Invariant Points

Example 1: Invariant Points – Reflections 

Shape A is shown below.

State the coordinates of any invariant points when shape A is reflected in the line x=1.

[1 mark]

In order to see if there are any invariant points in a transformation, we need to do the transformation.

First, we draw the line x=1. It is a horizontal line that passes through 1 on the x-axis.

Then, completing the reflection we get the picture shown below.

The reflected shape is shown in green.

As we can see, all points on the shape have moved to a different location except for one the one that was located on the reflection line.

Here we’ve marked this with a red X.

So, we can conclude that there is 1 invariant point, and the coordinates of it are

(1, 1)

Example 2: Invariant Points – Translation 

Perform the following two transformations to shape A, in the given order:

1. Rotate 180\degree about the point (1, 0)
2. Translate by the vector \begin{pmatrix} 2 \\ 2 \end{pmatrix}.

Label the final shape C and state the coordinates of any invariant points.

[2 marks]

Firstly, we need to perform the rotation. Here, we’ll mark a point at (1, 0) to note the centre of rotation, and then we’ll call the resulting shape B.

Then, we have to translate by the vector

\begin{pmatrix} 2 \\ 2 \end{pmatrix}

This means moving shape B 2 to the right, and then 2 up.

This completes the transformation, so we label the resulting shape C. The full picture is shown below.

We can now see one invariant point between C and A, with coordinates

(2, 1)

Here, marked with a red X.