Transformations
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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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Question
Which shapes are congruent?
Answer
Shapes A, B, E and G 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 vectorA vector describes a movement from one point to another. A vector quantity has magnitude (size) and direction.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
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.
Question
Describe the transformation of the shape ABC.
Answer:
The line y = 1 is a horizontal line which passes through 1 on the y-axis.
The shape has been reflected in the line y = 1.
Remember
There will be a mark for writing reflection and a mark for writing in the line y = 1
Question
Describe the transformation which maps shape S to shape T.
A reflection in the line y = 5
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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How to rotate a shape using tracing paper.
Different centres of rotation
The centre of rotation may not be at the origin (0,0).
Rotate the rectangle ABCD 90° clockwise about the point (0,-1).
Each corner of the image A'B'C'D' is the same distance from the centre of rotation as the original shape.
Enlargement
The base of A is 4 and the height is 5.
The base of B is 8 and the height is 10.
Shape B is an enlargement of shape A with scale factor 2.
Effect of Enlargement on Perimeter and Area
Example
The perimeter of rectangle P is 10
The perimeter of rectangle Q is 20
The perimeter has also been enlarged by scale factor 2
This is because perimeter is a length.
The area of P is 2 x 3 = 6 squares
The area of Q is 4 x 6 = 24 squares.
The area has been enlarged by scale factor 4.
This is the length scale factor squared because the sides are multiplied. 2² = 4
Example
The perimeter of rectangle S is 10
The perimeter of rectangle T is 30
The perimeter has also been enlarged by scale factor 3
This is because perimeter is a length.
The area of S is 1 x 4 = 4 squares
The area of T is 3 x 12 = 36 squares.
The area has been enlarged by scale factor 9.
This is the length scale factor squared because the sides are multiplied. 3² = 9
Test yourself
More on M6: Geometry and measures
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