### 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. Itβs 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 π΄π΅πΆ, then we denote the vertices of its image as π΄ prime, π΅ prime,
and πΆ prime. π΄ prime is the image of vertex π΄
after transformation, π΅ prime is the image of vertex π΅, and so on.

Letβs 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 π΄π΅πΆπ·?

Weβre asked to determine what
type of transformation has been applied to the given quadrilateral. We need to treat π΄π΅πΆπ· as
the object and π΄ prime π΅ prime πΆ prime π· 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, letβs 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
π΄π΅πΆπ· is a translation.

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

What kind of transformation is
shown in the figure?

Weβve 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, weβd 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 canβt
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, itβs 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 π₯, π¦ is mapped to the point π₯ prime, π¦ 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 π¦-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 π₯, π¦ is mapped to negative π₯, π¦. This means that for any given
point, the π¦-coordinate remains the same, but the π₯-coordinate changes sign. Itβs beyond the scope of this video
to recall this. But this does in fact correspond to
a reflection in the π¦-axis, as is illustrated on the coordinate grid.

Another example would be the rule
the point π₯, π¦ is mapped to the point π₯ plus three, π¦ plus two. This means that the π₯-coordinate
increases by three and the π¦-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.

Letβs 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 π΄π΅πΆ, where π΄ has coordinates one, three; π΅
has coordinates three, three; and πΆ has coordinates three, seven, after a
transformation π₯, π¦ is mapped to π₯, negative π¦? (a) π΄ prime negative one,
three; π΅ prime negative three, three; and πΆ prime negative three, seven. (b) π΄ prime negative one,
negative three; π΅ prime negative three, negative three; and πΆ prime negative
three, negative seven. (c) π΄ prime one, negative
three; π΅ prime three, negative three; and πΆ prime three, negative seven. Or (d) π΄ prime three, one; π΅
prime three, three; and πΆ prime seven, three.

Weβre given the rule that
describes this transformation. Every point π₯, π¦ is mapped to
the point π₯, negative π¦. In other words, the
π₯-coordinate stays the same, and the π¦-coordinate changes sign or is
multiplied by negative one. We can apply this mapping to
each vertex of triangle π΄π΅πΆ.

The point π΄ with coordinates
one, three is mapped to the point π΄ prime with coordinates one, negative
three. The point π΅ with coordinates
three, three is mapped to three, negative three. And the point πΆ 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
π΄π΅πΆ on a coordinate grid. If we also plot triangle π΄
prime π΅ prime πΆ prime, we can see that the two triangles appear to be
reflections of one another. The mirror line is the
π₯-axis. So, this tells us that we can
represent the transformation of reflection in the π₯-axis as the mapping π₯, π¦
is mapped to π₯, negative π¦, although itβs beyond the current scope to recall
this.

Letβs consider another example in
which weβll identify the type of transformation that has been performed from its
rule.

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

Weβre told that the vertices of
this square are transformed by the transformation π₯, π¦ is mapped to negative
π₯, π¦. So, for each vertex, the
π₯-coordinate changes sign, and the π¦-coordinate stays the same. Weβre 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. Letβs choose the points one,
one; one, two; two, two; and two, one to be the coordinates of π΄, π΅, πΆ, and
π·, respectively.

Under the given transformation,
point π΄ will be mapped to the point π΄ prime with coordinates negative one,
one. And we can add the image of
point π΄ to the coordinate grid. Point π΅ with coordinates one,
two will be mapped to negative one, two. Point πΆ will be mapped to
negative two, two. And point π· 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 π΄
and π΅ 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 itβs not required, we
can also identify that the position of the mirror line is along the π¦-axis. So the answer is option (c), a
reflection.

Letβs 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 π₯, π¦ is mapped to negative π¦,
π₯.

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

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

Although it isnβt 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: itβs now on its
side compared to its original orientation.

Letβs 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
π΄π΅πΆπ·, then the vertices of its image are denoted as π΄ prime π΅ prime πΆ prime
π· prime. We can use a rule to describe a
transformation. In general, we say that the point
with coordinates π₯, π¦ is mapped to the point with coordinates π₯ prime, π¦ prime,
where π₯, π¦ are the coordinates of the object and π₯ prime, π¦ 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.