1. Transformations

2. Reflections

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. 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.

5. 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.

6. 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).

7. Reflecting shapes
Use this activity to dynamically prove the result on the previous slide, regardless of the shape or the position of the mirror line.

8. Reflecting shapes
Use this activity to dynamically prove the result on the previous slide.

9. Reflect this shape
Modify one of the pentagons by dragging its vertices and define it as the image. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon. If necessary, use the pen tool (set to draw straight lines) to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

10. Finding a line of reflection
Construct the line that reflects shape A onto its image A’. A’
This is the line of reflection. A
Two lines are sufficient to define the line of reflection. A third line can be used to check the position of the line.
The mid-point can be found using a ruler to measure the length of the line, and then halving it.
Draw lines from any two vertices to their images.
Mark on the mid-point of each line.
Draw a line through the mid points.

11. Reflection on X-axis

12. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

13. Example 1
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Reflections

14. Reflection on Y-axis

15. Reflecting shapes
Use this activity to dynamically prove the result on the previous slide, regardless of the shape or the position of the mirror line.

16. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

17. Reflect the shape in a mirror line
1. Reflect this triangle in the y-axis
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18. 1. Reflect this triangle in the y-axis
2. Draw lines from each vertex (corner) at 900 to the mirror line (y-axis)
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Reflections

19. 3. Use these lines to find the reflected points
1. Reflect this triangle in the y-axis
2. Draw lines from each vertex (corner) at 900 to the mirror line (y-axis)
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Reflections

20. 3. Use these lines to find the reflected points
4. Draw the reflection
3. Use these lines to find the reflected points
1. Reflect this triangle in the y-axis
2. Draw lines from each vertex (corner) at 900 to the mirror line (y-axis)
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21. Example
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22. Reflection on the line Y =X

23. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

24. Reflection on the line Y = -X

25. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

26. Reflection on the line x = h

27. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

28. Reflection on the line y = k

29. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

30. Reflection Summary Reflection on X-axis: P(x, y) P’(x, -y)
Reflection on Y-axis: P(x, y) P’(-x, y)
Reflection on y= x: P(x, y) P’(y, x)
Reflection on y = - x: P(x, y) P’(-y, -x)
Reflection on x = h: P(x, y) P’(2h-x, y)
Reflection on y = k: P(x, y) P’(x, 2k-y)
Note: X-axis means y = 0
Y-axis means x = 0
Line parallel to X-axis means y = k
Line parallel to Y-axis means x = h

31. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

32. 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

33. 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

34. 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)

35. Reflect this shape
Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon.
If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

36. Reflection on a coordinate grid
Investigate the relationship between the coordinates of the object and its image for each line of symmetry. This can be done by revealing the coordinate of A and A’ say, moving the shape (or the point) around the grid and observing the change in the coordinates.
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.

37. What letter would you get if you reflected each shape in its corresponding mirror line?

38. What letter would you get if you reflected each shape in its corresponding mirror line?

39. What letter would you get if you reflected each shape in its corresponding mirror line?

40. What letter would you get if you reflected each shape in its corresponding mirror line?

41. What letter would you get if you reflected each shape in its corresponding mirror line?

42. e.g.

43. Reflections Using Tracing Paper
Mirror Line
Reflect the object shape in the mirror line shown.
object
Trace the shape and mirror line using tracing paper.
image
Turn tracing paper over and match up mirror lines/square corners.
Mark position of vertices, remove tracing paper and draw image of shape.

44. Reflections Using Tracing Paper
Mirror Line
Reflect the object shape in the mirror line shown.
Trace the shape and mirror line using tracing paper.
object
Turn tracing paper over and match up mirror lines.
Tracing 1

45. Reflections Using Tracing Paper
Mirror Line
Reflect the shape in the mirror line shown.
Trace the shape and mirror line using tracing paper.
object
Turn tracing paper over and match up mirror lines.
Mark position of vertices, remove tracing paper and draw image of shape.

46. Reflections Using Tracing Paper
Mirror Line
Reflect the shape in the mirror line shown.
Trace the shape and mirror line using tracing paper.
object
image
Turn tracing paper over and match up mirror lines.
Mark position of vertices, remove tracing paper and draw image of shape.

47. Reflections Using Tracing Paper
Mirror Line
Reflect the object shape in the mirror line shown.
object
Trace the shape and mirror line using tracing paper.
Turn tracing paper over and match up mirror lines.
Tracing 2

48. Reflections Using Tracing Paper
Mirror Line
Reflect the object shape in the mirror line shown.
object
Trace the shape and mirror line using tracing paper.
Turn tracing paper over and match up mirror lines.
Mark position of vertices, remove tracing paper and draw image of shape.

49. Reflections Using Tracing Paper
Mirror Line
Reflect the object shape in the mirror line shown.
object
image
Trace the shape and mirror line using tracing paper.
Turn tracing paper over and match up mirror lines.
Mark position of vertices, remove tracing paper and draw image of shape.