NZ Level 4

Combined Transformations

Lesson

A transformation is a change, so when we transform a shape, we change it in some way. There are three kinds of transformations: reflections, rotations and transformations. Let's recap these first.

We see reflections all the time- in mirrors, in pools of water and so on. A flip is a reflection over a line or axis. We can see in the picture below that the blue object has been reflected over the vertical axis to create the green image. Notice how they are exactly the same distance from the $y$`y`-axis?

A shape is rotated around a centre point in a circular motion. It does not have to be turned in a full circle, otherwise it would be back and the same point. We commonly see $90^\circ$90° turns (also known as quarter turns), $180^\circ$180° turns (half turns) and $270^\circ$270° turns (three-quarter turns). The triangle below has been rotated anticlockwise $90^\circ$90°.

The whole shape moves the same distance in the same direction, without being rotated or flipped. In the picture below, we can see the diamond has been translated (slid) to the right.

After any of those transformations (rotations, reflections and translations), the shape still has the same size, area, angles and line lengths. However, a shape may be transformed in more than one way. Let's look through some examples now.

What two transformations would be needed to get from Flag $A$`A` to Flag $B$`B`?

Rotation and translation

ATwo translations

BTwo reflections

CReflection and translation

DRotation and translation

ATwo translations

BTwo reflections

CReflection and translation

D

When the original image is rotated $90^\circ$90° clockwise about point $O$`O` and then translated $3$3 units up, what is the new image?

$J$

`J`A$H$

`H`B$N$

`N`C$K$

`K`D$J$

`J`A$H$

`H`B$N$

`N`C$K$

`K`D

A shape is translated, then rotated about its centre.

The same result can always be obtained by a rotation about its centre, followed by a translation.

True or false?

True

AFalse

BTrue

AFalse

B

Use the invariant properties of figures and objects under transformations (refl ection, rotation, translation, or enlargement)