EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Reflection and rotation
Transformations and constructions · Lesson 2 of 8
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. What are the coordinates of the origin?
(0, 0)
2. What is the equation of the y-axis?
x = 0
3. What is the equation of the x-axis?
y = 0
4. How many degrees in a half turn?
180
5. What is the reflection of (2, 3) in the y-axis?
(−2, 3)
Learning Objectives
1. Reflect a shape in a mirror line, including y = x.
2. Rotate a shape about a centre through 90° or 180°.
3. Describe a reflection fully.
4. Describe a rotation fully.
Congruent Transformations
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Object and image The original shape is the object; the shape after the transformation is the image. |
Congruent Reflection and rotation keep the size and shape the same, so the image is congruent to the object. |
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Orientation A reflection flips the shape; a rotation turns it. |
Describe fully Name the transformation and give all the details needed. |
A Reflection and a Rotation
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Each point of the image is the same distance from the mirror line, or the same distance from the centre of rotation. |
A triangle reflected in the y-axis on the left grid, and the same triangle rotated 90 degrees clockwise about the origin on the right grid.
Reflection or Rotation?
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REFLECTION |
ROTATION |
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▸ Flips the shape over a mirror line. ▸ Each point and its image are the same distance from the line, on opposite sides. ▸ Describe with the equation of the mirror line, e.g. x = 4, y = x. |
▸ Turns the shape about a fixed point, the centre. ▸ Each point and its image are the same distance from the centre. ▸ Describe with the angle, the direction (clockwise or anticlockwise) and the centre. |
Reflecting in a Line
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Triangle A has vertices (1, 1), (1, 3) and (3, 1). Reflect it in the line x = 4. |
1. Each point is 3, 3 and 1 away from the line x = 4 horizontally
(1, 1) is 3 to the left
2. Go the same distance to the right of the line
(7, 1)
3. Do the same for the other vertices
(1, 3) → (7, 3) and (3, 1) → (5, 1)
Answer: The image has vertices (7, 1), (7, 3) and (5, 1).
Rotating About the Origin
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Rotate triangle A, with vertices (1, 1), (1, 3) and (3, 1), through 90° clockwise about the origin. |
1. Rule for 90° clockwise about the origin
(x, y) → (y, −x)
2. Apply it
(1, 1) → (1, −1)
3. The other vertices
(1, 3) → (3, −1) and (3, 1) → (1, −3)
Answer: The image has vertices (1, −1), (3, −1) and (1, −3).
Describing a Rotation
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Triangle A has vertices (1, 1), (1, 3) and (3, 1). Triangle B has vertices (−1, −1), (−1, −3) and (−3, −1). Describe fully the single transformation that maps A onto B. |
1. The shape has turned upside down
A half turn
2. Each x and y has changed sign
Centre at the origin
3. Name the transformation with the three details
Rotation, 180°, centre (0, 0)
Answer: A rotation of 180° about the centre (0, 0).
Quick Rules About the Origin
Handy for checking your answers.
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Transformation |
Rule for (x, y) |
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Reflection in the y-axis |
(−x, y) |
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Reflection in the x-axis |
(x, −y) |
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Reflection in y = x |
(y, x) |
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Rotation 90° clockwise |
(y, −x) |
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Rotation 90° anticlockwise |
(−y, x) |
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Rotation 180° |
(−x, −y) |
Key Terms
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Object The original shape before a transformation. |
Image The shape after a transformation. |
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Congruent The same shape and size, possibly turned or flipped. |
Mirror line The line a shape is reflected in. |
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Centre of rotation The fixed point a shape turns about. |
Clockwise The direction the hands of a clock move. |
Your Task: Transformation Detective
12 minutes
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Triangle A has vertices (1, 1), (1, 3) and (3, 1). Work out the vertices of its image under (a) reflection in the x-axis (b) reflection in y = x (c) rotation 90° anticlockwise about the origin. Then say which of the three images touches the original at a point. 1. Apply the rule. 2. Plot the image. 3. Check distances from the mirror line. |
A good answer shows: (a) (1, −1), (1, −3), (3, −1). (b) (1, 1), (3, 1), (1, 3), which is the same triangle. (c) (−1, 1), (−3, 1), (−1, 3). Image (b) coincides with A because A is symmetrical about y = x.
Note: Ask why the reflection in y = x leaves A unchanged.
Can I...?
☐ Reflect in the axes.
☐ Reflect in a line such as x = 4.
☐ Reflect in y = x.
☐ Rotate about the origin.
☐ Rotate about another point.
☐ Describe a reflection fully.
☐ Describe a rotation fully.
☐ Use the rules for coordinates.
Summary
✓ Reflection: name it and give the mirror line equation.
✓ Rotation: give the angle, the direction and the centre.
✓ Distances from the mirror line or the centre are preserved.
✓ Rotations and reflections keep the shape congruent.
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EXAM FOCUS Describe fully the single transformation that maps triangle P onto triangle Q, where the vertices of P are (1, 1), (1, 3), (3, 1) and the vertices of Q are (7, 1), (7, 3), (5, 1). (3 marks) A full description needs the name of the transformation and all its details: for a reflection, the equation of the mirror line. |
Exam Practice: Reflection and Rotation
Answer all questions. Show your working. · 25 minutes
▸ Question 1 · 2 marks · Non-calculator. Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is reflected in the y-axis. Write down the coordinates of the vertices of the image.
▸ Question 2 · 3 marks · Non-calculator. Triangle A has vertices (1, 1), (1, 3) and (3, 1). Triangle B has vertices (−1, −1), (−1, −3) and (−3, −1). Describe fully the single…
▸ Question 3 · 3 marks · Non-calculator. The diagram shows triangles P and Q on a grid. Describe fully the single transformation that maps triangle P onto triangle Q.
▸ Question 4 · 2 marks · Non-calculator. The point (2, 5) is reflected in the line y = x. Write down the coordinates of the image.
▸ Question 5 · 3 marks · Non-calculator. The point P(3, 1) is rotated 90° anticlockwise about the origin. Write down the coordinates of the image of P.
▸ Question 6 · 3 marks · Non-calculator. The point P(4, 2) is rotated 90° clockwise about the point C(2, 1). Find the coordinates of the image of P.
Question 1 · 2 marks · Non-calculator
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“Triangle A has vertices (1, 1), (1, 3) and (3, 1). It is reflected in the y-axis. Write down the coordinates of the vertices of the image.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ Two vertices correct. B1
▸ All three correct. B1
▸ Model answer. (−1, 1), (−1, 3) and (−3, 1).
Question 2 · 3 marks · Non-calculator
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“Triangle A has vertices (1, 1), (1, 3) and (3, 1). Triangle B has vertices (−1, −1), (−1, −3) and (−3, −1). Describe fully the single transformation that maps triangle A onto triangle B.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 2 · mark scheme
3 marks available. Award a mark for each point made.
▸ Rotation. B1
▸ 180°. B1
▸ Centre (0, 0). B1
▸ Model answer. A rotation of 180° about the centre (0, 0).
Question 3 · 3 marks · Non-calculator
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The diagram shows triangles P and Q on a grid. Describe fully the single transformation that maps triangle P onto triangle Q. (3 marks) |
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Question 3 · mark scheme
3 marks available. Award a mark for each point made.
▸ Reflection. B1
▸ The line x = 4. B2
▸ Model answer. A reflection in the line x = 4.
Question 4 · 2 marks · Non-calculator
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“The point (2, 5) is reflected in the line y = x. Write down the coordinates of the image.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ One coordinate correct. M1
▸ (5, 2). A1
▸ Model answer. (5, 2).
Question 5 · 3 marks · Non-calculator
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“The point P(3, 1) is rotated 90° anticlockwise about the origin. Write down the coordinates of the image of P.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ A correct rotation method. M1
▸ One coordinate correct. A1
▸ (−1, 3). A1
▸ Model answer. The rule is (x, y) → (−y, x), so the image is (−1, 3).
Question 6 · 3 marks · Non-calculator
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“The point P(4, 2) is rotated 90° clockwise about the point C(2, 1). Find the coordinates of the image of P.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ Vector from C to P is (2, 1). M1
▸ Rotated vector (1, −2). M1
▸ (3, −1). A1
▸ Model answer. P is 2 to the right and 1 above C. After a 90° clockwise turn, the vector (2, 1) becomes (1, −2). The image is (2 + 1, 1 − 2) = (3, −1).