Lesson Video: Geometric Transformations Mathematics

Video Transcript

In this video, 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.

A geometric transformation refers to a change that has occurred to a two-dimensional shape that has altered its position, orientation, or size. A translation is when an object is moved by sliding it a set number of units horizontally and vertically to a new position. We call the object in its new position the image. For a translation, the object and its image look exactly the same but are just in different positions.

A rotation is when an object is turned around a fixed point by a set number of degrees, either in a clockwise or counterclockwise direction. The object and its image will be congruent, that is, identical, but will appear in different orientations. A reflection is when an object is flipped in a mirror line to give a new image which faces in the opposite direction. Notice that each vertex of the image is the same distance away from the mirror as the corresponding vertex on the object. It’s just on the other side.

When transforming objects, 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 𝐴𝐵𝐶, then we denote the vertices of its image as 𝐴 prime, 𝐵 prime, and 𝐶 prime. 𝐴 prime is the image of vertex 𝐴 after transformation, 𝐵 prime is the image of vertex 𝐵, and so on.

Let’s now consider some examples. In our first example, we will see how to determine what type of transformation has taken place when an object is mapped onto an image.

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

We’re asked to determine what type of transformation has been applied to the given quadrilateral. We need to treat 𝐴𝐵𝐶𝐷 as the object and 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime as its image following transformation. The types of transformation we need to consider are translation, rotation, and reflection. And in order to determine which of these has been applied, let’s consider the properties of the object and its image.

First, we can see that all corresponding vertices are in the same position on both shapes. Starting in the lower-left corner, the vertices on both are labeled in alphabetical order in the counterclockwise direction. We can also see that the object and its image are the same size and in the same orientation. This means that the transformation applied cannot be a rotation or a reflection, as for each of these, the object and its image would not be in the same orientation.

In the case of a reflection, the clockwise or counterclockwise labeling of the vertices would also not be preserved. In fact, the only thing that has changed is the position of the shape. Each vertex has moved the same distance in the same direction, which corresponds to a translation. So, we can conclude that the type of geometrical transformation that has been applied to the quadrilateral 𝐴𝐵𝐶𝐷 is a translation.

Let’s consider another example in which we determine the type of transformation that has been applied.

What kind of transformation is shown in the figure?

We’ve been given an object and its image following a transformation and asked to determine the type of transformation that has been performed. The three types of transformation we need to consider are translation, rotation, and reflection. If the transformation was a translation, then the image would be exactly the same size and shape as the object and in the same orientation. The only thing that would change is its position. However, we can see that the image does not look exactly the same as the object. It is in a different orientation. And so, this rules out a translation.

If the transformation was a rotation, then the image would be exactly the same shape and size, but in a different position and orientation. A point has been marked on the figure, so this is a possible point about which the shape has been rotated. In order for the image to appear in the correct position below the dotted line, we’d need to rotate the shape by 180 degrees about this point. But if we did so, the image of the shape would actually be in the same orientation as the object because this shape has rotational symmetry. The transformation therefore can’t be a rotation.

The final possibility is a reflection, and there is a dotted line drawn on the figure, which is a possible mirror line. We can see that corresponding vertices on the two shapes are the same distance away from this horizontal line, but in opposite directions. The shape has also been flipped, which we can see more easily if we color the corresponding sides. The pink side is originally at the top of the object and is now at the bottom of the image. But in both cases, it’s the side furthest from the mirror. We can conclude then that the type of transformation shown is a reflection in a horizontal mirror.

We can also apply transformations to objects, such as individual points, line segments, or shapes that are drawn on a coordinate grid. In this case, we can consider the effect of a given transformation on the coordinates of a point. In general, we say that the point with coordinates 𝑥, 𝑦 is mapped to the point 𝑥 prime, 𝑦 prime by a transformation. The arrow between the two pairs of coordinates indicates that a transformation has occurred. For example, for the transformation shown, which is a reflection in a mirror placed along the 𝑦-axis, the point negative four, five has been mapped to the point four, five.

We can also use this notation to describe transformations using rules. For example, a particular transformation might be defined by 𝑥, 𝑦 is mapped to negative 𝑥, 𝑦. This means that for any given point, the 𝑦-coordinate remains the same, but the 𝑥-coordinate changes sign. It’s beyond the scope of this video to recall this. But this does in fact correspond to a reflection in the 𝑦-axis, as is illustrated on the coordinate grid.

Another example would be the rule the point 𝑥, 𝑦 is mapped to the point 𝑥 plus three, 𝑦 plus two. This means that the 𝑥-coordinate increases by three and the 𝑦-coordinate increases by two, which does in fact correspond to a translation where each point is moved three units to the right and two units up.

Let’s now look at an example of how an object can be transformed onto its image using a rule to describe the mapping of its vertices.

Which of the following represents the image of triangle 𝐴𝐵𝐶, where 𝐴 has coordinates one, three; 𝐵 has coordinates three, three; and 𝐶 has coordinates three, seven, after a transformation 𝑥, 𝑦 is mapped to 𝑥, negative 𝑦? (a) 𝐴 prime negative one, three; 𝐵 prime negative three, three; and 𝐶 prime negative three, seven. (b) 𝐴 prime negative one, negative three; 𝐵 prime negative three, negative three; and 𝐶 prime negative three, negative seven. (c) 𝐴 prime one, negative three; 𝐵 prime three, negative three; and 𝐶 prime three, negative seven. Or (d) 𝐴 prime three, one; 𝐵 prime three, three; and 𝐶 prime seven, three.

We’re given the rule that describes this transformation. Every point 𝑥, 𝑦 is mapped to the point 𝑥, negative 𝑦. In other words, the 𝑥-coordinate stays the same, and the 𝑦-coordinate changes sign or is multiplied by negative one. We can apply this mapping to each vertex of triangle 𝐴𝐵𝐶.

The point 𝐴 with coordinates one, three is mapped to the point 𝐴 prime with coordinates one, negative three. The point 𝐵 with coordinates three, three is mapped to three, negative three. And the point 𝐶 with coordinates three, seven is mapped to the point three, negative seven. Looking carefully at the four options given, we can see that this set of coordinates is option (c).

We can also visualize the effect of this transformation graphically. Here, we have plotted triangle 𝐴𝐵𝐶 on a coordinate grid. If we also plot triangle 𝐴 prime 𝐵 prime 𝐶 prime, we can see that the two triangles appear to be reflections of one another. The mirror line is the 𝑥-axis. So, this tells us that we can represent the transformation of reflection in the 𝑥-axis as the mapping 𝑥, 𝑦 is mapped to 𝑥, negative 𝑦, although it’s beyond the current scope to recall this.

Let’s consider another example in which we’ll identify the type of transformation that has been performed from its rule.

The vertices of a square are transformed by the transformation 𝑥, 𝑦 is mapped to negative 𝑥, 𝑦. Which of the following geometric transformations is performed? (a) Translation, (b) rotation, or (c) reflection.

We’re told that the vertices of this square are transformed by the transformation 𝑥, 𝑦 is mapped to negative 𝑥, 𝑦. So, for each vertex, the 𝑥-coordinate changes sign, and the 𝑦-coordinate stays the same. We’re not given the coordinates of the vertices of this square. So we may find it helpful to choose some arbitrary coordinates ourselves to help with visualizing the transformation. Let’s choose the points one, one; one, two; two, two; and two, one to be the coordinates of 𝐴, 𝐵, 𝐶, and 𝐷, respectively.

Under the given transformation, point 𝐴 will be mapped to the point 𝐴 prime with coordinates negative one, one. And we can add the image of point 𝐴 to the coordinate grid. Point 𝐵 with coordinates one, two will be mapped to negative one, two. Point 𝐶 will be mapped to negative two, two. And point 𝐷 will be mapped to negative two, one.

Now we need to compare the two shapes to determine the type of transformation that has occurred. We can see that the square has been flipped over. For example, the vertices 𝐴 and 𝐵 are originally on the left of the square, and in its image, they are on the right. The type of transformation is therefore a reflection. Whilst it’s not required, we can also identify that the position of the mirror line is along the 𝑦-axis. So the answer is option (c), a reflection.

Let’s now consider one final example.

The given figure shows a triangle on the coordinate plane. Sketch the image of the triangle after the geometric transformation 𝑥, 𝑦 is mapped to negative 𝑦, 𝑥.

We are told that the geometric transformation we need to apply is at a general point with coordinates 𝑥, 𝑦 is mapped to the point with coordinates negative 𝑦, 𝑥. This means that the 𝑥- and 𝑦-coordinates swap around, and then we also change the sign of the new 𝑥-coordinate. We can apply this mapping to the coordinates of each vertex of triangle 𝐴𝐵𝐶 individually.

The coordinates of vertex 𝐴 are two, four. Applying the given mapping, so swapping the coordinates around and then changing the sign of the new 𝑥-coordinate, gives the point 𝐴 prime with coordinates negative four, two. We can plot this point on the coordinate grid to show the image of point 𝐴. The coordinates of point 𝐵 are three, one. Applying the given mapping gives the coordinates of 𝐵 prime as negative one, three, and then we can also plot this point. Finally, the coordinates of point 𝐶 are one, one, which under the given transformation is mapped to negative one, one. Plotting point 𝐶 prime and then connecting the three points together gives the image of triangle 𝐴𝐵𝐶 after the given transformation.

Although it isn’t required, we can also observe that the type of transformation that has been applied is a rotation, because the orientation of the triangle has changed: it’s now on its side compared to its original orientation.

Let’s now summarize the key points from this video. We can transform an object using the following transformations. Firstly, a reflection is when an object is flipped in a mirror line to obtain its image. Secondly, in a rotation, an object is turned a set number of degrees about a fixed point to obtain its image. The direction of rotation can be either clockwise or counterclockwise. And finally, a translation is when an object is moved by sliding it a number of units horizontally and/or vertically.

If we have an object with vertices 𝐴𝐵𝐶𝐷, then the vertices of its image are denoted as 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime. We can use a rule to describe a transformation. In general, we say that the point with coordinates 𝑥, 𝑦 is mapped to the point with coordinates 𝑥 prime, 𝑦 prime, where 𝑥, 𝑦 are the coordinates of the object and 𝑥 prime, 𝑦 prime are the coordinates of its image following transformation. We also saw some examples of the types of rules that we might encounter to describe transformations.