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Lesson Explainer: Geometric Transformations Mathematics

In this explainer, we will learn how to perform simple transformations on a grid and identify different geometric transformations, such as translation, reflection, and rotation, of some figures.

In geometry, a transformation refers to a change that has occurred to a 2D shape that has changed its position, orientation, or size.

Recall the types of transformations that we have met before: reflection, rotation, and translation. Let’s recap each of these.

A reflection is when an object is flipped in a mirror line to give a new image (which then faces the opposite direction), as shown in the figure below.

Notice that the vertices in the object are the same distance away from the mirror line as the corresponding vertices in the image, as shown below. Furthermore, the mirror line is the perpendicular bisector of a point and its image after bisection.

A rotation is when an object is moved around a fixed point by a set number of degrees either clockwise or counterclockwise to give a new image. In the figure below, the object has been rotated 90∘ counterclockwise to give the new image.

A translation is when an object is moved by sliding it by a set number of units vertically and horizontally. In the figure below, the object has been translated 2 units to the right and 1 unit down.

When transforming objects to their corresponding image, we use notation to denote how the vertices of an object have been transformed to the vertices of its image. For example, if the vertices of an object are 𝐴𝐡𝐢, then if it is transformed, its image would have vertices 𝐴′𝐡′𝐢′.

We can see the use of the notation in the example below, where a rectangle with vertices 𝐴, 𝐡, 𝐢, and 𝐷 has been translated one unit right and one unit down to give its image with vertices 𝐴′, 𝐡′, 𝐢′, and 𝐷′.

Using our understanding of an object and its image, we can define a transformation more formally as follows.

Definition: Geometric Transformations

A transformation is when any point 𝑃 in a plane is transformed to an image point 𝑃′ in the same plane.

In our first example, we will consider how to determine what type of transformation has taken place when an object is mapped onto an image.

Example 1: Identifying a Given Transformation

What type of geometrical transformation has been applied to the quadrilateral 𝐴𝐡𝐢𝐷?

Answer

To determine the type of transformation, we will consider which of the properties of both the object and the image have remained the same and which have been changed.

First, we can see that the vertices of the object occur in the same order as the vertices in the image. In other words, the first quadrilateral has vertices 𝐴𝐡𝐢𝐷 and the image has vertices 𝐴′𝐡′𝐢′𝐷′ in the same order counterclockwise. This means that it cannot be a reflection; otherwise, the vertices would have changed order.

Second, we can see that the vertices of the object occur in the same relative positions to one another as the vertices in the image. In other words, the bottom-left vertex in the object is 𝐴, and the bottom-left vertex in the image is 𝐴′. Similarly, the bottom-right vertex in the object is 𝐡, and the bottom-right vertex in the image is 𝐡′, and so on. This means that it is unlikely to be a rotation, unless the angle is small. We therefore need to consider the orientation.

Third, we can see that the object and the image are the same size and orientation, with the only thing changing being the position, as seen with the arrows on the diagram below.

The sizes and directions of the arrows are all the same; therefore, the transformation must be a translation.

In the next example, we will consider which transformation has been performed on an object to get its image.

Example 2: Identifying a Given Transformation

What kind of transformation is shown in the figure?

Answer

We can see from the figure that the object at the top has been reflected in the mirror line shown to give the image at the bottom. As such, the transformation must be a reflection, since it is a reflection in a mirror line.

When performing transformations on an object, we can use coordinates to describe how the object’s vertices are mapped to its image’s vertices. To do this, we use the notation (π‘₯,𝑦)⟢(π‘₯β€²,𝑦′), where (π‘₯,𝑦) represents the coordinates of the object, (π‘₯β€²,𝑦′) represents the coordinates of the image, and β†’ indicates a transformation has occurred.

In the next example, we will explore how an object can be transformed onto its image using a rule to describe the mapping of its vertices.

Example 3: Transforming a Shape Using Its Coordinates

Which of the following represents the image of △𝐴𝐡𝐢, where 𝐴(1,3), 𝐡(3,3), and 𝐢(3,7), after a transformation (π‘₯,𝑦)β†’(π‘₯,βˆ’π‘¦)?

  1. 𝐴(βˆ’1,3), 𝐡(βˆ’3,3), and 𝐢(βˆ’3,7)
  2. 𝐴(βˆ’1,βˆ’3), 𝐡(βˆ’3,βˆ’3), and 𝐢(βˆ’3,βˆ’7)
  3. 𝐴(1,βˆ’3), 𝐡(3,βˆ’3), and 𝐢(3,βˆ’7)
  4. 𝐴(3,1), 𝐡(3,3), and 𝐢(7,3)

Answer

To find out which of the options given represents the image of △𝐴𝐡𝐢, we substitute the coordinates of each of the vertices of △𝐴𝐡𝐢 into (π‘₯,𝑦)β†’(π‘₯,βˆ’π‘¦) to give us the vertices of the image.

For 𝐴(1,3), we get (π‘₯,𝑦)⟢(π‘₯,βˆ’π‘¦)(1,3)⟢(1,βˆ’3).

So, 𝐴′ has coordinates (1,βˆ’3).

For 𝐡(3,3), we get (π‘₯,𝑦)⟢(π‘₯,βˆ’π‘¦)(3,3)⟢(3,βˆ’3).

So, 𝐡′ has coordinates (3,βˆ’3).

For 𝐢(3,7), we get (π‘₯,𝑦)⟢(π‘₯,βˆ’π‘¦)(3,7)⟢(3,βˆ’7).

So, 𝐢′ has coordinates (3,βˆ’7).

Therefore, the vertices of the image have coordinates 𝐴′(1,βˆ’3), 𝐡′(3,βˆ’3), and 𝐢′(3,βˆ’7), which is option C.

Taking our previous example, we can demonstrate what transformation has taken place on △𝐴𝐡𝐢 by plotting its coordinates and the coordinates of its image. First, let’s plot the coordinates of △𝐴𝐡𝐢, which are 𝐴(1,3), 𝐡(3,3), and 𝐢(3,7).

If we also plot the coordinates of the image, which are 𝐴′(1,βˆ’3), 𝐡′(3,βˆ’3), and 𝐢′(3,βˆ’7), we then get the following.

By comparing the object with the image, we can see that the object has been reflected in the π‘₯-axis to give the image. We can show this by drawing the mirror line, as shown below.

This shows that we can use the transformation (π‘₯,𝑦)β†’(π‘₯,βˆ’π‘¦) to describe a reflection of an object in the π‘₯-axis to give its image. Similarly, we can use the transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,𝑦) to describe a reflection of an object in the 𝑦-axis to give its image.

In the next example, we will consider how to determine the type of transformation that has occurred when given a rule.

Example 4: Identifying the Type of a Transformation given Its Rule

The vertices of a square are transformed by the transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,𝑦). Which of the following geometric transformations is performed?

  1. Translation
  2. Rotation
  3. Reflection

Answer

We are told that the vertices of a square are transformed by the transformation (π‘₯,𝑦)β†’(βˆ’π‘₯,𝑦); therefore, in order to help us work out which transformation has taken place, it is helpful to assign coordinates to the vertices of the square.

Since we are not told any vertices, let’s consider a specific example by choosing vertices ourselves. As we have a square, we need to make sure each of the sides are the same length and at 90 degrees to one another. We will call the vertices 𝐴, 𝐡, 𝐢, and 𝐷. To make working out coordinates easier, we will choose vertices that have edges of length one unit, with the bottom-left vertex at 𝐴(1,1), the top-left vertex at 𝐡(1,2), the top-right vertex at 𝐢(2,2), and the bottom-right vertex at 𝐷(2,1). Plotting these gives us the following.

Next, we can apply the rule of the transformation, (π‘₯,𝑦)β†’(βˆ’π‘₯,𝑦), to each of our chosen coordinates to get the coordinates of the image with vertices 𝐴′𝐡′𝐢′𝐷′.

For 𝐴(1,1), we get (π‘₯,𝑦)⟢(βˆ’π‘₯,𝑦)(1,1)⟢(βˆ’1,1).

So, 𝐴′ has coordinates (βˆ’1,1).

For 𝐡(1,2), we get (π‘₯,𝑦)⟢(βˆ’π‘₯,𝑦)(1,2)⟢(βˆ’1,2).

So, 𝐡′ has coordinates (βˆ’1,2).

For 𝐢(2,2), we get (π‘₯,𝑦)⟢(βˆ’π‘₯,𝑦)(2,2)⟢(βˆ’2,2).

So, 𝐢′ has coordinates (βˆ’2,2).

For 𝐷(2,1), we get (π‘₯,𝑦)⟢(βˆ’π‘₯,𝑦)(2,1)⟢(βˆ’2,1).

So, 𝐷′ has coordinates (βˆ’2,1).

This then gives us the coordinates of the image of square 𝐴𝐡𝐢𝐷 as 𝐴′(βˆ’1,1), 𝐡′(βˆ’1,2), 𝐢′(βˆ’2,2), and 𝐷′(βˆ’2,1). Plotting these on the same coordinate grid as the original object, we get the following.

By comparing the coordinates of square 𝐴𝐡𝐢𝐷 with the coordinates of the image, we can see that they have been reflected in the 𝑦-axis. We can illustrate this by drawing the reflection line as follows.

Therefore, the geometric transformation that has been performed is option C, a reflection.

Next, we will discuss how to sketch the image of an object when a transformation has been performed.

Example 5: Sketching a Shape after a Transformation

The given figure shows a triangle on the coordinate plane. Sketch the image of the triangle after the geometric transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯). Which of the following matches your sketch?

Answer

To help us sketch the image of the object, we can use the coordinates of the object and the rule for the transformation, (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯).

Looking at the object below, we can see that the coordinates of the vertices are 𝐴(2,4), 𝐡(3,1), and 𝐢(1,1).

Having found the vertices, we can then apply the rule for the geometric transformation (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯) to each set of coordinates to find the coordinates of the image.

For 𝐴(2,4), to use the rule (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), we let π‘₯=2 and 𝑦=4 (since they are the π‘₯-coordinate and 𝑦-coordinate for 𝐴). Substituting this into (βˆ’π‘¦,π‘₯), we can see that the π‘₯-coordinate and 𝑦-coordinate have switched places and that there is a change of sign in the first coordinate. This then gives us the coordinates (βˆ’4,2), so the coordinates of 𝐴′ are (βˆ’4,2).

For 𝐡(3,1), we let π‘₯=3 and 𝑦=1. Substituting this into the rule (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯) we get (3,1)⟢(βˆ’1,3).

So, the coordinates of 𝐡′ are (βˆ’1,3).

For 𝐢(1,1), we let π‘₯=1 and 𝑦=1. Substituting this into the rule (π‘₯,𝑦)β†’(βˆ’π‘¦,π‘₯), we get (1,1)⟢(βˆ’1,1).

So, the coordinates of 𝐢′ are (βˆ’1,1).

Now that we have found the coordinates of the vertices of the image, 𝐴′(βˆ’4,2), 𝐡′(βˆ’1,3), and 𝐢′(βˆ’1,1), we can plot them on the set of axes. Doing so gives us the following.

Lastly, we join up the vertices with edges to obtain a sketch of the image. This gives us the following.

Comparing this with our options, we can see this is the same as the sketch in option A. Interestingly, we can see that this is a rotation.

In this explainer, we have learned about different types of geometric transformations, how to denote the image of an object that has been transformed, and how to use rules for transformations. Let’s recap the key points.

Key Points

  • We can transform an object using the following transformations:
    • A reflection is when an object is flipped in a mirror line to obtain its image.
    • A rotation is when an object is moved around a fixed point to obtain its image.
    • A translation is when an object is moved by sliding it a number of units left or right and up or down.
  • If we have an object with vertices 𝐴𝐡𝐢𝐷, then the vertices of its image are denoted 𝐴′𝐡′𝐢′𝐷′.
  • We can use a rule to describe the transformation that had taken place, which we generally denote by (π‘₯,𝑦)⟢(π‘₯β€²,𝑦′), where (π‘₯,𝑦) represents the coordinates of the vertices of the object and (π‘₯β€²,𝑦′) represents the coordinates of the vertices of the image.

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