11.03 Transformations

A transformation is when an object or point is moved, turned, flipped or changed in shape or size. We're going to consider a few types of transformations in which the shape and size of the image remain the same.

These transformations are known as:

• translations
• reflections
• rotations

Multiple transformations can be combined in order to create a unique final image. Inverse transformations are when the transformations are reversed in order for us to find the object in its original position and orientation. Let's consider the three transformations in which the shape and size of the object remain the same.

Translations

A translation is a type of transformation for which the size, shape and rotation of the object being transformed does not change. We can think of sliding a shape across a page or screen as a translation.

Play with the applet below by dragging the sliders to move the triangle around. Notice how the triangle does not change its orientation or size, it just slides from one place on the grid to another.

That was an example of a translation!

When translating a shape, it can be helpful to choose a point on the shape which will be our reference point. To describe the translation between the original shape and the translated image, we can compare the corresponding reference points on both shapes. This allows us to easily find the horizontal and vertical distance between them.

If we were to choose two points on the original and translated image that do not match, our description of the translation will be incorrect.

In the figure below, triangle $A$A has been translated $4$4 units to the right to produce triangle $B$B. We can see this most easily by comparing the location of the bottom vertex of each triangle.

Translation from $A$A to $B$B is $4$4 units right.

 

Worked example

example 1

Solve:

a) What is the translation of triangle $A$A to triangle $B$B?

Think: Look for a corresponding point on both triangles, like one of the vertices. Then we can count the units between the corresponding point on triangle $A$A and triangle $B$B to find the translation. First let's count how many units triangle $A$A has been moved horizontally, and then count how many units it has been moved vertically.

Do: We can count that triangle $A$A has been moved $5$5 units to the right, let's draw that on the diagram to keep track.

Triangle $A$A moved $5$5 units right.

From there we can count that triangle $A$A has been moved $2$2 units up to get to triangle $B$B.

Triangle $A$A moved $5$5 units right and $2$2 units up.

So the translation of triangle $A$A to triangle $B$B is $5$5 units right and $2$2 units up.

b) What is the inverse transformation of triangle $B$B to triangle $A$A?

Think: We need to do the reverse translations to get back to triangle $A$A from $B$B. Left is the opposite of right, and down is the opposite of up.

Do: Therefore, the inverse transformation is $5$5 units left and $2$2 units down.

Reflect: When describing a translation it is important to pay attention to the order of the movement we are describing.

Reflections

We can think of the reflection of an object or point as the original object being flipped over a line of reflection. This means that the original object and the reflected object have an equal perpendicular distance between the line of reflection. Note that the line of reflection does not have to be strictly horizontal or vertical and can be in any direction.

Play with the following applet by dragging the sliders, and consider the way that the image moves relative to the object.

That was a reflection! One way to think about this type of transformation is that we are flipping the object over the line of reflection to created the reflected image.

In the figure below, triangle $A$A has been flipped over the line of reflection, resulting in triangle $B$B. Notice that the corresponding points on $A$A and $B$B are equidistant from the line of reflection.

Reflecting triangle $A$A across the vertical line, resulting in triangle $B$B.

The inverse transformation of reflecting a shape across a line, is just to reflect the shape across the same line again. 

Worked example

example 2

Solve: Reflect the following triangle $A$A across the $y$y-axis.

 

Think: If we need to reflect across the $y$y-axis, we need to choose some reference points on triangle $A$A to transform. In order to reflect our shape, the transformation must maintain the same distance between the $y$y-axis. All we need to do is flip!

Do: First let's choose two reference points and measure their distance from the $y$y-axis. The vertices of the triangle are perfect reference points.

Counting the distance of the vertices from the $y$y-axis.

 

Now let's plot the corresponding points of the reflection, keep in mind they must be the same distance from the $y$y-axis as in the original triangle.

Plotting the points of reflection.

 

Now we can draw the rest of the shape. To double check if the image has been reflected correctly, we can draw the original triangle and its reflection on a piece of paper. Then if we fold the page across the line of reflection and all points on both images match up, then we've correctly reflected the triangle!

Triangle $B$B is the reflection of triangle $A$A across the $y$y-axis.

Rotations

Another type of transformation, known as a rotation comes from rotating an image about a fixed point. The fixed point the image is rotated about is known as the centre of rotation.

Play with the applet below to explore the rotation transformation. Try changing the shape and size of the original triangle, then use the slider to change the angle of rotation.

The centre of rotation does not always have to be a point on the image. Consider the figure below, which shows square $A$A being rotated about the point $O$O.

Square $A$A is rotated $135^\circ$135° clockwise, or $225^\circ$225° anticlockwise, about $O$O resulting in square $B$B.

 

We can use a protractor to measure the angle of rotation between the original object and the rotated object. We can also use a protractor to measure the correct angle of rotation so we can draw the transformation.

Worked example

example 3

Solve: Which is the correct image after triangle $A$A is rotated $90^\circ$90° anticlockwise about the point $O$O?

 

Think: What point is the image being rotated around and which direction is the image being rotated? We can draw some horizontal and vertical lines to help us visualise the rotation.

Do: First lets draw some horizontal and vertical lines so we can measure the angle of rotation.

Grid split up into four quadrants, each with an angle of $90^\circ$90°.

 

Since we know that each quadrant has an angle of $90^\circ$90°, all we need to do is rotate the triangle $A$A to the next quadrant in an anticlockwise direction.

Rotating triangle $A$A by $90^\circ$90° anticlockwise around point $O$O leaves us at triangle $D$D, therefore triangle $D$D is the transformed shape.

Reflect: If we were to instead rotate triangle $A$A by $90^\circ$90° clockwise, the correct image would then be triangle $B$B. This is the inverse transformation to rotating triangle  $A$A by $90^\circ$90° clockwise

Practice questions

question 1

question 2

question 3