Transcript Document

KS3 Mathematics
S4 Coordinates and
transformations 1
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Contents
S4 Coordinates and transformations 1
S4.1 Coordinates
S4.2 Reflection
S4.3 Reflection symmetry
S4.4 Rotation
S4.5 Rotation symmetry
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Coordinates
We can describe the position of any point on a 2-dimensional
plane using coordinates.
The coordinate of a point tells us where the point is relative to
a starting point or origin.
For example, when we write a coordinate
(3, 5)
x-coordinate
y-coordinate
the first number is called the x-coordinate and the second
number is called the y-coordinate.
y-coordinate.
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Using a coordinate grid
Coordinates are plotted on a grid of squares.
4
3
The x-axis and the y-axis
intersect at the origin.
y-axis
2
1
origin
x-axis
–4 –3 –2 –1 0
–1
–2
–3
1
2
3
4
The lines of the grid are
numbered using positive
and negative integers as
follows.
The coordinates of the
origin are (0, 0).
–4
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Quadrants
The coordinate axes divide the grid into four quadrants.
4
y
3
second
quadrant
2
first
quadrant
1
x
–4 –3 –2 –1 0
–1
third
–2
quadrant
–3
1
2
3
4
fourth
quadrant
–4
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What quadrant?
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Coordinates
The first number in the coordinate pair tells you how many
units to move along in the x-direction.
A positive number means move right from the origin and a
negative number means move left.
The second number in the coordinate pair tells you how
many units to move up or down in the y-direction.
A positive number means move up from the origin and a
negative number means move down.
Remember:
Along the corridor and up (or down) the stairs.
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Plotting points
Plot the point (–3, 5).
y
(–3, 5)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
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1 2 3 4 5 6 7x
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Plotting points
Plot the point (–4, –2).
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
(–4, –2)
–4
–5
–6
–7
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1 2 3 4 5 6 7x
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Plotting points
Plot the point (6, –7).
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
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1 2 3 4 5 6 7x
(6, –7)
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Making quadrilaterals
Where could we add a fourth point to make a parallelogram?
y
(–5, 4)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
(–5, –1)
–3
–4
–5
–6
–7
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(3, 2)
1 2 3 4 5 6 7x
(3, –3)
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Making quadrilaterals
Where could we add a fourth point to make a square?
y
(–2, 2)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
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(2, 6)
(6, 2)
1 2 3 4 5 6 7x
(2, –2)
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Making quadrilaterals
Where could we add a fourth point to make a rhombus?
y
(–2, 4)
(–7, 2)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
(–2, 0) –2
–3
–4
–5
–6
–7
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(3, 2)
1 2 3 4 5 6 7x
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Making quadrilaterals
Where could we add a fourth point to make a kite?
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
(–7, –1)
–3
–4
–5
–6
–7
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(2, 2)
1 2 3 4 5 6 7x
(5, –1)
(2, –4)
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Making quadrilaterals
Where could we add a fourth point to make an arrowhead?
y
7
(0, 6) 6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
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(6, 6)
(3, 3)
1 2 3 4 5 6 7x
(3, –2)
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Making quadrilaterals
Where could we add a fourth point to make a rectangle?
y
(–6, 3)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
(–3, –3)
–4
–5
–6
–7
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(2, 7)
(5, 1)
1 2 3 4 5 6 7x
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Don’t connect three!
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Finding the mid-point of a horizontal line
Two points A and B have the same y-coordinate.
A is the point (–2, 5) and B is the point (6, 5).
8
?
A(–2, 5)
M(xm?, 5).
B(6, 5)
What is the coordinate of the mid-point of the line
segment AB?
Let’s call the mid-point M(xm, 5).
xm is the point half-way between –2 and 6.
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Finding the mid-point of a horizontal line
Two points A and B have the same y-coordinate.
A is the point (–2, 5) and B is the point (6, 5).
8
?
M(xm?, 5).
A(–2, 5)
Either,
xm, = –2 + ½ × 8
B(6, 5)
or
xm, = ½(–2 + 6)
= –2 + 4
=½×4
=2
=2
The coordinates of the mid-point of AB are (2, 5).
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Finding the mid-point of a line
If A is the point (2, 1) and B is the point (8, 5),
what is the mid-point of the line AB?
Start by plotting points A and B on a coordinate grid.
y
The x-coordinate of point A
is 2 and the x-coordinate of
point B is 8.
7
B(8, 5)
6
5
4
3
2
A(2, 1)
1
0
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1
2
3
4
5
6
7
8
9 10
x
The x-coordinate of the midpoint is half-way between 2
and 8.
2+8
= 5
2
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Finding the mid-point of a line
If A is the point (2, 1) and B is the point (8, 5),
what is the mid-point of the line AB?
Start by plotting points A and B on a coordinate grid.
y
The y-coordinate of point A
is 1 and the y-coordinate of
point B is 5.
7
B(8, 5)
6
5
M(5, 3)
4
3
2
A(2, 1)
1
0
1
2
3
4
5
6
7
8
9 10
x
The y-coordinate of the midpoint is half-way between 1
and 5.
1+5
= 3
2
The mid-point of AB is (5, 3)
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Finding the mid-point of a line
We can generalize this result to find the mid-point of any line.
If the coordinates of A are (x1, y1) and the coordinates of B
are (x2, y2) then the coordinates of the mid-point of the line
segment joining these points are given by:
y
x1 + x2
2
,
x1 + x2
y1 + y2
,
2
2
y1 + y2
2
B(x2, y2)
A(x2, y2)
x
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x1 + x2
2
y1 + y2
2
is the mean of the
x-coordinates.
is the mean of the
y-coordinates.
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Contents
S4 Coordinates and transformations 1
S4.1 Coordinates
S4.2 Reflection
S4.3 Reflection symmetry
S4.4 Rotation
S4.5 Rotation symmetry
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Reflection
An object can be reflected in a mirror line or axis of
reflection to produce an image of the object.
For example,
Each point in the image must be the same distance from
the mirror line as the corresponding point of the original
object.
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Reflecting shapes
If we reflect the quadrilateral ABCD in a mirror line we label
the image quadrilateral A’B’C’D’.
A’
A
B’
B
object
image
C’
C
D
D’
mirror line or axis of reflection
The image is congruent to the original shape.
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Reflecting shapes
If we draw a line from any point on the object to its image
the line forms a perpendicular bisector to the mirror line.
A’
A
B’
B
object
image
C’
C
D
D’
mirror line or axis of reflection
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Reflecting shapes
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Reflecting shapes by folding paper
We can make reflections by folding paper.
Draw a random polygon at the top of a piece of
paper.
Fold the piece of paper back on itself so you
can still see the shape.
Place a piece of modeling clay behind the paper and pierce
through each vertex of the shape using a compass point.
When the paper is unfolded the vertices of the
image will be visible.
Join the vertices together using a ruler.
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Reflecting shapes using tracing paper
Suppose we want to reflect this
shape in the given mirror line.
Use a piece of tracing paper to
carefully trace over the shape and
the mirror line with a soft pencil.
When you turn the tracing paper
over you will see the following:
Place the tracing paper over the
original image making sure the
symmetry lines coincide.
Draw around the outline on the back of the tracing paper
to trace the image onto the original piece of paper.
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Reflect this shape
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Reflection on a coordinate grid
y
A’(–2, 6) 7
B’(–7, 3)
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
C’(–4, –1) –2
–3
–4
–5
–6
–7
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A(2, 6)
B(7, 3)
1 2 3 4 5 6 7 x
C(4, –1)
The vertices of a
triangle lie on the
points A(2, 6), B(7, 3)
and C(4, –1).
Reflect the triangle in
the y-axis and label
each point on the
image.
What do you notice
about each point
and its image?
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Reflection on a coordinate grid
y
A(–4, 6)
D(–5, 3)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
D’(–5, –3)
–2
–3
–4
–5
–6
–7
A’(–4, –6)
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B(4, 5)
C(2, –2)
1 2 3 4 5 6 7 x
C’(2, –2)
B’(4, –5)
The vertices of a
quadrilateral lie on
the points A(–4, 6),
B(4, 5), C(2, –2) and
D(–5, 3).
Reflect the quadrilateral
in the x-axis and label
each point on the image.
What do you notice
about each point
and its image?
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Reflection on a coordinate grid
B’(–1, 7)
C’(–6, 2)
y
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
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x=y
A’(4, 4)
A(4, 4)
1 2 3 4 5 6 7 x
B(7, –1)
C(2, –6)
The vertices of a
triangle lie on the
points A(4, 4), B(7, –1)
and C(2, –6).
Reflect the triangle in
the line y = x and
label each point on
the image.
What do you notice
about each point
and its image?
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Contents
S4 Coordinates and transformations 1
S4.1 Coordinates
S4.2 Reflection
S4.3 Reflection symmetry
S4.4 Rotation
S4.5 Rotation symmetry
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Reflection symmetry
If you can draw a line through a shape so that one half is a
reflection of the other then the shape has reflection or line
symmetry.
The mirror line is called a line of symmetry.
one line of
symmetry
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two lines of
symmetry
no lines of
symmetry
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Reflection symmetry
How many lines of symmetry do the following designs have?
one line of
symmetry
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five lines of
symmetry
three lines of
symmetry
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Make this shape symmetrical
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Planes of symmetry
Is a cube symmetrical?
We can divide the cube into two symmetrical parts here.
This shaded area is called a plane of symmetry.
How many planes of symmetry does a cube have?
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Planes of symmetry
We can draw the other eight planes of symmetry for a
cube, as follows:
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Planes of symmetry
How many planes of symmetry does a cuboid have?
A cuboid has three planes of symmetry.
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Planes of symmetry
How many planes of symmetry do the following solids have?
An equilateral
triangular prism
A square-based
pyramid
A cylinder
Explain why any right prism will always
have at least one plane of symmetry.
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Investigating shapes made from four cubes
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Contents
S4 Coordinates and transformations 1
S4.1 Coordinates
S4.2 Reflection
S4.3 Reflection symmetry
S4.4 Rotation
S4.5 Rotation symmetry
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Rotation
Which of the following are examples of rotation in real life?
Opening a door?
Walking up stairs?
Riding on a Ferris wheel?
Bending your arm?
Opening your mouth?
Opening a drawer?
Can you suggest any other examples?
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Describing a rotation
A rotation occurs when an object is turned around a fixed
point.
To describe a rotation we need to know three things:
The angle of rotation.
For example,
½ turn = 180°
¼ turn = 90°
¾ turn = 270°
The direction of rotation.
For example, clockwise or anticlockwise.
The centre of rotation.
This is the fixed point about which an object moves.
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Rotating shapes
If we rotate triangle ABC 90° clockwise about point O the
following image is produced:
B
object
90°
A
A’
image
B’
C
C’
O
A is mapped onto A’, B is mapped onto B’ and C is mapped
onto C’.
The image triangle A’B’C’ is congruent to triangle ABC.
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Rotating shapes
The centre of rotation can also be inside the shape.
For example,
90°
O
Rotating this shape 90° anticlockwise about point O
produces the following image.
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Determining the direction of a rotation
Sometimes the direction of the rotation is not given.
If this is the case then we use the following rules:
A positive rotation is an anticlockwise rotation.
A negative rotation is an clockwise rotation.
For example,
A rotation of 60° = an anticlockwise rotation of 60°
A rotation of –90° = an clockwise rotation of 90°
Explain why a rotation of 120° is
equivalent to a rotation of –240°.
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Inverse rotations
The inverse of a rotation maps the image that has been
rotated back onto the original object.
For example, the following shape is rotated 90° clockwise
about point O.
90°
O
What is the inverse of this rotation?
Either, a 90° rotation anticlockwise,
or a 270° rotation clockwise.
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Inverse rotations
The inverse of any rotation is either
A rotation of the same size, about the same point, but
in the opposite direction, or
A rotation in the same direction, about the same point,
but such that the two rotations have a sum of 360°.
What is the inverse of a –70° rotation?
Either, a 70° rotation,
or a –290° rotation.
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Rotations on a coordinate grid
C’(–4, 1)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
B’(–7, –3)
–5
–6
A’(–2, –6) –7
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A(2, 6)
B(7, 3)
1 2 3 4 5 6 7
C(4, –1)
The vertices of a
triangle lie on the
points A(2, 6), B(7, 3)
and C(4, –1).
Rotate the triangle 180°
clockwise about the
origin and label each
point on the image.
What do you notice
about each point and
its image?
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Rotations on a coordinate grid
A(–6, 7)
C(–4, 4)
B’(–4, 2)
7
6
5
4
3
2
1
–7 –6 –5 –4 –3 –2 –1 0
–1
–2
–3
–4
C’(–4, –4) –5
–6
A’(–7, –6)
–7
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B(2, 4)
1 2 3 4 5 6 7
The vertices of a
triangle lie on the
points A(–6, 7), B(2, 4)
and C(–4, 4).
Rotate the triangle 90°
anticlockwise about the
origin and label each
point in the image.
What do you notice
about each point and
its image?
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Contents
S4 Coordinates and transformations 1
S4.1 Coordinates
S4.2 Reflection
S4.3 Reflection symmetry
S4.4 Rotation
S4.5 Rotation symmetry
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Rotational symmetry
An object has rotational symmetry if it fits exactly onto
itself when it is turned about a point at its centre.
The order of rotational symmetry is the number of times
the object fits onto itself during a 360° turn.
If the order of rotational symmetry is one, then the object
has to be rotated through 360° before it fits onto itself again.
Only objects that have rotational symmetry of two or more
are said to have rotational symmetry.
We can find the order of rotational symmetry using tracing
paper.
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Finding the order of rotational symmetry
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Rotational symmetry
What is the order of rotational symmetry for the following
designs?
Rotational
symmetry
order 4
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Rotational
symmetry
order 3
Rotational
symmetry
order 5
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Finding the order of rotational symmetry
This shape has rotational symmetry of order 6 because it
maps onto itself in six distinct positions after rotations of 60°
about the centre point.
How can you prove that the central shape in
this diagram is a regular hexagon?
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Symmetry puzzle
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