Introduction
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. The transformations are able to be 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.
Exploration
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.
When an object is being translated, its image moves without changing size or shape, rotating or reflecting.
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 has been translated 4 units to the right to produce triangle B. We can see this most easily by comparing the location of the bottom vertex of each triangle.
The translation from A to B is 4 units right.
Examples
Example 1
Plot the translation of the point by moving it 11 units to the left and 9 units down.
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.
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.
Exploration
Play with the following applet by dragging the sliders, and consider the way that the image moves relative to the object.
The vertices of the triangle and its reflection have the same distance from the mirror line. The reflection is the mirror image of the original triangle on the other side of the line.
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 has been flipped over the line of reflection, resulting in triangle B. Notice that the corresponding points on A and B are equidistant from the line of reflection.
Examples
Example 2
Consider the triangle below. Draw the result of the reflection across the line x=-1.
A reflection is when we flip an object or shape across a line. Like a mirror, the object is exactly the same size, just flipped in position. Every point on the object or shape has a corresponding point on the image, and they will both be the same distance from the reflection line.
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.
Exploration
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 size and shape of the image of the object stays the same as it turns around the centre 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 being rotated about the point O.
Square A is rotated 135\degree clockwise, or 225\degree anticlockwise, about O resulting in square 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.
Examples
Example 3
Consider the shape below. Draw the result of a rotation by 180\degree clockwise about point A.
Rotation of a shape can be done by rotating it about a fixed point or the centre of rotation.