1. TRANSFORMATION GEOMETRY
REFLECT
ROTATE
TRANSLATE

2. REFLECTION

3. 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.

4. The image is congruent to the original shape.
Reflecting shapes
If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A B C D A’ B’
object
image C’
Explain that we call the original shape the object and the reflected shape the image. The image of an object can be produced by any transformation including rotations, translations and enlargements, as well as reflections.
Define the word congruent to mean the same shape and size.
Link:
S2 2-D shapes – congruence. D’
mirror line or axis of reflection
The image is congruent to the original shape.

5. 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 B C D A’ B’ C’ D’
object
image
mirror line or axis of reflection
Stress that the mirror line is always perpendicular (at right angles) to any line connecting a point to its image. The mirror line also bisects the line (divides it into two equal parts).

6. Reflecting shapes by folding paper
We can make reflections by folding paper.
Draw any 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.

7. 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.

8. Reflection on a coordinate gridy
The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1).
A’(–2, 6) 7
A(2, 6) 6
B’(–7, 3) 5
B(7, 3) 4 3 2
Reflect the triangle in the y-axis and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1
C’(–4, –1) –2
C(4, –1) –3
Pupils should notice that when a shape is reflected in the y-axis, the x-coordinate of each image point is the same as the x-coordinate of the original point × –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point. In other words, the x-coordinate changes sign and the y-coordinate stays the same. –4
What do you notice about each point and its image? –5 –6 –7

9. Reflection on a coordinate gridy
The vertices of a quadrilateral lie on the points A(–4, 6), B(4, 5), C(2, 0) and D(–5, 3).
A(–4, 6) 7 6
B(4, 5) 5 4 3
D(–5, 3) 2 1
C(2, 0)
Reflect the quadrilateral in the x-axis and label each point on the image. –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1
C’(2, 0)
D’(–5, –3) –2 –3
Pupils should notice that when a shape is reflected in the x-axis, the x-coordinate of each image point is the same as the x-coordinate of the original point and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. In other words, the x-coordinate stays the same and the y-coordinate changes sign. –4
What do you notice about each point and its image? –5
B’(4, –5) –6
A’(–4, –6) –7

10. Reflection on a coordinate gridy
x = y
B’(–1, 7)
The vertices of a triangle lie on the points A(4, 4), B(7, –1) and C(2, –6). 7 6
A’(4, 4) 5 4
A(4, 4) 3 2
C’(–6, 2)
Reflect the triangle in the line y = x and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7
B(7, –1) x –1 –2 –3
Pupils should notice that when a shape is reflected in the line x = y, the x-coordinate of each image point is the same as the y-coordinate of the original point and the y-coordinate of the image point is the same as the x-coordinate of the original point. In other words, the x- and y-coordinates are swapped around. –4
What do you notice about each point and its image? –5 –6 –7
C(2, –6)

11. ROTATION

12. 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.

13. Rotation Opening a door? Walking up stairs? Riding on a Ferris wheel?
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?
Anything that is fixed at a point and turns about that point is an example of a rotation. This is true even if a complete rotation cannot be completed, such as your jaw when opening your mouth.
Opening a drawer?
Can you suggest any other examples?

14. Rotating shapes B object A A’ image B’ C C’ O
If we rotate triangle ABC 90° clockwise about point O the following image is produced: B
object
90° A A’
image B’ C C’
Explain that if the centre of rotation is not in contact with the shape, we can extend a line from a point on the shape to the point O. A line extended from the corresponding point on the image will meet the centre of rotation at an angle equal to the angle of rotation. 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.

15. 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.

16. Rotating with tracing paper
The next slide reminds us how we can use tracing paper to rotate a shape
You may remember doing this last year

17. Rotate this shape anticlockwise through 1800 about the point c.

18. Rotations on a coordinate grid
The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). 7
A(2, 6) 6 5
B(7, 3) 4 3
C’(–4, 1) 2
Rotate the triangle 180° clockwise about the origin and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2
C(4, –1) –3
Pupils should notice that when a shape is rotated through 180º about the origin, the x-coordinate of each image point is the same as the x-coordinate of the original point × –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. –4
What do you notice about each point and its image?
B’(–7, –3) –5 –6
A’(–2, –6) –7

19. Rotations on a coordinate grid
The vertices of a triangle lie on the points A(–6, 7), B(2, 4) and C(–4, 4). 7
B(2, 4) 6 5
C(–4, 4) 4 3
B’(–4, 2) 2
Rotate the triangle 90° anticlockwise about the origin and label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 –3
Pupils should notice that when a shape is rotated through 90º anticlockwise about the origin, the x-coordinate of each image point is the same as the y-coordinate of the original point × –1. The y-coordinate of the image point is the same as the x-coordinate of the original point. –4
What do you notice about each point and its image?
C’(–4, –4) –5 –6
A’(–7, –6) –7

20. TRANSLATION

21. Translation image object object object object object object object
When an object is moved in a straight line in a given direction we say that it has been translated.
For example, we can translate triangle ABC 5 squares to the right and 2 squares up: C’ A’ B’
image C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object C A B
object
object
Introduce the key terms and ask pupils to give examples of translations from everyday situations. For example, walking in a straight line from one side of the room to the other.
Discuss how to define a translation. There are two ways to mathematically describe movement in a straight line in a given direction. One is to to give the direction as an angle and a distance (see S7 Measures – bearings), the other is to give the direction using a square grid. A movement in a straight line can be given as a number or units to the left or right and a number of units up or down.
Stress that when a shape is translated each vertex (or corner) is moved through the given translation.
The movement across is always given before the movement up or down (compare with coordinates).
Ask pupils how we could return the translated shape back to its original position.
Every point in the shape moves the same distance in the same direction.

22. Translations
When a shape is translated the image is congruent to the original.
The orientations of the original shape and its image are the same.
An inverse translation maps the image that has been translated back onto the original object.
What is the inverse of a translation 7 units to the left and 3 units down?
The inverse is an equal move in the opposite direction.
That is, 7 units right and 3 units up.

23. Describing translations
When we describe a translation we always give the movement left or right first followed by the movement up or down.
We can describe translations using vectors.
For example, the vector describes a translation 3 right and 4 down. –4 3
As with coordinates, positive numbers indicate movements up or to the right and negative numbers are used for movements down or to the left.
Compare a translation given as an angle and a distance to using bearings. For example, an object could be translated through 5cm on a bearing of 125º.
A different way of describing a translation is to give the direction as an angle and the distance as a length.

24. Translations on a coordinate gridy
A(5, 7)
The vertices of a triangle lie on the points A(5, 7), B(3, 2) and C(–2, 6).
C(–2, 6) 7 6 5 4 3 2
B(3, 2)
Translate the shape 3 squares left and 8 squares down. Label each point in the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1
A’(2, –1) –2
C’(–5, –2) –3
Point out that when we translate shapes on a coordinate grid we can find the coordinates of the vertices of the image shape by adding to the original’s x-coordinates the size of the translation to the right (or subtracting the size of a translation to the left); and adding to the original’s y-coordinates the size of the translation up (or subtracting the size of a translation down). –4
What do you notice about each point and its image? –5 –6
B’(0, –6) –7

25. Translations on a coordinate gridy
The coordinates of vertex A of this shape are (–4, –2). 7 6 5 4 3
When the shape is translated the coordinates of vertex A’ are (3, 2). 2
A’(3, 2) 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 –2
A(–4, –2)
What translation will map the shape onto its image? –3
Tell pupils that we can work this out by subtracting the coordinate of the original shape from the coordinate of the image.
The difference between the x-coordinates is 7 (3 – –4) and the difference between the y-coordinates is 4 (2 – –2). The required translation is therefore 7 right and 4 up. –4 –5 –6 –7
7 right
4 up

26. Translations on a coordinate grid1 2 3 4 5 6 –2 –3 –4 –5 –6 –7 –1 y x 7
The coordinates of vertex A of this shape are (3, –4).
When the shape is translated the coordinates of vertex A’ are(–3, 3).
A’(–3, 3)
What translation will map the shape onto its image?
The difference between the x-coordinates is –6 (–3 – 3) and the difference between the y-coordinates is 7 (3 – –4). The required translation is therefore 6 left and 7 up.
If required this slide can be copied and modified to produce more examples. To change the shapes, go to View/toolbars/drawing to open the drawing toolbar. Select the shape and using the drawing toolbar, click on draw/edit points to drag the points to different positions. Edit the new coordinates as necessary.
A(3, –4)
6 left
7 up