HW Chapter 4
HW Chapter 4
HW Chapter 4
4 GEOMETRY
TASK 4.1
Find the angles marked with letters.
1.
53 a 82 b
2.
d c 146
3.
e e
4.
44 67 f g h
112
5.
k j
6.
m l n 72
7.
94 109 124 o
8.
q q q q 156
76
9. a b c d
Write down the value of angle ADB. Give a reason for your answer. Find the value of angle CBD. Give full reasons for your answer.
A
10. a Find the value of angle QPR. b Give full reasons for your answer.
34
C S
123
Q P
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Geometry
TASK 4.2
1.
2.
b 39
3.
4.
110 e g
107 a
47 c d
5.
124 i h k j 143
6.
7.
s p 115 r q
8.
l 86 m n
54
30 u t 86 v
9. a Find the value of angle ABE. b Give full reasons for your answer.
C B 73 E D
45
Q 49 P
10. a Find the value of angle SQR. b Give full reasons for your answer.
68 S
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TASK 4.3
1. Copy and complete below. A pentagon can be split into _ _ _ _ triangles. Sum of interior angles = _ _ _ _ 180 = ____
2. Find the sum of the interior angles of an octagon. 3. Find the sum of the interior angles of a polygon with 15 sides. 4. Copy and complete below. This polygon can be split into _ _ _ _ triangles. Sum of interior angles = _ _ _ _ 180 = _ _ _ _ Add up all the given angles: 126 + 143 + 109 + 94 + 165 = _ _ _ _ angle x = _ _ _ _ In the questions below, find the angles marked with letters.
109 94
5.
6.
7.
145 167
142 c 146
173
TASK 4.4
1. An octagon has 8 sides. Find the size of each exterior angle of a regular octagon. 2. A decagon has 10 sides. a Find the size of each exterior angle of a regular decagon. b Find the size of each interior angle of a regular decagon.
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(9 sides)
4. Find the exterior angles of regular polygons with a 18 sides b 24 sides
5. Find the interior angle of each polygon in Question 4. 6. The exterior angle of a regular polygon is 24. How many sides has the polygon?
c 45 sides
interior angle 24
7. The interior angle of a regular polygon is 162. How many sides has the polygon? 8. In a regular polygon, each exterior angle is 140 less than each interior angle. How many sides has the polygon? 9. Part of a regular dodecagon (12 sides) is shown. O is the centre of the polygon. Find the value of x.
TASK 4.5
1. Prove that triangle QUT is isosceles. Give all your reasons clearly.
P S 116 W T Q 64 U X V R
2. Prove that the sum of the angles in a quadrilateral add up to 360. Give all your reasons clearly.
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3.
a
106 b 123
4.
49 c 58 d
5.
f 64 83 g 78 h
148
6.
58 k 48 j
152 i
7.
8.
B 24 m C AB = AC
A l
9. ABCD is a rectangle. Prove that triangle ABM is isosceles. Give all your reasons clearly.
A 2 cm D 2 cm B M 2 cm
B 2 cm C
10. Prove that angle ADC = angle BAC. Give all your reasons clearly.
TASK 4.6
Remember Pythagoras says a2 + b2 = c 2
b
You will need a calculator. Give your answers correct to 2 d.p. where necessary. The units are cm.
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Geometry
1.
2.
12
13
C M
4 2 x
x 19 4
x 3 4
14
10
x 6
4.
R
B 9 16
6 A P 7
TASK 4.7
You may use a calculator. Give answers to 2 d.p. 1. A rectangle has length 9 cm and width 7 cm. Calculate the length of its diagonal.
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20 m
12 m
3. A ladder of length 75 m reaches 55 m up a vertical wall. How far is the foot of the ladder from the wall? 4. A plane flies 100 km due south and then a further 150 km due east. How far is the plane from its starting point? 5. Calculate the area of this triangle.
39 cm
15 cm
6. A knight on a chessboard moves 2 cm to the right then 4 cm forwards. If the knight moved directly from its old position to its new position, how far would it move?
4 cm
7. Find x.
2 cm
19 m x
3m
8m
8. Calculate the length of the line joining (1, 3) to (5, 6). 9. Calculate the length of the line joining (22, 25) to (3, 7). 10. Find the height of each isosceles triangle below: a b
9 cm
9 cm
12 cm
12 cm
3 cm
3 cm
8 cm
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10 cm 15 cm 10 cm
6 cm
13 cm
10 cm
TASK 4.8
1. Copy the patterns below on squared paper. Shade in as many squares as necessary to complete the symmetrical patterns. The dotted lines are lines of symmetry.
2. Sketch these shapes in your book and draw on all the lines of symmetry.
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For each shape write the order of rotational symmetry (you may use tracing paper).
3.
4.
5.
10.
6.
7.
8.
9.
11. Draw your own shape which has an order of rotational symmetry of 3. 12. Draw a triangle which has no rotational symmetry.
TASK 4.9
1. How many planes of symmetry does this triangular prism have?
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10
TASK 4.10
1. Explain why these two triangles are congruent.
38 38
56 56
3. AB is parallel to DE. BC = CD. Prove that triangles ABC and CDE are congruent.
A C
D A x x X
4. a Prove that triangles ACX and ACY are congruent. b Explain why AY = CX.
B xx C
5. ABCD is a parallelogram. Prove that triangles ABD and CBD are congruent. 6. PR = RS. Prove that triangles PQR and RTS are congruent.
Q P R S T
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7. Triangle BCE is isosceles as shown. AB = ED. a Prove that triangles ABC and CED are congruent. b Explain why angle BAC = angle CDE.
B A C
E
8. PQRS is a kite. Use congruent triangles to prove that diagonal PR bisects angle SPQ:
S
9. Triangle ABC is isosceles with AB = BC. M and N are the midpoints of AB and BC respectively. PQBM and BRSN are both squares. a Prove that triangles BRM and BNQ are congruent. b Explain why MR = NQ.
P Q A M B C N R S
TASK 4.11
4 cm 50 55 5 cm
x 50 20 cm 55
3 cm
A 4 cm
B 14 cm
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8 cm C 5 cm z
20 cm D y 275 cm
5 cm x 2 cm
10 cm
5. Use similar triangles to find y.
12 cm 8 cm 6 cm
6. Use similar triangles to find x in each diagram below. x a b
15 cm 4 cm 10 cm
36 cm 45 cm
x
4 cm
TASK 4.12
1.
In Questions 1 to 3, find x.
3 cm
5 cm
2.
3 cm
3.
9 cm
7 cm
15 cm x
8 cm
x 6 cm x 14 cm
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1 cm E D 625 cm B 2 cm P R x 5 cm
6.
5. a Explain why triangles PQR and STR are similar. b Find the length of ST.
Q 3 cm
12 cm
T
Find x and y.
2x + 5
7.
Find x.
8.
Find x and y.
175 cm 315 cm x y 4 cm 14 cm
3y
4x 2
51 3 cm
x+1
25 cm
6 cm
4 cm
TASK 4.13
1.
2.
volume = 32 cm3 9 cm
These prisms are similar. The total surface area of the smaller prism is 19 cm2. Find the total surface area of the larger prism.
3 cm 45 cm
21 cm
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4. A factory makes two footballs and they charge a fixed price per square metre of leather that is used to cover the football. If they charge $12 for the football of radius 15 cm, how much do they charge for a football of radius 12 cm? 5. Two triangles are similar. The area of the larger triangle is 6 m2 and its base is 5 m. How long (to the nearest cm) is the base of the smaller triangle if its area is 1 m2? 6. The cost of similar bottles of milk is proportional to the volume. A bottle which has a radius of 36 mm costs 75c. a What is the price (to the nearest cent) of the bottle with a radius of 45 mm? b What is the radius (to 3 s.f.) of the bottle which costs $1? 7. These two containers are similar. The ratio of their diameters is 3 : 7. Find the capacity of the smaller container if the larger container has a capacity of 34 litres (give your answer to 3 s.f.)
8. A shop sells bars of soap in various sizes, all of which are similar to each other. The shop charges the same amount per cm3 of soap in each bar. If the bar which is 5 cm long costs 80c then what is the price (to the nearest cent) of the bar which is 8 cm long?
TASK 4.14
1. These two shapes are similar. If the volume of A is 47 cm3, find: a the area ratio b the length ratio c the volume ratio d the volume of B
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2. These two hemispheres are similar. If the surface area of Q is 912 cm3, find a the volume ratio b the length ratio c the area ratio d the surface area of P
3. Two hexagonal prisms are similar. One has a capacity of 54 l and the other has a capacity of 69 l. If the surface area of the smaller one is 19 m2, what is the surface area (to 3 s.f.) of the larger one? 4. A bottle has a surface area of 480 cm2 and a volume of 700 cm3. What is the surface area (to 3 s.f.) of a bottle whose volume is 500 cm3? 5. A towel has a volume of 6100 cm3. The towel shrinks in a tumble drier. It remains a similar shape but its surface area is reduced by 12%. Find the new volume of the towel (to 3 s.f.) 6. Two crystals are similar. Their surface areas are in the ratio 5 : 9. If the volume of the smaller crystal is 87 mm3, find the volume of the larger crystal (to 3 s.f.) 7. Two similar solid statues are made from the same material. The surface area of the larger statue is 2749 cm2 and the smaller statue is 1364 cm2. The cost of the material to make the larger statue is $490. What is the cost of the material to make the smaller statue (to the nearest pence)?
TASK 4.15
1.
Find the angles marked with letters. (O is the centre of each circle.)
a O 78
2.
b 31 42
3.
73 O
4.
58 O d
5.
e 26 O
6.
40
7.
8.
O h 75 g 130 i
32 106
k
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9. In this question, write down all the reasons for your answers. Find a angle OEG b angle OGF c angle OFG d angle FEO e angle EFG
F E 72 O 20 G
10. Find angle BCD. Write down the reasons for your answer.
B O A D C
11. Find x.
3x 20
x + 10
12. Find y.
130 y
TASK 4.16
1.
Find the angles marked with letters. (O is the centre of each circle)
a 78
2.
3.
132
109 e
4.
f 74 O
49 g
b 116
5.
h 28 i
6.
k 126 O j
7.
56 l O
8.
m 82
148
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9. In this question, write down all the reasons for your answers. Find a angle QRS b angle QOS c angle SQO
Q P 110 O R S
10. In this question, write down all the reasons for your answers. Find a angle CDO b angle COD c angle ABC d angle AOC e angle AOD
84 D E
72
O A
11. Find x.
76 x
4x 9
12. Find y.
2y 20
110
TASK 4.17
1.
Find the angles marked with letters. (O is the centre of each circle.)
2.
3.
e O d
4.
O a b 40 c
g f 23
62
35
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5. In this question, write down all the reasons for your answers. Find a angle BOC b angle BDC c angle CAB
O 29 B
6. In this question, write down all the reasons for your answers. Find a angle POR b angle PQO
P S 17 O R Q
7. Find x
3x + 20
2x + 30
8. In each part of the question below, find x giving each answer to 1 d.p. a b
O x 12 cm 7 cm x 14 cm
O 5 cm
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TASK 4.18
1. Prove that angle PST = x (ie. the exterior angle of a cyclic quadrilateral equals the opposite interior angle).
P Q x
2. O is the centre of the circle. Copy and complete the statements below to prove that the angle at the centre of a circle is twice the angle at the circumference. angle OPQ = angle POQ = (triangle OPQ is isosceles) (sum of angles in a triangle = 180)
Q n m O R
angle ORQ = (triangle ORQ is isosceles) P angle ROQ = (sum of angles in a triangle = 180) angle POR = 360 2 angle POQ 2 angle ROQ (sum of angles at a point add up to 360) POR = 360 2 ( ) 2 ( ) = 360 2 + 2 + = + = 2( + ) This proves that angle POR is twice the angle PQR, ie. the angle at the centre of the circle is twice the angle at the circumference.
TASK 4.19
1. Draw the angle ABC = 70. Construct the bisector of the angle. Use a protractor to check that each half of the angle now measures 35.
B
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2. Draw any angle and construct the bisector of this angle. 3. Draw a horizontal line AB of length 7 cm. Construct the perpendicular bisector of AB. Check that each half of the line measures 35 cm exactly.
4. Draw any vertical line. Construct the perpendicular bisector of the line. 5. Construct accurately the diagrams below: a Measure angle x and side y. b Measure angle x and angle y.
6 cm 55 5 cm 8 cm x 55 cm y 65 cm 5 cm
x 65 cm
5 cm
65 cm y P
6. a b c d
Draw PQ and QR at right angles to each other as shown. Construct the perpendicular bisector of QR. Construct the perpendicular bisector of PQ. The two perpendicular bisectors meet at a point (label this as S). Measure QS.
6 cm
8 cm
TASK 4.20
1. Construct an equilateral triangle with each side equal to 7 cm.
5 cm
3 cm
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b Construct an angle of 90 at A.
4. a Draw a line 10 cm long and mark the point B on the line as shown.
4 cm
6 cm
b Construct an angle of 45 at B. 5. Construct a right-angled triangle ABC, where the angle ABC = 90, BC = 6 cm and angle ACB = 60. Measure the length of AB. 6. Construct this triangle with ruler and compasses only. Measure x.
x 45 8 cm
7. Draw any line and any point A. Construct the perpendicular from the point A to the line.
60
TASK 4.21
You will need a ruler and a pair of compasses. 1. Draw the locus of all points which are less than or equal to 3 cm from a point A. 2. Draw the locus of all points which are exactly 4 cm from a point B. 3. Draw the locus of all points which are exactly 4 cm from the line PQ. 4. A triangular garden has a tree at the corner B. The whole garden is made into a lawn except for anywhere less than or equal to 6 m from the tree. Using a scale of 1 cm for 3 m, draw the garden and shade in the lawn.
5 cm B
18 m
A
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9m
Geometry
22
5. In a field a goat is attached by a rope to a peg P as shown. The rope is 30 m long. Using a scale of 1 cm for 10 m, copy the diagram then shade the area that the goat can roam in.
40 m
P 10 m 50 m
6. Draw the square opposite. Draw the locus of all the points outside the square which are 3 cm from the edge of the square.
4 cm
4 cm
7. Each square is 1 m wide. The shaded area shows a building. A goat is attached by a chain 5 m long to the point A on the outside of the building. Draw the diagram on squared paper then shade the region the goat can cover.
TASK 4.22
You will need a ruler and a pair of compasses. 1. Construct the locus of points which are the same distance from the lines AB and BC (the bisector of angle B).
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2. Faye wants to lay a path in her garden that is always the same distance from KL and KN. Using a scale of 1 cm for 10 m, draw the garden and construct a line to show where the path will be laid.
60 m
L 30 m
N
3. Construct the locus of points which are equidistant (the same distance) from M and N. 4. Draw A and B 7 cm apart.
M N
7 cm
A radar at A has a range of 150 km and a radar at B has a range of 90 km. Using a scale of 1 cm for every 30 km, show the area which can be covered by both radars at the same time. 5. Draw one copy of this diagram. a Construct the perpendicular bisector of FG and the bisector of angle FGH. b Make with a the point which is equidistant from F and G as well as the same distance from the lines FG and GH.
6 cm 70
G 5 cm H
6. Draw the line QR then draw the locus of all the points P such that angle QPR = 90.
6 cm R
7. Draw one copy of triangle ABC and show on it: a the perpendicular bisector of QR. b the bisector of angle PRQ. c the locus of points nearer to PR than to QR and nearer to R than to Q.
Q 5 cm P
7 cm
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TASK 4.23
You will need isometric dot paper. 1. Make a copy of each object below. For each drawing state the number of multilink cubes needed to make the object. a b
2. Draw a cuboid with a volume of 12 cm3. 3. How many more cubes are needed to make this shape into a cuboid?
4. a Draw a cuboid with length 8 cm, width 3 cm and height 1 cm. b Draw a different cuboid with the same volume. 5. Draw this object from a different view.
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TASK 4.24
1. 4 rabbits escape from their run and race off in the directions shown. On what bearing does each rabbit race?
North D 20 35 70 40 C B A
2. Use a protractor to measure the bearing of: a Kailua from Honolulu b Kailua from Wahiawa c Honolulu from Wahiawa d Wahiawa from Kailua e Honolulu from Kailua f Wahiawa from Honolulu
North Wahiawa
Kailua
Honolulu
Note The remaining questions need to be calculated. Do not use a protractor. Give answers to 1 d.p. when appropriate.
3. A plane flies 80 km north and 47 km west. What is the bearing from its original position to its new position?
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4. Baojia runs 5 km east from his home then 8 km north. On what bearing is he to run home if he is to take the shortest route?
North
8 km
home
5 km
5. B is on a bearing of 305 from C. a Find the length of BD. b Find the bearing of A from B if AD = 4 km.
North B 12 km
North P 143
9 km
15 km R
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