Module 8 (M8) – Geometry and measures - Transformations

Part ofMathsM8: Geometry and measures

Before reading this guide, it may be helpful to read the guides on transformations from Module 5, Module 6 and Module 7.

Transformations

Transformations change the size or position of shapes.

The 4 transformations of 2D shapes that you should know are covered in M5, M6 and M7:

  • translation – moving a shape in a straight line
  • reflection – flipping. a shape to create a mirror image
  • rotation – turning a shape
  • enlargement – changing the size of a shape by a scale factor

In the exam you may be asked to draw and/or describe transformations.

In Module 8 (M8), there may be questions on any of the transformations work at M5, M6 and M7 as well as:

  • enlargement using a negative scale factor
  • combined translations

Enlargement with a negative scale factor

An enlargement with a negative scale factor produces an image on the other side of the centre of enlargement. The image appears upside down.

Example

Enlarge triangle G by a scale factor of −2 about the centre (−2, 3).

Solution

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  1. An orange triangle labelled G on a grid. Coordinates of the triangle are (–6, 2), (–6, 5), (–4, 2).

    Step 1

An image showing the enlargement of a rectangle, ABCD, to A'B'C'D' by scale factor of −½.

The rectangle ABCD has been enlarged by a scale factor of \(–\frac{1}{2}\).

The lengths in rectangle A'B'C'D' are \(\frac{1}{2}\) times as long as rectangle ABCD. The distance from O to A'B'C'D' is half the distance from O to ABCD.

Question

Describe the transformation which maps triangle Q onto triangle P.

An image showing two triangles on a grid, triangle Q to triangle P. Triangle Q's coordinates are (–3, 8), (0, 8) and (–3, 6). Triangle P's coordinates are (5, 2), (5, –4) and (–4, –4).

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Combining transformations

Transformations can be combined by doing one transformation and then another.

Example

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  1. An image showing a quadrilateral shape, labelled V, on a grid. The shape's points are at coordinates (4, 1), (4, 4), (5, 2), (5, 3).

    Reflect the shape V in the line x = 2, followed by a rotation through 180° about the point (1,0).

Example

Question

Reflect triangle M in the line \(x = –2\) and then rotate the image 90° ACW about the point \((–2, 2)\).

  • Draw the final image after the two transformations
  • Describe the single transformation that maps the final image back onto triangle M.
An image of a triangle labelled M on a grid. Triangle M's coordinates are (–4, 0), (–3, 0), (–4, 2).

Solution

An image showing the transformation of a triangle labelled M on a grid. Triangle M's coordinates are (–4, 0), (–3, 0), (–4, 2). The image after reflection in the line x = –2 has coordinates (–1, 0), (0,0), (0, 2). The final image after rotation of image 90 degrees anti-clockwise, from the centre (–2, 2) has coordinates (0, 3), (0, 4), (–2, 4).

– The single transformation which maps the final image back to M is a reflection in the line \(y = –x\).

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Test yourself

Question

Describe a single transformation that is equivalent to a reflection in the \(x\)-axis followed by a reflection in the\(y\)-axis.

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