7 Geometry and measures: Transformations, constructions and loci 7.1 Congruent triangles Homework 7A State whether each pair of triangles is congruent and, if so, the condition of congruency it satisfies.
4 cm
4 cm 4 cm
7 cm 120°
7 cm
74cm cm 7 cm 120° 4 cm
120° 4 cm
7 cm
11
cm
6 cm
c20° 20° 130° 20° 130° 130° 8 cm 8 cm 8 cm
cm
7 cm 120°
9 cm
120°
6 cm
cm 20° 20° 11 8 cm 8 cm 8 cm 9 cm 30° 30° 9 cm 30°
cm
11
b6 cm
11 9 ccm m
a
191 cm cm
1
11
9 cm
120° 6 cm
6 cm
20°
6 cm
2
Draw a square ABCD. Draw in the diagonals AC and BD. Which triangles are congruent to each other?
3
Draw a kite EFGH. Draw in the diagonals EG and FH. Which triangles are congruent to each other?
4
Draw a rhombus ABCD. Draw in the diagonals AC and BD. Which triangles are congruent to each other?
5
Draw an equilateral triangle ABC. Draw the lines from each vertex to the midpoint of the opposite side. These three lines should all cross at the same point, T, inside the triangle. Which triangles are congruent to each other?
6
In the diagram, AB and CD are parallel and AB = CD.
A
B X
The lines AC and BD intersect at X. D
Prove that triangle ABX and triangle CDX are congruent.
7
Helen says that these two triangles are A congruent because the three angles are 50° the same. C Show that she is wrong.
C
C 36°
5 cm 50° B
P
P
A 5 cm
36°
5 cm B
R
50°
5 cm 36° 50° Q R
7.1 Congruent triangles
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Q
7.2 Rotational symmetry Homework 7B 1
Copy these shapes and write below each one the order of rotational symmetry. You can use tracing paper. a
2
b
d
e
Write down the order of rotational symmetry for each shape. a
3
c
b
c
d
e
Write down the order of rotational symmetry for each symbol. a
b
c
d
e
EE HH II LL EEEE HHHH II II LLLL NNNNQQ SS ZZ QQQQ SSSS ZZZZ
4
The upright capital letter A fits exactly onto itself only once. So, its order of rotational symmetry is 1. This means that it has no rotational symmetry. Copy these capital letters and write the order of rotational symmetry below each one. a
5
b
c
d
e
f
g
h
Draw two copies of the diagram on the right.
a On the first copy, shade in two more squares so that the diagram has rotational symmetry of order 2 and no lines of symmetry.
b On the second copy, shade in two more squares so that the diagram has rotational symmetry of order 1 and exactly 1 line of symmetry.
6
These patterns are taken from old Turkish coins. What is the order of rotational symmetry for each one? a
b
c
7
On a copy of this shape, shade in four more squares so that the shape has rotational symmetry of order 2.
8
Lizzie is drawing shapes that have rotational symmetry of order 3.
d
Here are some of her examples on the right. She says that all shapes that have rotational symmetry of order 3 must have three lines of symmetry. Draw an example to show that she is wrong.
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7.3 Transformations Homework 7C 1
Use vectors to describe these translations of the shapes on the grid. i A to B 6 5 4 3 2 1 0
ii A to C
iii A to D
iv B to A
v B to C
vi B to D
B
A D
C
0 1 2 3 4 5 6 7 8 9 1011121314
2
a Draw a set of coordinate axes with values of x and y from 0 to 10. Draw the triangle with coordinates A(4, 4), B(5, 7) and C(6, 5). 3 b Draw the image of ABC after a translation with vector . Label this P. 2 4 . Label this Q. c Draw the image of ABC after a translation with vector −3 −4 . Label this R. d Draw the image of ABC after a translation with vector 3 −3 e Draw the image of ABC after a translation with vector −2
3
4
. Label this S.
Look at your diagram from question 2. Describe the translation that will move: a P to Q
b Q to R
c R to S
d S to P
e R to P
f S to Q
g R to Q
h P to S.
A group of hikers walk between three points A, B and C using direction vectors, with distances in kilometres. B
C
A
−4 The direction vector from A to B is and the direction vector from B to C 3 −2 . is −5 a Draw a diagram on centimetre-squared paper, to show the walk. Use a scale of 1 cm represents 1 km. b Work out the direction vector from C to A.
7.3 Transformations
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5
Write down a series of translations which will take you from the Start/finish, around the shaded square without touching it, and back to the Start/finish. Make as few translations as possible. Start/finish
6
Joel says that if the translation from a point X to a point Y is described by the vector −3 2 , then the translation from the point Y to the point X is described by the 2 vector . −3 Is Joel correct? Show how you decide.
Homework 7D 1
a Draw a pair of axes. Label the x-axis from –5 to 5 and the y-axis from –5 to 5. b Draw the triangle with co-ordinates A(1, 1), B(5, 5) and C(3, 4). c Reflect triangle ABC in the x-axis. Label the image P. d Reflect triangle P in the y-axis. Label the image Q. e Reflect triangle Q in the x-axis. Label the image R. f Describe the reflection that will transform triangle ABC to triangle R.
2
y 4
Copy this diagram onto squared paper. a Reflect triangle A in the line x = 1.
3 2
Label the image B. b Reflect triangle B in the line y = −2. Label the image C.
4
3
2
A 1 0 1 0 1 2 3 4 1
5 6 7 x
2 3 4 5 6 7
3
A
A designer is making a logo for a company. She starts with a kite ABCD. She then reflects the kite in the line BD on top of the original kite to obtain the logo.
B
D
Draw any kite on squared paper to follow the designer’s method to obtain a logo.
4
A point X has coordinates (a, b).
C
Point X is reflected in the line x = 3 Find the coordinates of the image of point X.
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5
a Draw a pair of axes. Label the x-axis from –5 to 5 and the y-axis from –5 to 5. b Draw the triangle with co-ordinates A(2, 2), B(3, 4) and C(2, 4). c Reflect the triangle ABC in the line y = x. Label the image P. d Reflect the triangle P in the line y = –x. Label the image Q. e Reflect triangle Q in the line y = x. Label the image R. f Describe the reflection that will move triangle ABC to triangle R.
Homework 7E 1
y
Copy this diagram onto squared paper.
4
a Rotate the shape 90° clockwise about (0, 0). Label the image P.
3
b Rotate the shape 180° clockwise about (0, 0). Label the image Q. c Rotate the shape 90° anticlockwise about (0, 0). Label the image R.
2 1 0 –4
–3
–2
d What rotation takes R back to the original shape?
–1 0 1 –1
2
4 x
3
–2 –3 –4
2
Copy this diagram onto squared paper.
y
a Write down the coordinates of the vertices of the square ABCD.
4 3
b Rotate the square ABCD through 90° clockwise about (0, 0). Label the image S. Write down the coordinates of the vertices of the square S. c Rotate the square ABCD through 180° about (0, 0). Label the image T. Write down the coordinates of the vertices of the square T.
C
A
B
2 1 0 –4
–3
–2
d Rotate the square ABCD through 90° anticlockwise about (0, 0). Label the image U. Write down the coordinates of the vertices of the square U.
–1 0 1 –1
2
4 x
3
–2 –3
e What do you notice about the coordinates of the four squares?
3
D
–4
A
A designer is making a logo for a company.
B
She starts with a parallelogram ABCD. D She then rotates the parallelogram 90° clockwise about the point of intersection of the two diagonals to obtain the logo.
C
Draw any parallelogram on squared paper and follow the designer’s method to obtain a logo.
7.3 Transformations
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4
Copy the diagram and rotate the given triangle as described. a b
1 4 1 2
y
5 4
turn clockwise about (0, 0)
3
turn clockwise about (0, 2)
2
c 90° anticlockwise about (–1, 1)
1
d 180° about (0, 0)
0 –5
–4
–3
–2
0 1
–1
2
3
4
5
4
5
–1
x
–2 –3 –4 –5
5
Describe the rotation that takes the shaded triangle to:
y
5
a triangle A
4
b triangle B
3
A
c triangle C
2
d triangle D.
1
C
0 –5
–4
–3
–2
–1
0
1
2
3
–1
B
x
–2 –3
D
–4 –5
6
A point P has coordinates (a, b). a The point P is rotated 90° clockwise about (0, 0) to give a point Q. What are the coordinates of Q? b The point P is rotated 180° clockwise about (0, 0) to give a point R. What are the coordinates of R? c The point P is rotated 90° anticlockwise about (0, 0) to give a point S. What are the coordinates of S?
7
Triangle A, as shown on the grid, is rotated to form a new triangle B.
5
The coordinates of the vertices of B are (3, –1), (1, – 4) and (3, – 4).
3
y
4
A
Describe fully the rotation that maps triangle A onto triangle B. 5
4
3
2 1 0 2 1 01 1
2
3
4
5
x
2 3 4 5
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Homework 7F 1
Copy each shape with its centre of enlargement. Use the ray method enlarge it by the given scale factor.
+
Scale factor 2 Scale factor 3
4 cm
5 cm
3 cm
+
3 cm
4 cm 3 cm
2
Copy each of these shapes and grids onto squared paper. Enlarge them by scale factor 2 using the origin as the centre of enlargement. a
b
yyy 8 88 7 77
yyy 8 88 7 77
66 6 5 55
6 66 55 5
4 44 3 33
4 44 3 33
22 2 1 11 00 0
c
2 22 11 1 11 2 22 3 33 4 44 5 55 6 66 7 77 8 88 xxx 1
yyy 8 88 77 7
0 00 yy 88 y 8 7 77
d
6 66 5 55
6 66 5 55
44 4 3 33
4 44 33 3
2 22 1 11
2 22 1 11 0 00
3
1 11 2 22 3 33 4 44 5 55 6 66 7 77 8 88 xxx
1 11 2 22 3 33 4 44 5 55 6 66 7 77 8 88 xxx
0 00
1 11 2 22 3 33 4 44 5 55 6 66 7 77 8 88 xxx
Draw a letter T of any size. Now draw another letter T twice the size, as shown in the diagram. Use the ray method to find the centre of enlargement. Draw the rays as dotted lines to create a logo design.
4
Triangle A has coordinates (2, 2), (6, 2) and (6, 4). Triangle A is enlarged by a scale factor of
1 2
about the origin to give triangle B.
Find the coordinates of triangle B.
7.3 Transformations
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5
6 y
Triangle B is an enlargement of triangle A.
5
Which of the following describes the enlargement?
4
a an enlargement of scale factor –2 about (0, 0)
2 1 0
b an enlargement of scale factor –3 about (0, 0)
9
8
7
6
c an enlargement of scale factor –3 about (1, 2) d an enlargement of scale factor – about (1, 2)
A
3
5
4
3
2
1
1
01
2
3
4
5
x
2 B
3
1 3
4
5 6 7 8
Show how you decide.
7.4 Combinations of transformations Homework 7G 1
Describe fully the transformations that will result in the following movements. a T1 to T2
b T1 to T6
c T2 to T3
e T6 to T5
f T5 to T4
g T1 to T5
5 4
T6
T5
3 2
T1
1 0 –7 –6 –5 –4 –3 –2 –1 0 1 –1
T4
d T6 to T2
–2
2
3
4
5
6
7
8
9 10
T2
–3
2
T3
a Plot a triangle T with vertices (1, 1), (3, 1) and (3, 4). b Reflect triangle T in the x-axis and label the image Tb. c Rotate triangle Tb 90° clockwise about the origin and label the image Tc. d Reflect triangle Tc in the x-axis and label the image Td. e Describe fully the transformation that will move triangle Td to triangle T.
3
a The point P(2, 5) is reflected in the x-axis, then rotated by 90° clockwise about the origin. What are the coordinates of the image of P? b The point Q(a, b) is reflected in the x-axis, then rotated by 90° clockwise about the origin. What are the coordinates of the image of Q?
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4
a The point R(4, 3) is reflected in the line y = –x, then reflected in the x-axis. What are the coordinates of the image of R? b The point S(a, b) is reflected in the line y = –x, then reflected in the x-axis. What are the coordinates of the image of S?
5
Copy the diagram onto squared paper.
y
a Triangle A is translated by the 9 vector to give triangle B. −3 Triangle B is then enlarged by a scale factor –2 about the origin to give triangle C.
7 6 5
A
4 3 2 1 0
Draw triangles B and C on the diagram. b Describe fully the single transformation that maps triangle C onto triangle A.
9
8
7
6
5
4
3
2
1
1
0 1
2
3
4
5
x
2 3 4
5 6 7
7.5 Bisectors Homework 7H 1
Draw a line 8 cm long and bisect it. Check your accuracy by measuring each half.
2
a Draw any triangle. b On each side construct the line bisector. Your line bisectors should intersect at the same point. c Using this point as the centre, draw a circle that passes through the three vertices of the triangle.
3
a Draw a circle with a radius of about 4 cm. b Draw a quadrilateral inside the circle so that the vertices of the quadrilateral touch its circumference. c Bisect two of the sides of the quadrilateral. Your bisectors should meet at the centre of the circle.
4
a Draw any angle. b Construct the angle bisector. c Check your accuracy by measuring each half.
5
The diagram shows a park with two ice-cream sellers A and B. People always go to the ice-cream seller nearest to them. Shade the region of the park from which people go to ice-cream seller B.
B A
7.5 Bisectors
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6
Using only a pencil, a straight edge and a pair of compasses, construct: a an angle of 15 degrees b an angle of 75 degrees.
7
If I construct all the angle bisectors of any triangle, they will meet at a point. Show that if I draw a circle with this as the centre, the circle will just touch each side of the triangle.
7.6 Defining a locus Homework 7I 1
A is a fixed point. Sketch the locus of the point P when AP > 3 cm and AP < 6 cm.
2
A and B are two fixed points 4 cm apart. Sketch the locus of the point P for these situations: a AP < BP
b P is always within 3 cm of A and within 2 cm of B.
3
A fly is tethered by a length of spider’s web that is 1 m long. Describe the locus of the fly’s movement.
4
ABC is an equilateral triangle of side 4 cm. In each of the following loci, the point P moves only inside the triangle. Sketch the locus in each case. a AP = BP
b AP < BP
c CP < 2 cm
d CP > 3 cm and BP > 3 cm
5
A wheel rolls around the inside of a square. Sketch the locus of the centre of the wheel.
6
The same wheel rolls around the outside of the square. Sketch the locus of the centre of the wheel.
7
On a piece of plain paper, mark three points A, B and C, about 5 to 7 cm away from each other. Find the locus of point P when: a P is always closer to a point A than a point B b P is always the same distance from points B and C.
8
64
Sketch the locus of a point on the rim of a bicycle wheel as it makes three revolutions along a flat road.
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7.7 Loci problems Homework 7J For Questions 1 to 3, you should start by sketching the picture given in each question before drawing the locus accurately. The scale for each question is given.
1
Stake Fence
2
Fence
A goat is tethered by a rope, 10 m long, to a stake that is 2 m from each side of a field. What is the locus of the area that the goat can graze? Use a scale of 1 cm to 2 m.
A cow is tethered to a rail at the top of a fence 4 m long. The rope is 4 m long. Sketch the area that the cow can graze. Use a scale of 1 cm to 2 m. Fence
3
A horse is tethered to a corner of a shed, 3 m by 1 m. The rope is 4 m long. Sketch the area that the horse can graze. Use a scale of 1 cm to 1 m.
Tethered here Shed
4
Two ships, A and B, which are 7 km apart, both hear a distress signal from a fishing boat. The fishing boat is less than 4 km from ship A and less than 4.5 km from ship B. A helicopter pilot sees that the fishing boat is nearer to ship A than to ship B. Use accurate construction to show the region which contains the fishing boat. Shade this region.
5
The locus of a point is described as: 5 cm away from point A equidistant from both points A and B. Which of the following could be true? a The locus is an arc. b The locus is just two points. c The locus is a straight line. d The locus is none of these.
7.7 Loci problems
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For Questions 6 to 9, you should use a copy of the map on this page. For each question, trace the map and mark on those points that are relevant to that question.
Glasgow
Newcastle upon Tyne
North Sea York Leeds
Irish Sea
Manchester Sheffield
Norwich Birmingham
Bristol
London
Exeter
0 0
66
50 50
100
100 150
150 200
250
200 300
250 miles
350 km
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6
A radio station broadcasts from Birmingham with a range that is just far enough to reach York. Another radio station broadcasts from Glasgow with a range that is just far enough to reach Newcastle. a Sketch the area to which each station can broadcast. b Will the Birmingham station broadcast as far as Norwich? c Will the two stations interfere with each other?
7
An air traffic control centre is to be built in Newcastle. If it has a range of 200 km, will it cover all the area of Britain north of Sheffield and south of Glasgow?
8
There are plans to build a new radio transmitter so that it is the same distance from Exeter, Norwich and Newcastle. a Draw the perpendicular bisectors of the lines joining these three places and hence find its proposed location. b The radio transmitter will cause problems if it is built within 50 km of Birmingham. Will the proposed location cause problems?
9
Three radio stations pick up a distress call from a boat in the North Sea. The station at Norwich can tell from the strength of the signal that the boat is within 150 km of the station. The station at Sheffield can tell that the boat is between 100 and 150 km from Sheffield. If these two reports are correct, state the least and greatest possible distance of the boat from the helicopter station at Newcastle?
7.8 Plans and elevations Homework 7K 1
Draw the plan, front elevation and side elevation for each shape. a
b
7.8 Plans and elevations
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2
This 3D shape is made from cubes. B
A
Plan view
C D Front elevation
F
E
Side elevation
a Which diagram shows the plan view? b Which diagram shows the front elevation? c Which diagram shows the side elevation?
3
On an isometric grid, draw the 3D shape shown by this plan, front elevation and side elevation.
Plan
Front elevation
Side elevation
4
Draw an accurate plan, front elevation and side elevation for a 7 cm-long regular hexagonal prism with side length 3 cm.
5
The diagram shows the front elevation of a storage tank with a uniform cross-section. The storage tank is 4 metres deep. 3m
Draw the plan and side elevation of the tank using a scale of 2 cm to 1 m.
2m
3m
6
Draw an accurate plan, front elevation and side elevation these shapes. a
b
6 cm 6 cm
10 cm10 cm
2 cm 2 cm 2 cm 2 cm 2 cm 2 cm
cm
cm
4
2 cm 2 cm
4
cm 4
4
6 cm 6 cm
cm
8 cm 8 cm 2 cm 2 cm
2 cm 2 cm 2 cm 2 cm
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