Cambridge Essentials Mathematics Core 8 GM1.1 Answers GM1.1 Answers 1 a There is more than one valid reason for each statement; those given are the simplest. i Corresponding angles ii Vertically opposite angles iii Alternate angles iv Alternate angles v Vertically opposite angles vi Corresponding angles b i (b, h), (b, j), (d, h), (c, k), (e, k), (e, i) ii (a, k), (b, l), (c, i), (d, j), (e, g), (f, h) iii (a, c), (a, e), (c, e), (b, d), (b, f), (d, f), (l, h), (l, j), (h, j), (g, i), (g, k), (i, k) 2 There are many valid reasons for each answer; those given below are examples only. a y = 125 ; alternate angles are equal. b p = 130 ; the sum of angles about a point on a straight line is 180, so p + 50 = 180. q = 130 ; p and q are alternate angles, so p = q. r = 50 ; corresponding angles are equal. c a = 70 ; the sum of angles about a point on a straight line is 180, so a + 110 = 180. b = 110 ; vertically opposite angles are equal. c = 110 ; corresponding angles are equal, so c = b. d = 110 ; corresponding angles are equal, so d = c = b. e = 70 ; the sum of angles about a point on a straight line is 180, so d + e = 180. d j = 112 ; corresponding angles are equal. k = 112 ; alternate angles are equal, so k = j. l = 112 ; corresponding angles are equal, so l = k. m = 112 ; alternate angles are equal, so m = l. n = 68 ; the sum of angles about a point on a straight line is 180, so n + j = 180. p = 68 ; the sum of angles about a point on a straight line is 180, so p + l = 180. Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM1.1 Answers 3 a x + 100 + 50 = 180 (angle sum of a triangle), so x = 30. b p + 40 + 120 = 180 (angle sum of a triangle), so p = 20. c a + 142 = 180 (angles on a straight line), so a = 38. b + 66 + 38 = 180 (angle sum of a triangle), so b = 76. d l + 117 = 180 (angles on a straight line), so l = 63. m + 121 = 180 (angles on a straight line), so m = 59. n + 63 + 59 = 180 (angle sum of a triangle), so n = 58. 4 a q + q + q = 3q = 180 (angle sum of a triangle), so q = 60. b 2r + 90 = 180 (angle sum of a triangle), so r = 45. 5 a x + 100 = 140 (exterior angle = sum of interior opposite angles), so x = 40. y + 140 = 180 (angles on a straight line), so y = 40. b m + 125 = 180 (angles on a straight line), so m = 55. n + 50 = 125 (exterior angle = sum of interior angles), so n = 75. c p + 74 = 180 (angles on a straight line), so p = 106. q = r (symmetry of an isosceles triangle) q + r = 74 (exterior angle = sum of interior opposite angles), so q = r = 37. d v + 105 + 40 = 180 (angle sum of a triangle), so v = 35. w = 40 + 105 (exterior angle = sum of interior opposite angles) so w = 145. e d = 25 (symmetry of an isosceles triangle). e + 25 + 25 = 180 (angle sum of a triangle), so e = 130. f = 25 + 25 (exterior angle = sum of interior opposite angles), so f = 50. f v + 90 + 31 = 180 (angle sum of a triangle), so v = 59. w = 31 (symmetry of an isosceles triangle). x = 59 (symmetry of an isosceles triangle). y = 90 + 31 (exterior angle = sum of interior opposite angles), so y = 121. Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM1.1 Answers 6 a k + 95 + 60 + 88 = 360 (angle sum of a quadrilateral), so k = 117. b t + 125 + 90 + 90 = 360 (angle sum of a quadrilateral), so t = 55. c r + 45 = 180 (angles on a straight line), so r = 135. s + 135 + 84 + 90 = 360 (angle sum of a quadrilateral), so s = 51. d z + 62 + 90 + 90 = 360 (angle sum of a quadrilateral), so z = 118. e w = 70 (alternate angles are equal). x = 65 (alternate angles are equal). y + 65 = 180 (angles on a straight line), so y = 115. z + 70 = 180 (angles on a straight line), so z = 110. f a + 53 = 180 (angles on a straight line), so a = 127 b = 53 (alternate angles are equal). c = 53 (corresponding angles are equal). d + 53 = 180 (angles on a straight line), so d = 127. e + 127 + 53 + 127 = 360 (angle sum of a quadrilateral), so e = 53. 7 a j + 90 + 35 = 180 (angle sum of a triangle), so j = 55. k + 90 = 180 (angles on a straight line), so k = 90. l + 90 + 90 + 90 = 360 (angle sum of a quadrilateral), so l = 90. m = 35 (corresponding angles are equal). n + 90 = 180 (angles on a straight line), so n = 90. p = 90 + 35 (exterior angle = sum of interior opposite angles), so p = 125. b d = 35 (alternate angles are equal). e = 35 (symmetry of an isosceles triangle). f + 35 + 35 = 180 (angle sum of a triangle), so f = 110. Original material Cambridge University Press 2009 3
Cambridge Essentials Mathematics Core 8 GM1.2 Answers GM1.2 Answers 1 a Are all the angles the same? or Are all the angles 90?. b rectangle c kite, arrowhead, trapezium 2 a i square, rhombus ii square, rectangle iii square, rectangle, rhombus, parallelogram iv square, rectangle, rhombus, parallelogram, kite v square, rectangle, rhombus, parallelogram vi trapezium, isosceles trapezium vii square, rectangle, isosceles trapezium viii square, rhombus, kite, arrowhead ix kite, arrowhead, isosceles trapezium x rectangle, rhombus, parallelogram xi square, rectangle, rhombus xii square b i There is only one pair of parallel sides. ii It has rotational symmetry of order 2. 3 a (3, 3) b (0, 4) c (4, 0) d (1, 5) e ( 2, 1) f (4, 0) 4 a (2, 0) b (0, 1) c ( 0.5, 3.5) d ( 0.5, 0.5) 5 a (3, 6) b (3, 2) c ( 2, 0.5) d ( 2, 1) e (0.5, 1.5) f (0.5, 1.5) 6 a (9, 4) b (10, 3) c ( 1, 0) Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM1.3 Answers GM1.3 Answers 2 a, b B C A c The point of intersection is equidistant (approximately 6.5 units) from A, B and C. 3 5 Check that 70 angle is accurately drawn and bisected. 6 a, b Y X Z c All three bisectors intersect at a single point. Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM1.3 Answers 7 a, b Q P R Check that the hedge (dotted line) bisects QPS into two angles of 50. S 9 a, b R P R Q B Either of the dashed lines is a correct answer to part b. Check that PQ = QR = 4.7 units and PQR = 45. A 10 a AX = 6 cm b Area = 15 cm 2 11 a, b X Y P W Z c d See diagram. Length = 3.6 cm See diagram. Length = 3.6 cm Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM1.3 Answers 13 a, b, c Y C A X B d AC = 9.4 cm 14 a, b, c, d, e BC = 8.2 cm W Z S R P X Y Q e f Isosceles trapezium PS = QR = 5 cm Original material Cambridge University Press 2009 3
Cambridge Essentials Mathematics Core 8 GM2.1 Answers GM2.1 Answers 1 a 10 cm 2 b 16 mm 2 c 20 m 2 d 30 cm 2 e 13.5 cm 2 f 12.5 cm 2 2 a 5 cm b 10 mm 3 a 12 cm 2 b 24 cm 2 c x = 8 cm 4 a 6 cm 2 b 18 cm 2 c 9 cm 5 a 18 cm 2 b 72 cm 2 c 54 cm 2 6 a 30 cm 2 b 32 cm 2 c 32 cm 2 d 60 cm 2 e 4 cm 2 f 20 cm 2 7 a 26 cm 2 b 14 cm 2 c 36 cm 2 d 138 cm 2 8 a x = 5 b h = 6 cm c x = 20 cm d a = 8 cm 9 a 84 cm 2 b 76 cm 2 c 112 cm 2 d 90 cm 2 10 a 42 cm 2 b 52 cm 2 c 250 cm 2 d 24 cm 2 11 Area = 1 2 14 9 1 2 14 4 = 35 cm 2 12 Area = 2 1 (6 + 11) 8 + 2 1 (11 + 8) 4 = 106 m 2 13 a 38.5 cm 2 b 132 cm 2 c i 2x ii 2x iii 4(10 x) iv 40 m 2 Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM2.2 Answers GM2.2 Answers 1 a 48 cm 3 b 9.5 cm 3 c 54 m 3 d 125 cm 3 2 a 92 cm 2 b 40 cm 2 c 138 m 2 d 150 cm 2 3 a 1000 mm 3 b 1 cm 3 c Multiply the measurements by 1000. 4 a x = 4 cm b h = 8 cm c w = 0.2 cm or 2 mm d l = 7 cm 5 a 1200 cm 3 b 150 cm 2 c B: 40 cm 2 ; C: 120 cm 2 ; D: 80 cm 2 ; E: 40 cm 2 d 860 cm 2 6 a 108 cm 3 b 162 cm 2 7 a 220 cm 3 b 288 cm 2 8 a 68 cm 3 b 120 cm 2 9 a 8 cm 3 b 24 cm 2 c Volume of B = 64 cm 2 = 8 times the volume of A d Surface area of B = 96 cm 2 = 4 times the surface area of A Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM2.3 Answers GM2.3 Answers 1 1, 4, 6, 7, 8, 10, 11 and 15 are prisms. 2 A 11 B 2, 10 C 3 D 9 E 1 F 4, 6, 7 G 5 H 14 I 8 J 13 K 15 L 12 3 A 11 B 2, 7 C 1, 4, 6 D 3, 5, 12 E 9, 13 F 10 G 8, 15 H 14 4 a i ii b i ii c i ii d i ii 5 a A, D, E, F, G b A 5, D 6, E 3, F 6, G 3 Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM2.3 Answers 6 Any two correct nets. Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM2.4 Answers GM2.4 Answers 1 a 8 km = 8000 m b 1500 m = 1.5 km c 8.5 m = 850 cm d 70 cm = 700 mm e 0.65 m = 650 mm f 560 cm = 0.0056 km g 2 hectares = 20 000 m 2 h 5400 cm 2 = 0.54 m 3 i 875 g = 0.875 kg j 1.305 kg = 1305 g k 3.5 litres = 3500 cm 3 l 950 litres = 0.95 m 3 2 Approximate measure Types of measure Item 2 m length door a 1 tonne mass car b 750 ml capacity bottle c 7500 m 2 area football pitch d 100 g mass apple e 10 m length tree f 480 mm 2 area postage stamp g 200 litres capacity bathtub h 1 kg mass bag of sugar i 50 cm 2 area playing card j 15 cm length pencil 3 a i 666 667 hours ii 27 778 days iii 76 years b 19 or 20 leap years (or 18 leap years if the lifetime spans the turn of a century) 4 1 day 8 hours 5 a 21.6 cm b 3240 cm 2 or 0.324 m 2 6 a To the nearest 10 km b To the nearest 10 minutes 7 a 12 gallons b 17 gallons c 18 gallons 8 a i 2.2 lb ii 3.3 lb iii 1.1 lb b i 0.9 kg ii 2.5 kg c i bananas 73p, mushrooms 1.98, grapes 1.25, apples 1.16, potatoes 1.21 ii bananas 80p, mushrooms 1.94, grapes 1.15, apples 1.13, potatoes 1.33 d Greene s Grocers 9 The mile is longer by 100 m. (Accept 110 m.) Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM3.1 Answers GM3.1 Answers For questions 1, 2 and 3 there are many possibilities; those shown below are examples only. 1 2 3 4 A, B, D, F, H, J 5 b c These are parallelograms since opposite sides are equal. These are arrowheads since they have one line of symmetry and one obtuse internal angle. This is a kite since it has one line of symmetry and no obtuse internal angle. Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM3.1 Answers 6 b c This is a rectangle since all its angles are right angles. These are parallelograms since opposite sides are equal. These are isosceles triangles, since each has one line of symmetry. This is a kite since it has one line of symmetry. Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM3.2 Answers GM3.2 Answers 1 a b 2 a b 3 a b Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM3.2 Answers 4 a b 5 a Rotation 180 about (0, 2) b Rotation 90 clockwise about (0, 0) 6 a b 7 a b This is the difference between the translation X to A and the translation X to B. Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM3.2 Answers 8 a b c d 9 180 rotation about (0, 0) 10 a b Original material Cambridge University Press 2009 3
Cambridge Essentials Mathematics Core 8 GM3.2 Answers c d 11 180 rotation about (0, 0) 12 a b 180 rotation about (1, 2) 13 a b 180 rotation about (3, 3) Original material Cambridge University Press 2009 4
Cambridge Essentials Mathematics Core 8 GM3.2 Answers 14 a, b c AA = 2M 1 M 2 d Translation of 2M 1 M 2 to the right 15 a ii rectangle iii All angles are equal and opposite sides are equal. b ii parallelogram iii Opposite angles and opposite sides are equal. 16 (0, 6) 17 a b c Original material Cambridge University Press 2009 5
Cambridge Essentials Mathematics Core 8 GM3.2 Answers 18 a b c The order of the transformations effects the position of the image in these cases. 19 There are many possible answers for each part; below are some general forms (pupils are expected to give a particular example only). 2a a Reflection in the line x = 1 a, followed by translation 3 2b or translation followed by reflection in the line x = 1 + b. 3 3 b Reflection in the line y = c followed by translation 3 2c 3 or translation followed by reflection in the line y = d. 2d 3 p q + 2 c Rotation 90 anticlockwise about (p, q) followed by translation p q 4 or a translation followed by rotation [there are many possibilities, the simplest being 4 translation followed by rotation 90 anticlockwise about (0, 0)]. 2 d Reflection in x = 2.5 followed by reflection in y = 0, or vice versa 5 2r or rotation 180 about (r, s) followed by translation 2s or a translation followed by a rotation (there are many possibilities). Original material Cambridge University Press 2009 6
Cambridge Essentials Mathematics Core 8 GM3.2 Answers 20 Pupils own words to describe the following transformations. a A single rotation about the same centre through the sum of the two individual angles of rotation (taking angles of clockwise rotation as negative) b A single translation that is the vector sum of the two individual translations c A translation in a direction perpendicular to the axes of reflection d A rotation of 180 about the point of intersection of the two axes of reflection 21 Number of lines of symmetry 0 1 2 Rotation symmetry None G D, E, F Order 2 B A, C 22 order 4 order 3 order 6 order 2 order 1, no rotation symmetry order 2 23 a order 5 b order 8 c order 24 a b Original material Cambridge University Press 2009 7
Cambridge Essentials Mathematics Core 8 GM3.3 Answers GM3.3 Answers 1 a enlargement factor 2.2 b enlargement factor 4 2 a b 3 a b 4 a i centre (1, 3) ii enlargement factor 2 b i centre (9, 2) ii enlargement factor 3 5 a A 1 (1, 5), A 2 (2, 10), A 3 (3, 15), A 4 (4, 20), A 5 (5, 25) b A 6 (6, 30), A 10 (10, 50), A n (n, 5n) c B 1 (3, 1), B 2 (6, 2), B 3 (9, 3), B 4 (12, 4), B 5 (15, 5) d B 6 (18, 6), B 10 (30, 10), B n (3n, n) e scale factor 15 f Each side in image n is n times the length of the corresponding side in object 1. g Corresponding angles in object 1 and all the images are the same. Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM4.1 Answers GM4.1 Answers 1 a 4.8 m b 3 m c 2.1 m d 0.4 m 2 a 2.5 m b 2.75 m c 200 m d 42.5 m e 1.5 m f 150 m 3 a 4 cm b 1.25 cm c 24 cm d 36 cm e 51 cm f 10 cm 4 Check that the sides of the rectangle have length 5 cm and 2.5 cm. 5 4.5 cm 3.5 cm 6 cm 13 cm 6 0.75 cm 7 a, b R 3.2 cm 4.8 cm S Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM4.1 Answers 8 a, b W X Z Y c 12.7 m Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM4.2 Answers GM4.2 Answers 1 a i Pupils constructions; check that PQ = QR = PR = 6 cm. ii equilateral triangle b i Pupils constructions; check that XZ = YZ = 6 cm and XY = 4cm. ii isosceles triangle c i Pupils constructions; check that AB = 10 cm, BC = 12 cm and AC = 5 cm. ii scalene triangle 2 b It is impossible to construct a triangle with these side-lengths, because length LM is greater than the sum of lengths LN and MN. 3 ABC, DEF and MNO can be constructed. 4 a, b Pupils constructions c BD = 8.5 cm 5 a, b Pupils constructions c XZ = 4.1 cm 6 a, b Pupils constructions 7 2.5 cm 5 cm 8 cm 8 a E 6.4 cm F b GH = 4.3 cm c GH = 5.4 m 4 cm 5.6 cm H 4.3 cm G Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM4.2 Answers 9 a X b XY = 2.4 m 6.9 cm 8 cm Y Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM4.3 Answers GM4.3 Answers 1 X 5 cm 2 O 3 cm 3 Inner circle has radius 2 cm, outer circle has radius 5cm. A 4 B C 3 cm A D Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM4.3 Answers 5 a, b Q 1 cm R 2 cm P X S 6 a, b Y 3 cm X Z 7 P Q 8 M L 9 a b B 5.2 cm A 5 cm C c The point of intersection is also equidistant from B and C. Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM4.3 Answers 10 a d O 4 cm A B e equilateral triangle 11 Pupils own constructions. 12 a, b, d, e B 22.5 C 45 22.5 Y A X D c f XD = 3 cm YD = 1.1 cm 13 5 cm Original material Cambridge University Press 2009 3
Cambridge Essentials Mathematics Core 8 GM4.3 Answers 14 4 cm 8 cm 4 cm (Not to scale, pupils diagrams should be size 4 cm indicated) 15 a b 8.75 cm 3.75 cm X 6.25 cm 8.75 cm 1.25 cm 5 cm 6.25 cm 6.25 cm Y 16 a, b A B 1.5 2 cm 6 1.5 D C 17 a, b 1.5 cm 3 cm 2 cm 2.5 cm 4 cm Original material Cambridge University Press 2009 4
Cambridge Essentials Mathematics Core 8 GM4.4 Answers GM4.4 Answers 1 a 135 b 075 c 212 d 214 2 a 315 b 255 c 032 d 034 3 a N b 117 A c 297 B 4 a Q b 060 N c 240 P 5 a Pupils own constructions. b 200 6 a Pupils own constructions. b 142 7 a Pupils own constructions. b KM = 10.9 cm, LM = 5.8 cm c MLK = 180 45 = 135 (angles on a straight line) LMK = 180 MLK LKM (angle sum of triangle) = 180 135 22 = 23 d 225 Original material Cambridge University Press 2009 1
Cambridge Essentials Mathematics Core 8 GM4.4 Answers 8 a X Y Z b XZ = 6.0 cm, YZ = 7.7 cm c YXZ = 65 d XZY = 70 9 a, b d e Pupils own constructions. 252, 348 168, 072 c Two points marked on the constructed line, both points to be 4.5 cm from V and W. 10 a Pupils own constructions. BA should be 5 cm long, CA should be 3.5 cm long. b 047 c 2.9 cm d 3.8 km 11 a Carlos b 039 c 184 m 7 cm Ahmed 4.5 cm Brian Original material Cambridge University Press 2009 2
Cambridge Essentials Mathematics Core 8 GM4.4 Answers 12 a, b c LB = 5.6 cm, MB = 7.8 cm B d 28 km from L, 39 km from M L 6.5 cm M 13 a b XY = 1.7 cm 5.5 cm c 2.6 km P X d 010 6 cm Y 3.2 cm e f 020, 7.4 km 108, 8.4 km Q Original material Cambridge University Press 2009 3