Maths

Transformations

Transforming an object means changing its appearance. There are many ways to do this, but we are going to concentrate on just four of them.

In each of these sections, we refer to the original shape as the object and the transformed shape as the image.

Translations

If we translate an object, we move it up or down or from side to side. But we do not change its shape, size or direction.

Question

Which of the following triangles (P, Q or R) is a translation of triangle ABC?

image: a graph with 4 triangles plotted

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Answer

Q is a translation of triangle ABC. Triangle ABC has been translated 3 squares to the right and 2 squares up.

P is not a translation of triangle ABC because the object and its image are facing in opposite directions.

R is not a translation of triangle ABC because the object and its image have different side lengths. They are different shapes.

Remember

When we translate an object, every vertex (corner) must be moved in the same way.

Example:

image: a graph with 2 triangles plotted

Triangle PQR has been translated 3 squares down and 4 squares to the right. All of the vertices have been translated in the same way, and the object and its image are exactly the same shape and size.

Using vectors to describe translations

The line AB has been translated 4 units to the right and 2 units upwards. We say that the displacement vector is 4 over 2.

image: a graph with two lines plotted, the first at 1,1 and 2,4 the second at 5,3 and 6,6

We always write the horizontal displacement at the top of the vector and the vertical displacement at the bottom.

A move downwards or to the left is indicated by a - sign:

image: a graph with x plotted at two locations, 1,0 and 3,3

The object X has been translated by the vector -2 over -3 to give its image, X'.

Question

P' (7, 2) is the image of P (2, 11) after a translation. What vector describes this translation?

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Answer

Did you get 5 over -9 ?

If so - well done.

To work out the horizontal movement after the translation, subtract the x-coordinates for p from p':

7 - 2 = 5

To work out the vertical movement after the translation, subtract the y-coordinates for p from p':

2 - 11 = - 9

The vector that describes the translation is:5 over -9

If you are not sure, try drawing a sketch and marking the points P and P' on it.

The shape has been translated 5 units to the right and 9 units downwards.

Remember that any move to the left or downwards needs a - sign.

Reflections

When an object is transformed by a reflection the object and its image are always the same perpendicular distance from the mirror line.

Perpendicular means 'at right angles to'.

Examples:

image: a graph with two triangles plotted with the mirror line labelled

A and A' are the same perpendicular distance from the mirror line, as are B and B' and C and C'.

The object (ABC) has been reflected in the mirror line to give the image (A'B'C').

image: 4 lines of varying length with a mirror line down the middle of each

P and P' are the same perpendicular distance from the mirror line, as are Q and Q', R and R' and S and S'.

The object (PQRS) has been reflected in the mirror line to give the image (P'Q'R'S').

Reflections in y = x, y = - x, y = c and x = c

In your exam, you might have to reflect a shape in a given line. You will need to remember that:

y = c describes a horizontal line through c.

x = c describes a vertical line through c.

A point and its image are always the same perpendicular distance from the mirror line. Have a look at the reflections below:

The object is triangle ABC



reflection in x = -1transformation 2



reflection in y = 1transformation 3



reflection in y = xtransformation 4



reflection in y = -xtransformation 5



Look at this table:

 

 Coordinates of verticesReflection in y = xReflection in y = -x
A(1, 2)(2, 1) (-2, -1)
B(1, 5) (5, 1) (- 5, -1)
C(3, 2) (2, 3) (-2, -3)

Can you see the pattern?

After a reflection in the line y = x, the x and y coordinates swap over.

After a reflection in the line y = - x, the x and y coordinates swap over and change signs.

Activity

Creat your own reflections using the transformations activity.

Rotations

We can rotate an object 1/4 turn, 1/2 turn or 3/4 turn around a given centre.

Example

Activity

You may also be asked to rotate an object around the origin. Practise rotations using the transformations activity.

Rotations about any point

Remember that a point and its image are always the same distance from the centre of rotation.

Example

Describe the rotation which moves ABCD to A'B'C'D' in this diagram.

image: a graph with two squares plotted the first: A (1,1), B (1,4), C (3,4), D (3,1). The second A' (2, minus 2), D' (2, minus 4), B' (5, minus 2), C' (5, minus 4)

The centre of rotation is (0, -1).

We can see that A is the same distance from (0, -1) as A'.

D is the same distance from (0, -1) as D'.

The angle of rotation is 90° clockwise.

It is possible to use constructions to find the centre of rotation, but this can be time-consuming and complicated. Try to find the centre by observation, but remember that each point and its image must be the same distance from the centre of rotation.

Enlargements

When working out enlargements, you will need to know the scale factor and centre of enlargement.

  • The scale factor tells us by how much the object has been enlarged.
  • The centre of enlargement tells us where the enlargement is being measured from.

Example:

image: two triangles, the smaller is 1 3rd the size of the larger triangle. The smaller triangle is marked Q, P and R. The larger triangle is marked Q', P' and R'.

P'Q' = 3 × PQ

Q'R' = 3 × QR

P'R' = 3 × PR

Therefore, we say that the transformation is an enlargement with scale factor 3.

You may be asked to enlarge a shape with a given centre of enlargement. Look through this slideshow to see an example.

To enlarge an object, scale factor 2.

The object is triangle ABCenlargement 1

The centre of enlargement is O.



O is the centre of enlargementenlargement 2

Draw lines from the centre of enlargement.



The image points are A' B' C'enlargement 3

OA' is twice as far as OA, because the scale factor is 2.



The image is triangle A'B'C'enlargement 4

Draw the image shape.



To enlarge an object, scale factor 3.

This method is identical to the previous one, but

OA' = 3 × OA

OB' = 3 × OB

OC' = 3 × OC

The centre of enlargement is not always outside the object. Look at the following examples:

The next diagram shows an enlargement, scale factor 3. The centre of enlargement, X, is also a point of the object WXYZ. The result is that X' is in the same place as X.

image: an example of an enlargement with a scale factor of 3

The next diagram shows an enlargement, scale factor 2, centre O, where O is inside the object. The result is that the image surrounds the object.

image: an example of an enlargement with a scale factor of 2

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