Chapter 15: Transformation and Similarity

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15.1 Reflections

Some objects possess mirror symmetry, or reflection symmetry.

2D objects have a mirror line, dividing the object into two matching (congruent: the same shape and size) halves, one a mirror image of the other.

3D images have a plane of symmetry

(http://www.bbc.co.uk/schools/gcsebitesize/maths/images/sequence_transformations_5.gif)(http://cribbd.com/data/911352832431.6542_57cef45955d5890d46a2a1742d0857f9_468x161.jpg)

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15.2 Rotations

These are objects that are turned to face in a different direction while remaining the same shape and size. The pivot for the rotation is the centre of rotation. In order to answer a rotation question, include:

  • Angle of rotation
  • Clockwise/anticlockwise
  • Centre of rotation

(http://www.bbc.co.uk/staticarchive/911a3ee547582f779519e4f9817da90da6aac041.gif)

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15.3 Combining Transformations

Reflection, rotation and translation preserve congruence. Translations consist of sliding an object up, down, left or right. You write this in a column vector:

(http://www.autograph-maths.com/newsletter/issue7/screenshots/diagnostic-question-large.png)

This would be B, because the shape moves 5 units right and 2 units down.
Sometimes, you will have to combine transformations to 'map' one object onto another

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15.4 Enlargements

When enlarging, you must specify a centre of enlargement and a scale factor. It often helps to draw rays from the centre of enlargement

  • Scale factor greater than 1: simple enlargement- object gets bigger
  • Scale factor between 0 and 1: the enlargement is a reduction- object gets smaller
  • Negative scale factor: object is enlarged/reduced and inverted

(http://www.bbc.co.uk/staticarchive/1d8267406b12430267ac0d34cf8c7d8f1efdd4b8.gif)

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15.5 Similar shapes and solids

Similarity: objects are the same shape but one is an enlargement of the other. If the enlargement factor is n:

  • Corresponding lengths are in the ratio 1:n
  • Corresponding areas are in the ratio 1:n(squared)
  • Corresponding volumes are in the ratio 1:n(cubed)

(http://www.bbc.co.uk/staticarchive/6a40a3b4bfa47c16e6ae749cfb1ff0d45336e949.gif)

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