Cambridge Essentials Mathematics Extension 8 GM1.1 Homework Answers
Original material © Cambridge University Press 2009 1
GM1.1 Homework Answers
1 a a = 58° (corresponding angles are equal)
b = 58° (alternate angles are equal or vertically opposite angles are equal)
c = 122° (angles on a straight line add up to 180°)
b d = 57° (angles in a triangle add up to 180°)
e = 123° (angles on a straight line add up to 180° or
an exterior angle of a triangle equals the sum of the interior opposite angles)
c f = 66° (angles on a straight line add up to 180°)
g = 66° (an isosceles triangle has one pair of equal angles)
h = 48° (angles in a triangle add up to 180°)
2 m = 36° (alternate angles are equal)
n = 96° (angles on a straight line add up to 180°)
p = 48° (angles in a triangle add up to 180°)
3 a = 55° (angles in a triangle add up to 180°)
b = 125° (angles on a straight line add up to 180°)
c = 30° (angles in a triangle add up to 180°)
4 x = 60° (the angles of equilateral triangle ABC are equal and add up to 180°)
y = 25° (the angles of triangle ACD add up to 180°; ∠CAD = ∠CDA)
z = 35° (triangle ABC is equilateral, so ∠BAC = 60° = z + y)
5 p = 133° (angles in a quadrilateral add up to 360°; angles on a straight line add up to 180°)
6 a = 60° (the angles of an equilateral triangle are equal and add up to 180°)
b = 10° (each angle in a square is 90°, so b + 60° + 20° = 90°)
c = 70° (the angles of a triangle add up to 180°, so c + 20° + 90° = 180°)
d = 70° (vertically opposite angles are equal)
Cambridge Essentials Mathematics Extension 8 GM1.1 Homework Answers
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7 p = 55° (p and 35° together make the right angle)
q = 102° (angles round a point add up to 360°)
r = 151° (angles on a straight line add up to 180°)
s = 52° (angles of in quadrilateral add up to 360°)
8 Sum of the interior angles of a pentagon = (5 – 2) × 180° = 540°
So a = 540 ÷ 5 = 108°.
Sum of angles on a straight line = 180° = a + b
So b = 180° – 108° = 72°.
(Or b is an exterior angle or a regular pentagon, so b = 360° ÷ 5 = 72°.)
Sum of angles in a triangle = 180° = b + b + c (isosceles triangles have two equal angles).
So c = 180° – (2 × 72°) = 36°.
Cambridge Essentials Mathematics Extension 8 GM1.2 Homework Answers
Original material © Cambridge University Press 2009 1
GM1.2 Homework Answers
1 a Name of
quadrilateral Do the diagonals always intersect at right angles?
Number of axes of symmetry
Order of rotational symmetry
rectangle no 2 2
rhombus yes 2 2
parallelogram no 0 2
kite yes 1 1
square yes 4 4
trapezium no 0 1
b Name of quadrilateral symmetry rotational oforder
symmetryofaxesofnumber
rectangle 1
rhombus 1
parallelogram 0
kite 1
square 1
trapezium 0
The ratio is always 0 or 1. The number of axes of symmetry is either zero or it is equal
to the order of rotational symmetry.
c i
Isosceles trapezium
ii One axis of symmetry
iii Order 1
Cambridge Essentials Mathematics Extension 8 GM1.2 Homework Answers
Original material © Cambridge University Press 2009 2
2 a E (3, 3) b F (1, 0) c G (1, 5) or (5, –9) or (–3, 1)
3 a (5, 7) b (–1, –3)
4 (3, 4)
5 (9, –10)
6 Pupils’ drawings
Cambridge Essentials Mathematics Extension 8 GM1.3 Homework Answers
Original material © Cambridge University Press 2009 1
GM1.3 Homework Answers
1 a–d A
B C
X
e AX = 3.2 cm
2 M
NL
3 a–c X
ZY35°
P
Q
d YQ = 4.3 cm
Cambridge Essentials Mathematics Extension 8 GM1.3 Homework Answers
Original material © Cambridge University Press 2009 2
4
F H
G
5 Construction of circumcircle of triangle of pupil’s choice.
6
4 cm
5 cm
12 cm
Cambridge Essentials Mathematics Extension 8 GM2.1 Homework Answers
Original material © Cambridge University Press 2009 1
GM2.1 Homework 1 Answers
1 a 12 cm2 b 25 cm2 c 13.5 cm2 d 27.5 cm2
2 18 cm
3 a 6 cm2 b 30 cm2 c 12 cm
d i True. Area of R = Area of Q = half the area of the 5 cm by 12 cm rectangle
ii False. Area of S = (4 × 12) cm2 − area of P = (48 − 6) cm2 = 42 cm2
iii True. P of AreaS of Area =
2
2
cm6
cm 42 = 7
iv True. Q and P of Area
rectangle of Area = 2
2
cm36
cm 108 = 3
4 a i 35 cm2 ii 36 cm2
b i 70 cm2 ii 30 cm2
5 a x = 7; the marked length is 7 cm.
b x = 6; the marked length is 6 cm.
c x = 2; the marked lengths are 2 cm and 10 cm.
Cambridge Essentials Mathematics Extension 8 GM2.1 Homework Answers 2
Original material © Cambridge University Press 2009 1
GM2.1 Homework 2 Answers
1 a 57 cm2 b 222 cm2 c 87 cm2 d 41 cm2
2 a 900 mm2 b 459 mm2 c 47 000 mm2 d 2.9 mm2
3 a 160 cm2 b 49 cm2 c 0.55 cm2 d 3.8217 cm2
4 Jodie is not correct. 1 m = 100 cm, but 1 m2 = 1002 cm2 = 10 000 cm2
5 a i 39 600 mm2 ii 396 cm2
b i 88 250 mm2 ii 882.5 cm2
c Sarah is not correct. letter large a of area maximumletter normal a of area maximum = 396
882.5 ≈ 0.42 < 0.5
Cambridge Essentials Mathematics Extension 8 GM2.2 Homework Answers
Original material © Cambridge University Press 2009 1
GM2.2 Homework Answers
1 a 105 cm3 b 64 cm3 c 6 cm3
2 a x = 5; marked side length is 5 cm.
b x = 5; marked side lengths are both 5 cm.
c x = 5; marked side length is 5 cm.
3 a 158 cm2 b 90 cm2 c 150 cm2
4 a Volume = 120 cm3
b Area of A = 30 cm2
c Area of B = 40 cm2
Area of C = 8 cm2
Area of D = 20 cm2
d Total surface area = 196 cm2
5 a 729 cm3 b 486 cm2 c PofvolumeQ of volume = 27 d
P of area surfaceQ of area surface = 9
6 a Cross-sectional area = 21 × 10 cm × 12 cm = 60 cm2
Volume = 60 cm2 × 40 cm = 2400 cm3
Surface area = 2 × 60 cm2 + 40 cm × (13 cm + 13 cm + 10 cm) = 1560 cm2
b Cross-sectional area = 36 cm × 5 cm + 6 cm × 5 cm = 210 cm2
Volume = 210 cm2 × 20 cm = 4200 cm3
Surface area = 2 × (210 cm2 + 20 cm × 36 cm + 10 cm × 36 cm) = 2260 cm2
7 a 9000 mm3 b 1800 mm3 c 340 000 mm3 d 42 mm3
8 a 36 cm3 b 8.3 cm3 c 0.055 cm3 d 0.82561 cm3
9 a Pupils’ own estimates.
b Answer to part a × 1000000
c Answer to part b × 1000
Cambridge Essentials Mathematics Extension 8 GM2.3 Homework Answers
Original material © Cambridge University Press 2009 1
GM2.3 Homework Answers
1 A 3 B 2 C 3 D 5 E 2 F 6
G 3 H 4 I 4 J 7 K 1
2 a F, K b C c A d H
3 a i
ii
iii
or
b i
ii
iii
4 There are many possible nets; two examples are shown below.
2 cm
3 cm
5 cm
2 cm
3 cm
5 cm
Cambridge Essentials Mathematics Extension 8 GM2.3 Homework Answers
Original material © Cambridge University Press 2009 2
5 Yes, he can draw the following net on to the 10 cm by 10 cm card.
3 cm2 cm
9 cm
10 cm
5 cm
6 a, b
Base
Back
Top
Front
Side Side
X
X
X
Cambridge Essentials Mathematics Extension 8 GM2.4 Homework Answers
Original material © Cambridge University Press 2009 1
GM2.4 Homework Answers
1 a 3 km = 3000 m b 1800 m = 1.8 km c 4 m = 400 cm
d 320 cm = 3.2 m e 30 cm = 300 mm f 0.26 m = 260 mm
g 780 cm = 0.0078 km h 38 mm = 3.8 cm i 625 mm = 0.625 m
j 1.6 km = 160000 cm k 2.1 m = 2100 mm l 0.481 km = 481 m
2 a 4 kg = 4000 g b 2 tonnes = 2000 kg c 275 g = 0.275 kg
d 5.81 kg = 5810 g e 3.5 tonnes = 3500 g f 1384 g = 1.384 kg
3 a 3 m2 = 30 000 cm2 b 4 cm2 = 400 mm2 c 1.5 hectares = 15 000 m2
d 175 mm2 = 1.75 cm2 e 4.2 litres = 4200 ml f 36 000 m2 = 3.6 hectares
g 2 cm3 = 2000 mm3 h 5.26 litres = 5260 cm3 i 5 m3 = 5000 litres
4 a metres b tonnes c hectares d cubic centimetres e millilitres
f kilograms g litres h centimetres i square millimetres
5 a 63 years 7 months 2 days
b i 763.1 months ii 23211 days
6 a 43.75 m2
b 656.25 m3
c 656 250 litres
7 120 km/h ≈ 75m.ph, so Fabrice is breaking the speed limit.
9 a £5.04
b 49p
c 42p
Cambridge Essentials Mathematics Extension 8 GM3.1 Homework Answers
Original material © Cambridge University Press 2009 1
GM3.1 Homework Answers
1 There are four other ways.
2 a
b
c One possibility is
d One possibility is
3 A and J
F and H
I and K
L and O
Cambridge Essentials Mathematics Extension 8 GM3.2 Homework 1 Answers
Original material © Cambridge University Press 2009 1
GM3.2 Homework 1 Answers
1
2
Cambridge Essentials Mathematics Extension 8 GM3.2 Homework 1 Answers
Original material © Cambridge University Press 2009 2
3 a N b K c M d L
4
A′(2, 1), B′(1, 6), C′(7, 4)
A′′(–2, 3), B′′(–1, 8), C′′(–7, 6)
A′′′(3, –6), B′′′(8, –7), C′′′(6, –1)
Cambridge Essentials Mathematics Extension 8 GM3.2 Homework 2 Answers
Original material © Cambridge University Press 2009 1
GM3.2 Homework 2 Answers
1 a, b
c Reflection in the line y = x
2 a, b c Reflection in the line x = 4
ab
Cambridge Essentials Mathematics Extension 8 GM3.2 Homework 2 Answers
Original material © Cambridge University Press 2009 2
3
c (–6, –1)
4 Number of lines of symmetry
0 1 2 3 4 6 None D A Order 2 J G, I Order 3 E B Order 4 F
Rot
atio
n Sy
mm
etry
Order 6 C H
5 Examples of correct answers are given below.
a
b
c
6
Cambridge Essentials Mathematics Extension 8 GM3.3 Homework Answers
Original material © Cambridge University Press 2009 1
GM3.3 Homework Answers
1 Scale factor 3
2
3 a, b c A′ (–2, –2), B′ (10, –2),
C′ (2, 10), D′ (–2, 6)
4 a Scale factor 3 b (2, 10)
5
P
Cambridge Essentials Mathematics Extension 8 GM4.1 Homework Answers
Original material © Cambridge University Press 2009 1
GM4.1 Homework Answers
1 a i 4.5 m ii 3.15 m
b The dishwasher will not fit.
The dishwasher width is 60 cm; actual length of gap = 1.8 × 30 = 54 cm.
2 a 1.2 m b 2.1 m c 10 200 m
3 a 2 cm b 20 cm c 1.6 cm
4 a 600 000 cm2 b 60 m2
5 Please check that lengths on pupils’ diagrams match the given dimensions.
8.4 cm
7.8 cm
2.4 cm 2.4 cm
3.3 cm
2.1 cm
2.1 cm
Cambridge Essentials Mathematics Extension 8 GM4.2 Homework Answers
Original material © Cambridge University Press 2009 1
GM4.2 Homework Answers
Check measurements on pupils’ diagrams are as given throughout. 1 a i
3 cm 4 cm
5 cmA B
C
ii Right-angled triangle
b i
X
Y
Z
7 cm 4 cm
10 cm
ii Scalene triangle
iii Obtuse angle
2
A B
C must lie on thearc of points 6 cmfrom point A.
C must lie on thearc of points 5 cmfrom point B.
It is impossible to construct triangle ABC, since the two arcs in the diagram do not
intersect: there is no point that is both 6 cm from A and 5 cm from B.
This is because AB > AC + BC; that is, 12 cm > 6 cm + 5 cm.
Cambridge Essentials Mathematics Extension 8 GM4.2 Homework Answers
Original material © Cambridge University Press 2009 2
3 a, b
8 cm
5 cm5 cm
5 cm 5 cm
P
Q
R
S
c Rhombus
d QS = 6 cm
4
A
Cambridge Essentials Mathematics Extension 8 GM4.3 Homework Answers
Original material © Cambridge University Press 2009 1
GM4.3 Homework Answers
1
7 cm
3 cm
5 cm
P Q
2
X2 cm
4 cm
Cambridge Essentials Mathematics Extension 8 GM4.3 Homework Answers
Original material © Cambridge University Press 2009 2
3 a
P
X
YZ
b XP = 4.6 cm
4 Check pupil’s diagrams. The line OZ
bisects the angle XOY.
The diagram shown is an illustrative
example and OX and OY are not the
correct length.
X
YO
Z
5
10 cm
8 cm
6 cm 4.6 cm
Goat Atetheredhere
Goat Btetheredhere
a
b
b
a The area shaded in grey can be reached by both goats.
b Areas shaded with checks can be reached by neither goat. They lie outside both arcs.
Cambridge Essentials Mathematics Extension 8 GM4.3 Homework Answers
Original material © Cambridge University Press 2009 3
6 NewcastleUpon Tyne
Sheffield
4 cm3 cm
5 cm4 cm
7.6 cm
7 a, b
A
O B
X
Peter’sroute
Archie’sroute
7 cm
4 cm
c On the scale diagram the distance Peter had walked when he meets Archie is 7.8 cm.
The actual distance is 10.9 km.
Cambridge Essentials Mathematics Extension 8 GM4.3 Homework Answers
Original material © Cambridge University Press 2009 4
8
sea
P
Q
R
location of boat
Cambridge Essentials Mathematics Extension 8 GM4.4 Homework Answers
Original material © Cambridge University Press 2009 1
GM4.4 Homework Answers
1 a 128° b 247° c 232° d 324°
2 a 308° b 067° c 052° d 144°
3 a P = 106°
b Bearing of Y from X = 254°
c Bearing of X from Y = 180° – 106° = 074°
d 180°
4 a Check pupils’ diagrams. Measurements should be as indicated. This diagram is not
actual size.
b AC = 64 mm = 6.4 cm
BC = 103 mm = 10.3 cm
c ∠CAB = 180° – 80° = 100°
d Bearing of A from C = 180° + 80° = 260°
Cambridge Essentials Mathematics Extension 8 GM4.4 Homework Answers
Original material © Cambridge University Press 2009 2
5 a, b
5 cm
5 cm
3 cm
8 cm
165°
30°
N
M
E
B
The area coveredby Manchester Airport radar
is the interior of this circle.
The area coveredby Exeter Airportradar is the interiorof this circle.
b ii Yes. B lies within the circle of radius 5 cm, centre M.
c i 5.6 cm
ii 224 km
Cambridge Essentials Mathematics Extension 8 GM4.4 Homework Answers
Original material © Cambridge University Press 2009 3
6 a
X
Y
L P70°
N
300°
N
15°
3 cm
6 cm
4 cm
b LP = 3.4 cm
c 510 m
d 271°
e L is 705 m from Y on a bearing of 329°.
f P is 945 m from X on a bearing of 082°.