Module 6 (M6) - Geometry and measures - Transformations

Transformations

It may be useful to look at the following topics

Transformations change the size or position of shapes.
Scale factors can change the size of shapes.
Congruent shapes are identical in shape and size but may be reflected or rotated.

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Two of these triangles are congruent, which means they are identical in shape and size.
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Triangles ABC and GHI are congruent. Triangle GHI is the same as ABC but has been rotated 90° anticlockwise. Triangle DEF is a different size.
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Line segments AB and GH are the same length. BC and HI are the same length. AC and GI are the same length.
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Angle A and angle G are equal. Angle B and angle H are equal. Angle C and I are equal and are both right angles.
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These two rectangles are not congruent.
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Rotating rectangle EFGH and placing it on top of rectangle ABCD shows they are not the same size.
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These two quadrilaterals are congruent.
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EFGH is a reflection of ABCD on a vertical mirror line. Line segments AB and EH are the same length. They correspond to each other, which means they are in the same place in two different shapes. BC and GH are the same length. CD and FG are the same length. AD and EF are the same length.
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Angles in congruent shapes are equal. Angle A and angle E are equal. They correspond to each other. Angle B and angle H are equal. Angle C and G are equal. Angle D and F are equal.

Question

Which shapes are congruent?

There are 4 transformations of 2D shapes that you should know:

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.

Translation

When a shape is translated, it is moved up or down and/or left or right. Every point in the shape must be moved in exactly the same way.

Describing translations

Column are used to describe translations.

\(\left( \matrix{ 4 \cr -3 \cr} \right)\) means translate the shape 4 squares to the right and 3 squares down.

\(\left( \matrix{-2 \cr 1 \cr} \right)\) means translate the shape 2 squares to the left and 1 square up.

Vectors are given in the form \(\binom{x}{y}\) where \(x\) is the movement horizontally and \(y\) is the movement vertically. A positive value of \(x\) means a movement to the right and a negative value of \(x\) means a movement to the left. A positive value of \(y\) means a movement upwards and a negative value of \(y\) means a movement downwards.

Example

Triangle (PQR) moved down 3 squares and right 4 squares

Solution:

Triangle PQR has been translated 4 squares right and 3 squares down.This would be described as a translation by the vector \(\left( \matrix{ 4 \cr -3 \cr} \right)\)

There will be a mark for writing translation and a mark for writing (\(\left( \matrix{ 4 \cr -3 \cr} \right)\)

Example

Translate the shape using the vector \(\left( \matrix{ 5 \cr -5 \cr} \right)\) and label it B.

Solution

The shape will be moved 5 to the left and 5 up.

Reflection

When a shape is reflected it is flipped. The line of reflection could be the x axis or the y axis or a line parallel to either axis that is a horizontal or vertical line.

To reflect a shape

Draw the line of reflection.
Choose a point on the shape and count how far it is from the line of reflection.
Count the same amount on the other side of the line and plot the point.
Repeat steps 2 and 3 for each point in the shape and join the points.

Example

Reflect the shape in the line x = -1

Solution:

Draw the line of reflection.
The line x= -1 is a vertical line which passes through -1 on the x axis.
Choose a point on the shape and count how far it is from the line of reflection.
The line is across 2 squares from A
Count the same amount on the other side of the line and plot the point.
The reflection of A is 2 squares to the left of the reflection line.
Repeat steps 1 and 2 for each point in the shape and join the points.

Triangle (ABC) reflected in the line, x=-1

Question

Describe the transformation of the shape ABC.

Triangle (ABC) reflected in the line, y=1

Question

Describe the transformation which maps shape S to shape T.

Rotation

When a shape is rotated it is turned about a fixed point, the centre of rotation.

Three pieces of information are necessary to rotate a shape:

The centre of rotation, at M6 this could be anywhere on the grid.
The angle of rotation, 90° or 180°
The direction of rotation, clockwise CW or anticlockwise ACW.

You can use tracing paper to help.

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The shape has been rotated 90° (a quarter turn) clockwise about the centre of rotation
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The shape has been rotated 180° (a half-turn) about the centre of rotation
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The shape has been rotated 90° (a quarter turn) anticlockwise about the centre of rotation