Reflection and rotation.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reflection and rotation

Transformations and constructions

Lesson 2 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 2 of 8

Answer each one, then check.

1

What are the coordinates of the origin?

2

What is the equation of the y-axis?

3

What is the equation of the x-axis?

4

How many degrees in a half turn?

5

What is the reflection of (2, 3) in the y-axis?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 2 of 8

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)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 2 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Congruent Transformations

Transformations and constructions

Lesson 2 of 8

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.

Orientation

A reflection flips the shape; a rotation turns it.

Describe fully

Name the transformation and give all the details needed.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Reflection and a Rotation

Transformations and constructions

Lesson 2 of 8

Each point of the image is the same distance from the mirror line, or the same distance from the centre of rotation.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reflection or Rotation?

Transformations and constructions

Lesson 2 of 8

REFLECTION

• 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.

ROTATION

• 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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reflecting in a Line

Transformations and constructions

Lesson 2 of 8

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).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Rotating About the Origin

Transformations and constructions

Lesson 2 of 8

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).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Describing a Rotation

Transformations and constructions

Lesson 2 of 8

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).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Quick Rules About the Origin

Transformations and constructions

Lesson 2 of 8

Handy for checking your answers.

Transformation

Rule for (x, y)

Reflection in the y-axis

(−x, y)

Reflection in the x-axis

(x, −y)

Reflection in y = x

(y, x)

Rotation 90° clockwise

(y, −x)

Rotation 90° anticlockwise

(−y, x)

Rotation 180°

(−x, −y)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 2 of 8

Object

The original shape before a transformation.

Image

The shape after a transformation.

Congruent

The same shape and size, possibly turned or flipped.

Mirror line

The line a shape is reflected in.

Centre of rotation

The fixed point a shape turns about.

Clockwise

The direction the hands of a clock move.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 2 of 8

YOUR TASK

Transformation Detective

12 minutes

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.

WHAT 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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 2 of 8

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.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

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.

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.

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