes on a coordinate grid using vectors. Key words column vector congruent image object translate You already know that when you translate a 2D shape on a coordinate grid, you move it up or down and right or left. You can describe this movement with a column vector. This is an example of a column vector: 2 5 The top number tells you how many units to move the shape right (positive number) or left (negative number). The bottom number tells you how many units to move the shape up (positive number) or down (negative number). For example: 2 5 means ‘move the shape 2 units right and 5 units up’. − − 2 3 means ‘move the shape 2 units left and 3 units down’. If the scale on the grid is one square to one unit, the numbers tell you how many squares to move the object up/down and across. When a shape is translated, only its position changes. Its shape and size stay the same. This means that the object and its image are always congruent. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
305 14.3 Translating 2D shapes Worked example 14.3 The diagram shows triangle T on a coordinate grid. Draw the image of triangle T after each translation. a 3 2 b 2 −1 c − 3 1 d − − 1 3 Answer a Move triangle T 3 squares right and 2 squares up. b Move triangle T 2 squares right and 1 square down. c Move triangle T 3 squares left and 1 square up. d Move triangle T 1 square left and 3 squares down. x y 2 1 0 –1 –4 –1–2–3 4321 –2 –3 –4 3 4 T x y 2 1 0 –1 –4 –1–2–3 43 –2 –4 3 4 21 a b c T d –3 Exercise 14.3 1 The yellow cards show translations. The blue cards show column vectors. Match each yellow card with the correct blue card. The first one is done for you: A and iii A 4 squares left, 1 square up B 4 squares right, 1 square down C 4 squares left, 1 square down D 4 squares right, 1 square up i 4 −1 ii 4 1 iii − 4 1 iv − − 4 1 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
306 14 Position and transformation 2 The diagram shows triangle P on a coordinate grid. Copy the grid, then draw the image of triangle P after each translation. a 3 2 b 2 −2 c − 1 3 d − − 2 1 3 The diagram shows shape A on a coordinate grid. Copy the grid, then draw the image of shape A after each translation. a 3 2 b 4 −2 c − 2 2 d − − 1 2 4 This is part of Fin’s homework. Question A triangle ABC is translated using the column vector 3 –2 The image of ABC is A'B'C'. Write the column vector that translates A'B'C' back to ABC. Answer 2 –3 a Is Fin’s answer correct? Explain your answer. b How could Fin check whether his answer is correct? x y 6 5 4 3 2 1 0 876543210 7 8 P x y 2 1 –4 –3 4321 –2 –3 –4 3 4 –2 0 –1 –1 A Think like a mathematician 5 Look at this question in pairs or groups, then discuss the answers to parts a, b and c. Read what Zara says. If I translate a shape using the column vector 2 3 , I can translate the shape back to its original position using the column vector − − 2 3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
307 14.3 Translating 2D shapes 6 The diagram shows triangle DEF. ∠DEF=90°, ∠DFE=45° and ∠EDF=45° DF has a length of 4 units. a Copy the grid, then draw the image of the triangle after the translation 3 −2 Label the triangle D′E′F′. b Copy and complete these statements. ∠D′E′F′=......°, ∠D′F′E′ =......° and ∠E′D′F′ =......°. D′F′ has a length of ...... units. c Copy and complete these statements. Choose from the words in the box. When you compare an object and its image after a translation: • corresponding lengths are ............... • corresponding angles are ............... • the object and the image are ................ 7 The diagram shows two shapes, P and Q. Choose the column vector (A, B or C) that translates a shape P to shape Q A 2 3 B − 2 3 C 2 −3 b shape Q to shape P A 2 3 B − 2 3 C 2 −3 x y 5 4 3 2 1 0 876543210 D E F different shorter equal longer congruent smaller not congruent bigger x y 5 4 3 2 1 0 6543210 P Q Continued a Show that Zara is correct. b Write the column vectors that translate a shape back to its original position after these translations. i −4 7 ii 3 −5 iii − − 2 8 c When a shape is translated using a column vector, it can be translated back to its original position. Write a general rule for finding the column vector that will translate a shape back to its original position. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
308 14 Position and transformation 8 The diagram shows shapes L, M, N, P and Q on a coordinate grid. Write the column vector that translates a shape N to shape L b shape N to shape P c shape N to shape Q d shape N to shape M e shape L to shape P f shape P to shape M. –1–2–3–4 –2 –1 1 0 2 3 4 1 2 3 4 x y M N Q P L Think like a mathematician 9 The diagram shows triangle JKL. Marcus and Arun translate triangle JKL using the column vector 5 −4 They label the image J′K′L′. Read what Marcus and Arun say. a Explain why Marcus is correct and Arun is incorrect. b Use Marcus’s method to calculate the coordinates of K′ and L′. Use the diagram to check your answers are correct. c Discuss your methods and answers to parts a and b with other learners in your class. x y J9 K9 L9 2 1 –3 431 –2 –3 3 4 –2 0 –1 –1 J K L 2 I can calculate the coordinates of J′ like this: − + = − + + − = ( ) − ( ) ( ) 3 4 3 5 4 4 2 0 5 4 , , , I think you should write: − + = = ( ) − − + + − 3 4 5 4 3 5 4 4 2 0 , We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
309 14.3 Translating 2D shapes 10 A rectangle ABCD has vertices at the points A (−2, 3), B (4, 3), C (4, −2) and D (−2, −2). ABCD is translated using the column vector 8 5 a Calculate the coordinates of A′, B′, C′ and D′. b Check your answers are correct by drawing a diagram and translating rectangle ABCD. c Compare and discuss your working for part a with that of a partner. Have you used the same methods? Are both sets of working easy to understand? 11 This is part of Joule’s classwork. She has spilt some juice on her work. Question A square EFGH has vertices at the points E (–5, –1), F (3, –1), G (3, 7) and H (–5, 7) EFGH is translated using column vector 8 5 to E'F'G'H'. Work out the coordinates of the vertices of E'F'G'H'. Answer E' (–8, 6), F' (0, 6), G' (0, 14), H' (–8, 14) a Work out the coordinates of vertices i F′ ii G iii H b Explain how you worked out the answers to part a. Summary checklist I can translate shapes on a coordinate grid using vectors. I can work out the vector of a translation given the object and the image. I can work out the coordinates of the image of a shape given the vector. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
310 14 Position and transformation 14.4 Reflecting shapes In this section you will ... • reflect shapes on a coordinate grid given the equation of the mirror line • identify a reflection and its mirror line. Key words mirror line reflect You already know how to reflect a shape when you use the x-axis or y-axis as the mirror line. You must also be able to reflect a shape on a coordinate grid in other mirror lines. To do this, you need to know the equation of the mirror line. Some examples are shown on the grid on the right. All vertical lines are parallel to the y-axis and have the equation x=‘a number’. All horizontal lines are parallel to the x-axis and have the equation y = ‘a number’. –1–2–3 –3 –2 –1 1 0 2 3 x = –1 y = 1 y = –2 1 2 3 x y x = 2 Worked example 14.4 Draw a reflection of this triangle in the lines a x=4 b y=4 Answer a x y 6 5 4 3 2 image x = 4 object 1 0 76543210 First draw the mirror line x=4 on the grid. Take each vertex of the object in turn and plot its reflection in the mirror line. Use a ruler to join the reflected points with straight lines to make the image. x 5 6 y 4 3 2 1 0 0 1 2 3 4 5 6 7 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
311 14.4 Reflecting shapes Continued b x y 6 5 4 3 2 image y = 4 object 1 0 76543210 First draw the mirror line y=4 on the grid. Take each vertex of the object in turn and plot its reflection in the mirror line. Use a ruler to join the reflected points with straight lines to make the image. Notice that the vertices at (5, 4) and (7, 4) are the same on the object and the image. Exercise 14.4 1 Copy each diagram and reflect the shape in the mirror line with the given equation. a mirror line x=3 b mirror line x=4 c mirror line x=2.5 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 5 4 3 2 1 0 0 7654321 7 d mirror line y=4 e mirror line y=3 f mirror line y=3.5 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 5 4 3 2 1 0 0 7654321 7 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
312 14 Position and transformation 2 Copy each diagram and reflect the shape in the mirror line with the given equation. a mirror line x=4 b mirror line x=3 c mirror line x=2 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 7 5 4 3 2 1 0 0 7654321 x y 6 5 4 3 2 1 0 0 7654321 7 d mirror line y=3 e mirror line y=2 f mirror line y=4 x y 6 5 4 3 2 1 0 0 7654321 7 x y 6 5 4 3 2 1 0 76543210 7 x y 6 5 4 3 2 1 0 6543210 7 8 9 10 3 This is part of Gille’s homework. Question Reflect shape A in the line y=–1. Label the shape A’. Answer –5 –4 –1–2–3 –3 –4 –2 –1 1 0 2 3 y = –1 1 2 3 4 x y Mirror line y = −1 A9 A a Explain the mistake Gille has made. b Copy the diagram of shape A and draw the correct reflection. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
313 14.4 Reflecting shapes 4 The diagram shows shape B on a coordinate grid. Draw the image of shape B after reflection in the line a x=−1 b y=−2 c x=0.5 d y=−0.5 5 This is part of Oditi’s homework. Question Draw a reflection of the orange triangle in the line x=4. Explain your method. Answer Reflected triangle drawn on grid in green. I reflected each corner of the triangle in the line x=4, then I joined the three corners together. x y 4 3 2 x = 4 1 0 76543210 a Make a copy of this grid. Use Oditi’s method to draw these reflections. i Reflect the triangle in the line x=4 ii Reflect the parallelogram in the line y=5 iii Reflect the kite in the line x=8 b What do you think of Oditi’s method? Is it easy to follow? Can you think of a better method to use to reflect shapes when the mirror line goes through the shape? Explain your answer. x y 2 1 –1 –4 –1–2–3 4 5321 0 –2 –3 –4 –5 –6 –7 3 4 B x y 4 5 6 3 2 1 0 6543210 7 8 9 10 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
314 14 Position and transformation Think like a mathematician 6 Work with a partner, or in a small group, to answer this question. The diagram shows a rectangle. It also shows the line y=x. a Reflect the rectangle in the line y=x. b Compare your answer with other groups in your class and discuss the methods used. c Use your preferred method to reflect these shapes in the line y=x. i x y 5 6 4 3 2 1 0 6543210 y = x ii x y 5 4 3 2 1 0 543210 y = x iii x y 5 6 4 3 2 1 0 543210 x y 5 6 4 3 2 Mirror line y = x 1 0 6543210 y = x Think like a mathematician 7 Work with a partner, or in a small group, to answer this question. Alicia reflects trapezium ABCD in the line y=x. The diagram shows the object, ABCD, and its image, A’B’C’D’. a The table shows the coordinates of the vertices of the object and its image. Copy and complete the table. Object A (3, 6) B (3, 4) C ( , ) D ( , ) Image A’ ( , ) B’ ( , ) C’ ( , ) D’ ( , ) b What do you notice about the coordinates of ABCD and its image A’B’C’D’? c Write a rule you can use to work out the coordinates of the image of a shape when it is reflected in the line y=x. d Does your rule in part c work for any shape reflected in the line y=x? Explain your answer. x y 5 6 4 3 C C9 D9 B9 A9 D B A 2 1 0 6543210 y = x We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
315 14.4 Reflecting shapes 8 The diagram shows shape ABCD on a coordinate grid. It also shows the line y=x. a Write the coordinates of the points A, B, C and D. When shape ABCD is reflected in the line y=x, the image is A′B′C′D′. b Use your rule from Question 7, part c to write the coordinates of the points A′, B′, C′ and D′. c Copy the diagram. Reflect shape ABCD in the line y=x. d Check the coordinates of the points A′, B′, C′ and D’ you worked out in part b are correct. If any of the coordinates are incorrect, check your answers with a partner. 9 The diagram shows shape ABCD on a coordinate grid. It also shows the line y=−x. a Make a copy of the diagram. Reflect ABCD in the line y=−x and label the image A′B′C′D′. b The table shows the coordinates of the vertices of the object and its image. Copy and complete the table. Object A (−1, 2) B (−1, 4) C ( , ) D ( , ) Image A’ ( , ) B’ ( , ) C’ ( , ) D’ ( , ) c What do you notice about the coordinates of ABCD and its image A′B′C′D′? d Write a rule you can use to work out the coordinates of the image of a shape when it is reflected in the line y=−x. e Does your rule in part d work for any shape reflected in the line y=−x? Explain your answer. 10 The diagram shows triangle PQR on a coordinate grid. It also shows the line y=−x. a Write the coordinates of the points P, Q and R. When shape PQR is reflected in the line y=−x, the image is P′Q′R′. b Use your rule from Question 9, part d to write the coordinates of the points P′, Q′ and R′. c Copy the diagram. Reflect shape PQR in the line y=−x. d Check the coordinates of the points P′, Q′ and R′ you worked out in part b are correct. If any of the coordinates are incorrect, check your answers with a partner. x y y = x A D B C 4 5 6 3 2 1 0 76543210 x y y = –x 1 –1 –1–2–3–4–5 0 321 –2 –3 A D 2 B C 3 4 x y y = –x 1 –1 –1–2–4 Q R P 21 0 –2 2 3 4 –3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
316 14 Position and transformation Activity 14.4 Work with a partner for this question. Make a copy of these coordinate axes on a piece of squared paper. Draw a rectangle inside the shaded region. Exchange your diagram with a partner. Reflect their rectangle in the line y=x. Label it A. Reflect their rectangle in the line y=−x. Label it B. Exchange back and mark each other’s work. Discuss any mistakes. x –1 –4–5 –1–2–3 4 5321 0 –2 –3 –4 –5 y 2 1 3 4 5 11 The diagram shows shapes J, K, L, M, N and P. Choose the correct equation of the mirror line for each of these reflections. a J and K A x=1 B y=3 C x=3 b J and M A x=4 B y=4 C x=5 c M and N A y=3 B y=4 C y=3.5 d K and L A x=3 B y=1 C y=2 e L and P A x=5.5 B y=5.5 C x=8 12 The diagram shows eight triangles, labelled A to H. Identify which of the following are reflections. For each one that is a reflection, write the equation of the mirror line. a triangle A to triangle B b triangle A to triangle C c triangle B to triangle F d triangle B to triangle E e triangle D to triangle E f triangle G to triangle E g triangle C to triangle E h triangle F to triangle G i triangle D to triangle H j triangle E to triangle H x y 5 6 4 3 2 1 0 43210 5 6 7 8 9 M N P J K L x y 5 6 7 8 4 3 2 1 0 43210 5 6 7 8 A C D E GF H B We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
317 14.5 Rotating shapes Look back at Question 1a. a Write the steps you took to draw the reflection of the shape. You might begin with: Step 1: Draw the mirror line. Step 2: b Look back at Question 6ciii. Write the steps you took to draw the reflection of this shape. c Which steps were the same or different for these two questions? Explain why. Summary checklist I can reflect shapes on a coordinate grid in the lines x=‘a number’ and y=‘a number’. I can reflect shapes on a coordinate grid in the lines y=x and y=−x. I can identify a reflection and its mirror line. 14.5 Rotating shapes In this section you will ... • rotate shapes on a coordinate grid and describe a rotation. Key words anticlockwise centre of rotation When you carry out a rotation, or describe a rotation, you need three clockwise pieces of information: • the angle of the rotation • the direction of the rotation (clockwise or anticlockwise) • the coordinates of the centre of rotation. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
318 14 Position and transformation Exercise 14.5 1 Copy each diagram and rotate the shape using the given information. a 90° clockwise centre (2,1) x y 4 3 2 1 0 543210 b 90° anticlockwise centre (−2, 2) x y 4 3 2 1 0 1–1–2–3–4 c 180° centre (−1, −2) x y 1 –1 –1–2–3–4 21 0 –2 2 Worked example 14.5 a Draw the image of this shape after a rotation 90° clockwise about the centre of rotation (−2, −1). x y 1 –1 –1–2–3–4–5 321 0 –2 2 3 4 b Describe the rotation that takes shape A to shape B. 6 4 5 2 3 0 1 0 1 2 3 4 5 x y 6 A B Answer a y 1 –1 –1–2–3–4–5 –2 (–2, –1) 2 3 4 0 321 x × Start by tracing the shape, then put the point of your pencil on the centre of rotation. Turn the tracing paper 90° clockwise, then make a note of where the image is. Draw the image onto the grid. b Rotation is 180° The centre of rotation is at (3, 3). When you describe a rotation, give the number of degrees and the coordinates of the centre of rotation. Note that when the rotation is 180° you do not need to say clockwise or anticlockwise as both give the same result. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
319 14.5 Rotating shapes Think like a mathematician 2 This is how Milosh rotates a shape when the centre of rotation is not on the shape and he doesn’t have tracing paper. Question Rotate the shape 90° clockwise about centre (1, 2). Answer Draw a vertical line from the shape to the centre of rotation. x y 4 5 3 2 1 × 0 43210 Rotate the line 90° clockwise and draw in the new shape. x y 4 5 3 2 1 × 0 43210 90° clockwise x y 4 5 3 2 1 0 43210 Discuss the answers to these questions in pairs or groups. a What do you think of Milosh’s method? Do you think it makes it easier to rotate a shape? Do you think you could use this method? Do you think it would work for all rotations? b Use Milosh’s method or your own method to rotate each shape 90° clockwise about centre (2, 4). Remember, you must not use tracing paper. x y 4 5 3 2 1 0 4 53210 x y 4 5 3 2 1 0 4 5 63210 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
320 14 Position and transformation 3 a Copy each diagram and rotate the shape using the given information. Do not use tracing paper. i x y 4 5 3 2 1 0 4 53210 90° clockwise centre (2, 2) ii x y 1 –1 –1–2–3 0 21 –2 2 5 3 4 90° anticlockwise centre (1, 1) iii x –1 –1–2–3 4 5321 0 –2 –3 –4 –5 y 2 1 5 3 4 180° centre (1, 0) b Use tracing paper to check your answers to part a. 4 This is part of Rohan’s classwork. Question Rotate this parallelogram 90° anticlockwise about centre (2, 3). Answer I have used a dotted line to show the image. x –1 –4 –1–2–3 4321 0 –2 –3 y 2 1 3 4 x –1 –4 –1–2–3 4321 0 × –2 –3 y 2 1 3 4 a What is wrong with Rohan’s answer? b Copy the object onto squared paper and draw the correct image. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
321 14.5 Rotating shapes Think like a mathematician 5 This is part of Marcus’s homework. Question Describe the rotations that take shape A to shape B. a x y –1 2 0 1 3 4 5 5 1 2 A B 3 4 b x y 4 5 3 2 1 0 4 5 63210 A B Answer a Rotation 90° clockwise b Rotation 180° anticlockwise, centre (3, 2) Read what Sofia and Zara say to Marcus. Discuss the answers to these questions in pairs or groups. a Are Sofia and Zara correct? Explain your answers. b Look again at part a of Marcus’s homework. You can see that the centre of rotation is at (1, 1) because this point is the same on both the image and the object. How can you work out the centre of rotation when no point is the same on both the image and the object (for example, in part b of Marcus’s homework)? Tip Try drawing lines to corresponding vertices on the object and the image. You haven’t given enough information in part a. You have given too much information in part b. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
322 14 Position and transformation 6 The diagram shows seven triangles. Match each rotation with the correct description. a A to B i 90° clockwise, centre (3, 5) b B to C ii 180°, centre (4, 1) c C to D iii 180°, centre (6, 5) d C to E iv 90° anticlockwise, centre (3, 8) e F to G v 180°, centre (4, 4) 6 7 8 4 5 2 3 0 1 0 1 2 3 4 5 x y 6 7 8 C A B D G E F Continued c Use your answer to part b to work out the centre of rotation in each diagram. Notice that both rotations are 180°. i x −1 −4 −3 4321 0 −2 y 2 1 3 4 A B −2 −1 ii x y 4 5 3 2 1 0 3210 4 5 6 7 8 9 A B d Complete this sentence: ’I can find the centre of a 180° rotation by ...................’ e Does your method for finding the centre of a 180° rotation work for a 90° rotation? Test your answer on these two diagrams. i x y 4 5 3 2 1 0 4 53210 A B ii x y 4 5 3 2 1 0 3210 4 5 6 7 B A f Describe a method you can use to work out the centre of rotation for a 90° rotation. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
323 14.5 Rotating shapes 7 The diagram shows triangles R, S, T, U, V and W on a coordinate grid. Describe the rotation that transforms a triangle R to triangle S b triangle S to triangle T c triangle T to triangle U d triangle U to triangle V e triangle V to triangle W. 8 The diagram shows seven shapes labelled A to G. Here are seven cards labelled i to vii. Each card shows a rotation of one shape to another. For example, card i means rotate shape A to shape B. i A to B ii A to C iii C to B iv B to D v E to B vi G to A vii D to F a Put the cards into groups using one property of the rotations. Describe the property of each group. b Sort the cards into different groups using a different property of the rotations. Describe the property of each group. 9 Describe the rotation that transforms S to T in each diagram. a x y 4 5 3 2 1 0 4 5 63210 T S b –1–2–3–4–5 1 0 2 3 4 1 y x 5 S T c –1–2–3–4–5 1 0 2 3 4 1 2 3 4 y x S T –1–2–3–4 –3 –4 –2 –1 1 0 R T V S W U 2 3 4 1 2 3 4 x y x y 4 5 6 7 3 2 1 0 3210 4 5 6 7 8 9 10 11 B A C E D F G Tip You could use the angle, the direction, or the centre of rotation. Summary checklist I can rotate shapes on a coordinate grid. I can describe rotations on a coordinate grid. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
324 14 Position and transformation 14.6 Enlarging shapes In this section you will … • enlarge shapes using a positive whole number scale factor from a centre of enlargement. Key words centre of enlargement enlargement An enlargement of a shape is a copy of the shape that changes the scale factor lengths but keeps the same proportions. In an enlargement, all angles stay the same size. Look at these two rectangles. object 2cm 4 cm 1cm 2cm image The image is an enlargement of the object. Every length on the image is twice as long as the corresponding length on the object. The scale factor is 2. The centre of enlargement tells you where to draw the image on a grid. In this case, as the scale factor is 2, not only must the image be twice the size of the object, it must also be twice the distance from the centre of enlargement. You can check you have drawn an enlargement correctly by drawing lines through the corresponding vertices of the object and image. All the lines should meet at the centre of the enlargement. This is also a useful way to find the centre of enlargement if you are only given the object and the image. 3cm 6cm centre of enlargement centre of enlargement We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
14.6 Enlarging shapes 325 Worked example 14.6 Draw enlargements of the following triangles using the given scale factors and centres of enlargement, marked with a red dot. a scale factor 2 b scale factor 3 Answer a Start by looking at the corner of the triangle that is closest to the centre of enlargement (COE). This corner is 1 square to the right of the COE so, with a scale factor of 2, the image will be 2 squares to the right of the COE. Plot this point on the grid, then complete the triangle. Remember to double all the lengths. b One of the corners of this triangle is on the centre of enlargement, so this corner doesn’t move. Look at the bottom right corner of the triangle. This corner is 1 square to the right and 1 square down from the COE. With a scale factor of 3, the image will be 3 squares to the right and 3 squares down from the COE. Plot this point on the grid, then complete the triangle. Remember to multiply all the other lengths by 3. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
326 14 Position and transformation Exercise 14.6 1 Copy each shape onto squared paper. Enlarge each shape using the given scale factor and centre of enlargement. a scale factor 2 b scale factor 3 c scale factor 4 d scale factor 2 e scale factor 3 f scale factor 4 2 This is part of Geraint’s homework. Question Enlarge this triangle using a scale factor of 2 and the centre of enlargement shown. Answer a Explain Geraint’s mistake. b Make a copy of the triangle on squared paper. Draw the correct enlargement. Tip Make sure you leave enough space around your shape to complete the enlargement. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
327 14.6 Enlarging shapes 3 The vertices of this triangle are at (2, 2), (2, 3) and (4, 2). a Copy the diagram on squared paper. Mark with a dot the centre of enlargement at (1, 1). Enlarge the triangle with scale factor 3 from the centre of enlargement. b Write the coordinates of the vertices of the image. Think like a mathematician 4 Marcus and Arun enlarge this square using scale factor 3. Marcus uses a centre of enlargement at (1, 1). Arun uses a centre of enlargement at (0, 1). Read what Marcus and Arun say. Work with a partner or in a small group to answer these questions. a Make two copies of the grid above and enlarge the square using scale factor 3 with i Marcus’s centre of enlargement ii Arun’s centre of enlargement. b Look at the diagrams you draw for part a. What do you think Marcus and Arun mean by ’invariant points’? c Describe where a centre of enlargement must be, for you to have one invariant point. d Describe where a centre of enlargement must be, for you to have no invariant points. Discuss your answers with other groups in your class. x 0 1 2 3 4 5 0 1 2 3 4 5 6 7 y There is one invariant point on my object and image. There are no invariant points on my object and image. x 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 11 y We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
328 14 Position and transformation 5 The vertices of this trapezium are at (3, 2), (7, 2), (5, 4) and (4, 4). a Copy the diagram onto squared paper. Mark with a dot the centre of enlargement at (5, 2). Enlarge the trapezium with scale factor 2 from the centre of enlargement. b Write the coordinates of the vertices of the image. c Write the coordinates of the invariant point. Activity 14.6 Work with a partner for this question. Read the instructions before you start. a On a coordinate grid, draw a quadrilateral of your choice. b Ask your partner to enlarge your quadrilateral by a scale factor of your choice. Give them the coordinates of the centre of enlargement, which must be somewhere on the perimeter of the quadrilateral. You must make sure the enlarged shape will fit on the coordinate grid. c Check each other’s work and discuss any mistakes. 6 Each diagram shows an object and its image after an enlargement. For each part, write down the scale factor of the enlargement. a x 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 y image object b x 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 y image object x 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 y We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
329 14.6 Enlarging shapes 7 The diagram shows shape ABCD and its image A′B′C′D′. a Write the scale factor of the enlargement. Read what Marcus and Zara say: b Who is correct? Explain how you worked out your answer. Think like a mathematician 8 Work with a partner or in a small group to answer these questions. Zara drew a triangle with vertices at (1, 1), (2, 1) and (1, 3). She enlarged the shape by a scale factor of 3, centre (0, 0). Read what Zara said. a Show, by drawing, that in this case Zara is correct. Read what Arun said. b Use a counter-example to show that Arun is wrong. c What are the only coordinates of the centre of an enlargement where you can multiply the coordinates of the vertices of the object to get the coordinates of the vertices of the image? This means that, for any enlargement, with any scale factor and centre of enlargement, I can multiply the coordinates of each vertex by the scale factor to work out the coordinates of the enlarged shape. Tip A counterexample is just one example that shows a statement is not true. –1–2–3–4–5 –2 –3 –4 –5 –1 1 0 2 3 4 5 1 2 3 4 5 y x A A9 B9 C9 D9 B C D Summary checklist I can enlarge shapes using a positive whole number scale factor from a centre of enlargement. I think the centre of enlargement is at (−3, −4). I think the centre of enlargement is at (−4, −3). If I multiply the coordinates of each vertex by 3 it will give me the coordinates of the enlarged triangle, which are at (3, 3), (6, 3) and (3, 9). We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
330 14 Position and transformation Check your progress 1 Draw diagrams to show these bearings of B from A. For each one, start with the diagram on the right. a 045° b 170° c 215° d 340° 2 Work out the midpoint of the line segment joining each pair of coordinates a (3, 7) and (9, 7) b (2, 8) and (11, 6) 3 The diagram shows shapes M, N, P and Q on a coordinate grid. Write the column vector that translates: a shape M to shape N b shape N to shape Q c shape Q to shape P d shape P to shape M. 4 Make two copies of this diagram. a On one copy, reflect the shape in the line x=7. b On the other copy, reflect the shape in the line y=3. 5 The diagram shows shapes A, B and C. Describe the rotation that takes a A to B b A to C. 6 Copy this shape onto squared paper. Enlarge the shape using scale factor 3 and the centre of enlargement shown. N A x –1–2–3 –2 –3 –1 1 0 2 3 4 1 2 3 y P N Q M x 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 y x –1–2–3–4 –2 –3 –4 –1 1 0 2 3 4 1 2 3 y B A C We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
331 Getting started 1 Work out these calculations. a 1 8 × 32 b 5 8 × 32 c 1 5 × 15 d 8 5 × 15 2 Work out the area of each shape. a 12cm 5 cm b 8m 7m 3 The diagram shows a cuboid. Work out a the volume of the cuboid b the surface area of the cuboid. 6cm 4 cm 3cm 15 Distance, area and volume The metric system that is used today was developed in France, in the late 18th century, by Antoine Lavoisier. At that time, different countries used different units for measuring, which was very confusing. The modern metric system is called the International System of Units (SI) and is now used by about 95% of the world’s population. However, some countries that use the metric system still use some of their old units as well. For example, in the UK, Liberia and the USA, distances and speeds on road signs are shown using miles, not kilometres. Rulers are often marked in both inches and centimetres. Antoine Lavoisier, 1743–1794. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
332 15 Distance, area and volume In this section you will … • convert between miles and kilometres. 15.1 Converting between miles and kilometres In some countries, such as the USA, Liberia and the UK, distances are measured in miles rather than kilometres. A kilometre is a shorter unit of measurement than a mile. One kilometre is about 5 8 of a mile. If the blue line below represents a distance of 1 mile, then the red line represents a distance of 1 kilometre. 1 mile 1 kilometre To convert a distance in kilometres to a distance in miles, multiply by 5 8 . To convert a distance in miles to a distance in kilometres, multiply by 8 5 . Key words kilometre mile Worked example 15.1 a Which is greater, 20 miles or 20km? b Convert 72 kilometres into miles. c Convert 50 miles into kilometres. d Which is further, 200km or 120 miles? Answer a 20 miles 1 mile is greater than 1km, so 20 miles is greater than 20km. b 72÷8=9 9×5=45 miles To multiply 72 by 5 8 , first divide 72 by 8, then multiply the answer by 5. c 50÷5=10 10×8=80km To multiply 50 by 8 5 , first divide 50 by 5, then multiply the answer by 8. d 200÷8=25 25×5=125 miles 200km is further Convert 200km into miles (or 120 miles intokm) so the units are the same. 125 miles is greater than 120 miles, so 200km is further than 120 miles. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
333 15.1 Converting between miles and kilometres Exercise 15.1 1 Write true (T) or false (F) for each statement. a 15 miles is further than 15km. b 100km is exactly the same distance as 100 miles. c 2.5km is further than 2.5 miles. d 6km is not as far as 6 miles. e In one hour, a car travelling at 70 miles per hour will travel a shorter distance than a car travelling at 70 kilometres per hour. 2 Read what Zara says. Is Zara correct? Explain your answer. 3 Copy and complete these conversions from kilometres to miles. a 64km 64÷8=8 8×5= miles b 40km 40÷8= ×5= miles c 56km 56÷ = × = miles 4 Copy and complete these conversions from miles to kilometres. a 55 miles 55÷5=11 11×8= km b 20 miles 20÷5= ×8= km c 85 miles 85÷ = × = km Think like a mathematician 5 Read what Sofia says. Discuss a strategy Sofia could use to help her decide when she should multiply by 5 8 and when she should multiply by 8 5 6 Convert each distance to miles. a 24km b 48km c 96km d 176km 7 Convert each distance to kilometres. a 10 miles b 100 miles c 125 miles d 180 miles My brother lives 35 km from my house. My sister lives 35 miles from my house. I live closer to my brother than to my sister. When I convert between miles and kilometres, I never know whether to multiply by 5 8 or 8 5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
334 15 Distance, area and volume Think like a mathematician 8 Look at this question: Which is further, 107km or 70 miles? Discuss with a partner or in a small group: a Do you think it is easier to change 107km into miles or 70 miles into km without using a calculator? Explain why. b If you could use a calculator, would this change your answer to part a? Explain why. c When you compare a number of km and a number of miles, explain how you would decide which unit to convert. d Test your answer to part c on these questions: i Which is further, 90 miles or 150km? ii Which is further, 51 miles or 80km? 9 Use only the numbers from the cloud to complete these statements. a 120km= miles b 105 miles= km c km= miles d miles= km 115 140 75 224 184 168 Activity 15.1 Hamza and Inaya use different methods to convert 23 miles into kilometres. This is what they write. Hamza 23 × 8 5 =23÷5 × 8 23÷5=4 3 5 4 3 5 × 8=4 × 8+ 3 5 × 8 = 32+ 24 5 = 32 +4 4 5 =36 4 5 km Inaya 23 × 8 5 =23 × 1 3 5 = 23 × 1+23 × 3 5 = 23+ 69 5 = 23+13 4 5 = 36 4 5 km a Whose method do you prefer, Hamza’s or Inaya’s? Explain why. b Can you think of a better method? c Discuss your answers to parts a and b with other learners in your class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
335 15.1 Converting between miles and kilometres 10 Work out the missing numbers in these conversions. Use your preferred method. a 17 miles= km b 33 miles= km c 54 miles= km d 28km= miles e 42km= miles f 75km= miles 11 Every car in the USA is fitted with a milometer. The milometer shows the total distance a car has travelled. Evan is a salesman. This is the reading on his car’s milometer at the start of one week. 125465 miles This is the reading on his car’s milometer at the end of the week. 126335 miles a How many kilometres has Evan travelled in this week? b Evan is paid 20 cents for each kilometre he travels. This is to pay for the fuel he uses. Evan works out that, in this week, he will be paid more than $250 for the fuel he uses. Is Evan correct? Explain your answer. Tip Give each answer as a mixed number in its simplest form. Summary checklist I can convert between miles and kilometres. In this section you have learned to convert between miles and kilometres. a Match each statement to the correct method. A Convert from miles tokm i × 5 8 B Convert from km to miles ii × 8 5 b Explain to a partner how you remember these methods. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
336 15 Distance, area and volume Look at this parallelogram. Imagine you cut off a triangle from the left end of the parallelogram and move it to the right end. You can see that you have made a rectangle. So the area of the parallelogram is the same as the area of the rectangle with the same base and perpendicular height. You can write the formula for the area of a parallelogram as: area=base×height or simply A=bh base height Now look at this trapezium. The lengths of its parallel sides are a and b. Its perpendicular height is h. a b h Two trapezia can be put together like this to make a parallelogram with a base length of (a+b) and a height h. The area of the parallelogram is: area=base×height=(a+b)×h The area of one trapezium is half the area of the parallelogram. So, the area of a trapezium is: A=1 2 ×(a+b)×h a + b h 15.2 The area of a parallelogram and a trapezium In this section you will … • derive and use the formulae for the area of a parallelogram and a trapezium. Key word trapezia We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
337 15.2 The area of a parallelogram and a trapezium Worked example 15.2 Work out the area of each shape. a 7 cm 5 cm b 18mm 12mm 8mm Answer a A=bh=7×5 = 35cm2 Write the formula, then substitute the values of b and h. Work out the answer. Remember to include the units (cm2 ). b A = + a b = + = = = 1 2 1 2 1 2 12 18 8 30 8 15 8 120 2 × × × × × × × ( ) ( ) h mm Write the formula. Substitute the values of a, b and h. Work out 12+18=30 first. Then work out 1 2 of 30=15 Finally work out 15×8 Remember to include the units (mm2 ) with your answer. Exercise 15.2 1 Copy and complete the workings to find the area of each parallelogram. a 8cm 4cm b 6m 1.5m A=bh=8×4= cm2 A=bh= × = m2 2 Copy and complete the workings to find the area of each trapezium. a 8 cm 5cm 6cm A=1 2 ×(a+b)×h=1 2 ×(6+8)×5 = 1 2 × ×5= ×5 = cm2 b 12mm 7mm 4mm A=1 2 ×(a+b)×h=1 2 ×( + )×7 = 1 2 × ×7= ×7 = mm2 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
338 15 Distance, area and volume 3 This is part of Bembe’s homework. Question Work out the area of this parallelogram. Answer Area=bh = 7 × 5 = 35 cm2 7cm 4 cm 5 cm a Explain the mistake Bembe has made. b Work out the correct answer. 4 Sofia, Marcus and Zara are discussing the methods they use to find the area of a trapezium. I work out a+b, then divide by 2, then multiply by h. I work out half of h, then work out a+b, then multiply my two answers together. I work out a+b, then multiply by h, then divide by 2. Will they all get the same answer? Explain why. Think like a mathematician 5 Work with a partner to answer this question. Look back at the methods used by Sofia, Marcus and Zara in Question 4. Whose method would it be best to use to find the areas of these trapezia? Explain why. a a=6cm, b=4cm, h=3cm b a=7cm, b=4cm, h=6cm c a=2m, b=3m, h=5m d a=16mm, b=14mm, h=12mm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
339 15.2 The area of a parallelogram and a trapezium 6 Work out the area of each trapezium using the method shown. Look back at Question 4 to check the method. 7 This is part of Zalika’s homework. Question What is the difference in area between these two shapes? A 15mm 12mm B 10cm 9cm 6cm Answer Area A=b × h=15 × 12=180 Area B= 1 2 × (a+b) × h= 1 2 × (6+10) × 9 = 1 2 × 16 × 9=8 × 9=72 Difference=180 – 72=108 a Explain the mistake Zalika has made. b Work out the correct answer. a Sofia’s method 3m 5m 7m b Marcus’s method 9cm 8cm 6cm c Zara’s method 9mm 6mm 5mm Think like a mathematician 8 Work with a partner or in a small group to discuss this question. Read what Zara says. Is Zara correct? Explain your answer. If you double the base length of a parallelogram and double the height of the parallelogram, the area of the parallelogram will be doubled. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
340 15 Distance, area and volume 9 The diagram shows a trapezium. a Work out an estimate of the area of the trapezium. b Use a calculator to work out the accurate area of the trapezium. 10 Here are four shapes, A, B, C and D. Here are five area cards. i 9.86 cm2 ii 18.81 cm2 iii 24.48cm2 iv 15.54 cm2 v 11.07 cm2 a Using estimation only, match each shape with the correct area card. b Use a calculator to check your answers to part a. c Sketch a shape that has an area equal to the area on the card you have not matched. 11 A parallelogram has an area of 832mm2 . It has a perpendicular height of 2.6 cm. What is the length of the base of the parallelogram? Tip To work out an estimate, round all the numbers to one significant figure. 9.8cm 4.6 cm 2.3cm A 4.5cm 3.8 cm 5.4cm B 4.2cm 3.7 cm C 2.9cm 3.4cm D 2.7cm 8.2cm Tip Be careful with the units. Summary checklist I can derive and use the formula for the area of a parallelogram. I can derive and use the formula for the area of a trapezium. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
15.3 Calculating the volume of triangular prisms 341 You already know how to work out the volume of a cuboid by multiplying the length (l) by the base (b) by the height (h). You can multiply the dimensions in any order. A cuboid is a rectangular prism. A prism is a 3D shape which has the same 2D shape throughout its length. This 2D shape is called the cross-section of the prism. When you work out the volume of a cuboid, if you start with b×h, you find the area of the rectangular cross-section of the cuboid. When you multiply this area by the length l, you get the volume of the cuboid. The diagram shows a triangular prism. You can see that the cross-section of the prism is a triangle. You can work out the volume of the prism using the formula: volume=area of cross-section×length l h b b l h b l h 15.3 Calculating the volume of triangular prisms In this section you will … • derive and use the formula for the volume of a triangular prism. Key words cross-section prism We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
342 15 Distance, area and volume Exercise 15.3 1 Copy and complete the workings to find the volume of each triangular prism. a 6cm 10cm 8cm Area of cross-section=1 2 ×b×h = 1 2 ×6×8 = cm2 Volume=area of cross-section×length = ×10 = cm3 b 3m 7m 4m Area of cross-section=1 2 ×b×h = 1 2 × × = m2 Volume=area of cross-section×length = ×7 = m3 2 Work out the volume of each triangular prism. a 9cm 30 cm 12cm b 7m 2m 6m Worked example 15.3 Work out the volume of this triangular prism. Answer area cm = = = 1 2 1 2 4 6 12 2 × × × × b h First, work out the area of the triangular cross-section. Substitute in the values. The lengths are all in cm, so the area is in cm2 . volume area length cm = = = × ×12 15 180 3 Multiply the area of the cross-section by the length. Substitute in the values. The area is in cm2 and the length is in cm, so the volume is in cm3 . 4cm 15 cm 6 cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
343 15.3 Calculating the volume of triangular prisms Think like a mathematician 3 Yari and Mike use different methods to work out the volume of this triangular prism. This is what they write. Yari Area of cross-section= 1 2 × b × h = 1 2 × 5 × 7=17.5mm2 Volume of prism=area × length = 17.5 × 20 = 350mm3 Mike Volume of cuboid=length × base × height = 20 × 5 × 7=700mm2 Volume of prism=volume of cuboid÷2 = 700÷2 = 350mm3 Discuss the answers to these questions with a partner or in a small group. a How does Mike’s method work? Why does it give the same answer as Yari’s method? b Which method do you prefer? Explain why. c Can you think of a different method you can use to work out the volume of a triangular prism? Discuss your answers with other groups in the class. 20mm 5mm 7mm 4 This is part of Vin’s homework. Question Work out the volume of this triangular prism. Answer Area of cross-section= 1 2 × b × h = 1 2 × 7 × 8=28 cm2 Volume=area × length=28 × 120=3360 cm3 8 cm 120mm 7cm Vin has got the answer wrong. Explain Vin’s mistake and work out the correct answer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
344 15 Distance, area and volume 5 The table shows the base, perpendicular height and length of four triangular prisms. Copy and complete the table. Base Height Length Volume a 4cm 8cm 5mm cm3 b 2cm 15mm 8mm mm3 c 7m 9m 10cm m3 d 30mm 6cm 200mm cm3 Tip Make sure the length, width and height are all in the same units before you work out the volume. Think like a mathematician 6 The diagram shows a compound prism. The compound prism is made of a triangular prism and a cuboid. a Show that the volume of the compound prism is 1920cm3. b Discuss with other learners in the class the method you used to work out the volume. c What do you think is the easiest method to use to work out the volume of a compound prism? Explain why. 8cm 7 cm 10 cm 20 cm 7 Work out the volume of each compound prism. a 4m 6m 3m 2.5m b 15mm 12mm 7mm 8mm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
345 15.3 Calculating the volume of triangular prisms 8 The diagram shows a triangular prism. The volume of the prism is 96 cm3 . a Work out the area of the shaded triangle. b Copy and complete these possible dimensions for the shaded triangle: Option 1: base= cm and height= cm Option 2: base= cm and height= cm c Compare your answers to part b with those of a partner. Did you have the same base and height measurements, or were they different? Discuss the number of different possible combinations. 9 The diagram shows a triangular prism. The volume of the prism is 168m3 . a Work out the height of the triangle. b Compare the method you used to answer part a with other learners in the class. Which method do you think is best to use to answer this type of question? Explain why. 10 A triangular prism has a base of 10 cm, a height of 6 cm and a length of 15 cm. a Work out the volume of the triangular prism. b Work out the dimensions of three other triangular prisms with the same volume. 8 cm base height Tip Choose your own values for the base and height of the triangle that will give the area you found in part a. 4m 12m height Activity 15.3 Work with a partner to answer this question. On a piece of paper, draw two triangular prisms like those in question 1. Make sure you write all the dimensions on your prisms. a On a different piece of paper, work out the volume of each prism. b Exchange pieces of paper with your partner and work out the volume of each of their prisms. c Exchange back and mark each other’s work. Discuss any mistakes. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
346 15 Distance, area and volume 11 The diagram shows a triangular prism made from silver. Jan is going to melt the prism and make the silver into cubes. The side length of each cube is 8mm. Jan thinks he can make nine cubes from this prism. Is Jan correct? Explain your answer. Show all your working. 30mm 12mm 25mm Summary checklist I can derive and use the formula for the volume of a triangular prism. In this lesson you have looked at volumes of triangular prisms. What do you think is the most important thing to remember when working out volumes of triangular prisms? In this section you will … • calculate the surface area of triangular prisms and pyramids. You already know how to draw the net of a cube or cuboid to help you work out the surface area of the shape. You can use the same method to help you work out the surface area of triangular prisms and pyramids. 15.4 Calculating the surface area of triangular prisms and pyramids Key words net surface area We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
347 15.4 Calculating the surface area of triangular prisms and pyramids Worked example 15.4 The diagram shows a triangular prism. a Sketch a net of the prism. b Work out the surface area of the prism. Answer a 6cm A E D 4 cm B C 5cm 8cm 5cm The prism has a rectangular base (A), measuring 8 cm by 6cm. It has two rectangular faces (B and C) that measure 8 cm by 5 cm. It has two triangular faces (D and E), each with base length 6cm and perpendicular height 4cm. b Area A=l×w=8×6 = 48cm2 Work out the area of rectangle A. Area B =l×w=8×5 = 40cm2 Work out the area of rectangle B. Note that C has the same area as B. Area D =1 2 bh=1 2 ×6×4 = 12cm2 Work out the area of triangle D. Note that E has the same area as D. Surface area=48+40×2+12×2 = 48+80+24 = 152cm2 Add the areas together. Remember to include 40×2 and 12×2. Remember the units (cm2 ). 5 cm 8cm 6cm 5cm 4cm Exercise 15.4 1 Copy and complete the workings to find the surface area of each shape. b 14 cm 10 cm a 8 cm 6cm 12cm 10 cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
348 15 Distance, area and volume 12 cm 12cm 10cm 8cm 6cm 8 cm 12 cm A D B E C Area of A=8×12=96cm2 Area of B= × = cm2 Area of C= × = cm2 Area of D=1 2 ×6× = cm2 Area of E=Area of D Surface area=96+ + +2× = cm2 A C B E D 14cm 10cm 10cm Area of A=10× = cm2 Area of B=1 2 ×10× = cm2 Area of C, D and E=Area of B Surface area= +4× = cm2 2 For each of these solids i sketch a net ii work out the surface area. a triangular prism (isosceles) 13 cm 30cm 24cm 5cm b triangular prism (right-angled triangle) 9cm 10 cm 8cm 6 cm c square-based pyramid (all triangles equal size) 18 cm 12 cm d triangular-based pyramid (all triangles equal size) 15cm 13 cm 3 The diagram shows a triangular prism and a cube. Which shape has the greater surface area? Show your working. 15 cm 6 cm 7cm 5cm 4cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
349 15.4 Calculating the surface area of triangular prisms and pyramids 4 The diagram shows a triangular-based pyramid and a cuboid. In the triangular-based pyramid, all triangles are the same size. Show that the surface area of the triangular-based pyramid is 8m2 more than the surface area of the cuboid. 2m 1.5m 2m 4m3.5 m Think like a mathematician 5 This square-based pyramid has a base side length of xcm. a Write an expression for the area of the base of the pyramid. The perpendicular height of each triangular face is double the base side length. b Write an expression for the area of one of the triangular faces of the pyramid. c Write a formula for the surface area of the pyramid. d In Pyramid A x=5. In Pyramid B x=7. Use your formula to work out the difference in surface area between the two pyramids. e Compare and discuss your answers to parts c and d with the rest of the class. x height Activity 15.4 When you have answered this question, you will swap your solution with a partner. They will follow the method you have used and check your working. Make sure you set out your solution so it is easy for your partner to follow. Once you have checked each other’s solutions, discuss each other’s work and give feedback on the methods used. The surface area of this triangular-based pyramid is 249.6cm2. Work out the height of the triangular face (all triangles are the same size). 12cmheight We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE