Video Transcript
In this video, we will learn how to
carry out and describe combinations of transformations. The four different types of
transformation are translation, reflection, rotation, and dilation. So, letโs begin by recalling what
each of these transformations would look like.
The first transformation we can
look at is translation. Translation is when we move a
shape, but its size, shape, and orientation wonโt change. If, for example, we translated this
triangle five units right and two units down, then the image would appear here. Image is the word that we give for
a shape after a transformation. To perform a translation, we take
each of the vertices. Here, we were asked to translate
five units to the right and two units down. So, doing this for each vertex
gives us the location for each of the new vertices.
If weโre looking at a shape and its
image and we donโt know which transformation it is, we can spot a translation
because the object and its image will be the same shape and size. And notably, translations will
always produce an object and its image in the same orientation. And then to describe a translation,
we would need to say how the shape has moved, either in the form of words โ for
example, five units right and two units down โ or in the form of notation, such as
vector notation.
And the second type of
transformation is reflection, which is when we flip or reflect an object across a
line. For example, if we reflected this
rectangle in the ๐ฆ-axis, then the image would appear on the other side. So, if weโre looking at a
transformation and weโre trying to establish which one it is, a reflected object and
its image will be the same size, but one will look like a reflected or mirror image
of the other. When describing a reflection, we
need to state the line of reflection. In our example, this wouldโve been
the ๐ฆ-axis.
Next step, letโs have a look at
rotation, which is when we turn an object about a point or center of rotation. For example, if we wanted to rotate
our blue triangle 90 degrees clockwise about the point marked ๐ด, then the image
would appear as shown. To identify a rotation, the object
and its image would be the same size, but the image would look rotated. To describe a rotation, we need to
say a few more things. We need to state the center of
rotation about which the object is rotated. This can either be by describing a
point denoted by a letter or by using a coordinate in a grid. We also need to give the angle of
rotation and the direction, for example, clockwise or counterclockwise.
The final transformation is
dilation, sometimes referred to as enlargement. This is when we expand or contract
an object by a given scale factor. For example, if we take our blue
triangle and dilate it by scale factor two about this point ๐ด, then the image would
look as shown. We can see that each of the lengths
on the image are twice as long as they are on the original shape. We can easily spot a dilation
because the object and its image are different size, unless the scale factor is one
or negative one. In a dilation, the object and its
image are usually the same orientation. But be careful as negative scale
factors often make a shape look like itโs been flipped or rotated. To describe a dilation, we need to
give the center of dilation and the scale factor, remembering that scale factors can
be fractional and negative values too.
So, when it comes to combining
transformations, that means we perform one transformation and then another
transformation and maybe even another one after that. So, if weโre carrying out a
transformation on a shape and we find the image after this transformation, then one
of the things we need to be careful of when weโre carrying out a second
transformation is whether this transformation comes from transforming the image or
from transforming the original shape. Weโll now look at some questions on
combining transformations. In the first question, weโll
actually carry out these transformations.
๐ด๐ต๐ถ๐ท is reflected in the
๐ฅ-axis and then translated five units to the right. What is the image of point ๐ต?
Here, we have a combination of
transformations, firstly reflection and then translation. We should always perform them in
the order in which weโre given. Weโll, therefore, begin with
reflection. And weโre told that the line of
reflection is the ๐ฅ-axis. Starting with vertex ๐ต, we can see
that this is six units up from the ๐ฅ-axis. So, when we reflect vertex ๐ต, it
will also be six units away from the ๐ฅ-axis but this time in the negative
direction. We can call the reflected vertex ๐ต
prime.
Vertex ๐ด is seven units away from
the ๐ฅ-axis. So, when we reflect it, it will
also be seven units away but in the negative direction. And we can create the reflected
vertex ๐ด prime. Vertices ๐ถ and ๐ท can be reflected
in the same way, giving us the images of ๐ถ prime and ๐ท prime. We can then join the vertices to
give the image. Notice that our two shapes ๐ด๐ต๐ถ๐ท
and ๐ด prime ๐ต prime ๐ถ prime ๐ท prime are the same size, but they look
reflected.
The second transformation that we
need to carry out is the translation. Weโre told that ๐ด๐ต๐ถ๐ท is
reflected and then translated. So, weโll be carrying out the
translation on the image rather than on the original shape ๐ด๐ต๐ถ๐ท. Each vertex will be translated or
moved five units to the right. So, letโs start with ๐ถ prime. If we count five hops or jumps to
the right, then we can see that the image of ๐ถ prime, written as ๐ถ double prime,
will lie as shown. We can find the images of ๐ด prime
and ๐ท prime. And the image of ๐ต prime will be
๐ต double prime, and we note that it lies on top of ๐ถ prime. We have then created a second
image.
We were asked for the image of
point ๐ต after these two transformations. So, we can give negative two,
negative six as our answer. In this type of question, we
couldโve simply transformed point ๐ต. But often, itโs helpful to draw the
whole shape as a check that weโve performed the transformations correctly.
Weโll now look at a question where
we need to describe a combination of transformations.
The triangle ๐ด๐ต๐ถ has been
transformed onto triangle ๐ด prime ๐ต prime ๐ถ prime which has been transformed onto
triangle ๐ด double prime ๐ต double prime ๐ถ double prime. Describe the single transformation
that maps ๐ด๐ต๐ถ to ๐ด prime ๐ต prime ๐ถ prime. Describe the single transformation
that maps ๐ด prime ๐ต prime ๐ถ prime to ๐ด double prime ๐ต double prime ๐ถ double
prime. Hence, are triangles ๐ด๐ต๐ถ and ๐ด
double prime ๐ต double prime ๐ถ double prime similar?
In this question, we have a series
or combinations of transformations which begin with triangle ๐ด๐ต๐ถ. The first transformation takes us
to the second smaller triangle. And the second transformation takes
us to the larger triangle of ๐ด double prime ๐ต double prime ๐ถ double prime. We can begin by finding the first
transformation between the two smaller triangles.
We can recall that the four types
of transformation are translation, reflection, rotation, and dilation. If we look at the two triangles
๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime, we can see that theyโre the same size. Therefore, this is unlikely to be a
dilation, as this usually changes the size. We can see that our two triangles
are at different orientation. So, we can rule out translation as
this moves the shape but keeps it the same way up.
The two triangles are not a mirror
image of each other. So, we can rule out reflection. Letโs see if we could describe this
transformation as a rotation. Starting with triangle ๐ด๐ต๐ถ, if
we rotated this in a clockwise direction, we could then work out the angles for
which this must be rotated. Between ๐ด and ๐ด prime, thereโs a
right angle of 90 degrees. Between ๐ถ and ๐ถ prime, we can
also see a 90-degree angle. This will confirm that we have a
rotation of 90 degrees.
Notice that weโve found this
rotation by moving our vertices through the same point or coordinate. This will be the center of
rotation. So, in order to fully describe this
transformation, we need to put together the facts that weโve discovered โ the center
of rotation, the angle, and the direction. We can then give our answer to the
first part as a 90-degrees clockwise rotation about the origin. We could, of course, also have
described this as a 270-degree counterclockwise rotation about the origin. Giving the coordinate zero, zero
instead of the origin would also have been valid.
Letโs look at the second
question. Describe the single transformation
that maps ๐ด prime ๐ต prime ๐ถ prime to ๐ด double prime ๐ต double prime ๐ถ double
prime. We need to be careful that weโre
using the second smaller triangle and asking how we go from this to the larger
triangle. If we look at our list of possible
transformations, the first three โ translation, reflection, and rotation โ keep the
object and its image the same size. As the triangles here are different
sizes, that means thereโs just one possible transformation, dilation.
To describe a dilation, we need to
find the center of dilation and the scale factor. We can find the scale factor
relatively easily by looking at how the length on the image have increased from the
length on the original shape. We can compare the lengths of ๐ด
double prime ๐ต double prime and ๐ด prime ๐ต prime. We can see that on ๐ด prime ๐ต
prime, the length is two units, and on the top length of ๐ด double prime ๐ต double
prime, this is four units long. So, it looks like weโll have a
scale factor of two. But itโs always worth checking some
of the lengths on the other sides, just to be sure. The length ๐ต prime ๐ถ prime is
three units long and the length ๐ต double prime ๐ถ double prime is six units
long. And as thatโs twice as large, then
weโve confirmed that the scale factor is two.
Thereโs a nice, easy way to find
the center of dilation. To do this, we create a ray between
each vertex and its image. Here, we have a ray between ๐ต
prime and ๐ต double prime and ๐ด prime and ๐ด double prime. We can do the same between ๐ถ prime
and ๐ถ double prime. And the place where the rays
converge will be the center of dilation, which once again will be the origin or the
coordinate zero, zero. We put our answer into this
statement form that this will be a dilation from the origin by a scale factor of
two.
Our final question asks if our
triangles ๐ด๐ต๐ถ and ๐ด double prime ๐ต double prime ๐ถ double prime are
similar. We can recall that similar means
the same shape, but a different size. The angles remain the same, but the
sides will be in proportion. So, in our first transformation
from ๐ด๐ต๐ถ to ๐ด prime ๐ต prime ๐ถ prime, we didnโt change the size of these
triangles, which means that they are congruent. And when we transformed ๐ด prime ๐ต
prime ๐ถ prime to ๐ด double prime ๐ต double prime ๐ถ double prime, the image here
did get larger. So, these two triangles would not
be congruent.
Each of the lengths in the image of
๐ด double prime ๐ต double prime ๐ถ double prime was in proportion to those in the
triangles of ๐ด prime ๐ต prime ๐ถ prime. All corresponding pairs of angles
are congruent. So, our triangles ๐ด๐ต๐ถ and ๐ด
double prime ๐ต double prime ๐ถ double prime are similar. So, our answer for the final part
of this question is: yes, these triangles are similar.
Weโll now look at one final
question on describing a combination of transformations.
A triangle with vertices three,
three; seven, zero; and 10, five was transformed to one, eight; five, five; and
eight, 10 and then to one, eight; negative two, four; and three, one. Which of the following describes
these transformations?
And weโre given four answer
options. It will be very difficult to answer
this question without drawing out the coordinates. So, letโs get some grid paper and
draw the vertices of the original triangle. When weโre planning our graph, it
can be very helpful to make sure that we have enough room to cover the highest and
lowest of our ๐ฅ- and ๐ฆ-values. Here, we have our first three
coordinates drawn for the original triangle. We can then draw the second set of
three coordinates to find the image of the triangle, which will look like this.
We now need to work out which
transformation will take the original blue triangle to the triangle drawn in
pink. We can recall that the four types
of transformations are translation, reflection, rotation, and dilation. Looking at our two triangles, we
can see that these are the same size. So, we can eliminate dilation as a
transformation option. In a dilation, the shape and its
image are usually a different size. In the diagram, we can see that the
two triangles are in the same orientation. In other words, theyโre not a
reflection or mirror image, and theyโre not a rotation of each other.
This leaves us with translation,
which is when we simply move a shape to the right or left and up or down. In order to work out how a shape
has been translated, we compare the vertices. Looking at the top vertex of each
triangle and beginning with the blue triangle, we could see that there is a movement
of two units to the left and five units upwards. We must be very careful to work
with the original triangle and the image. Otherwise, instead of correctly
describing this as a translation two units to the left and five units upwards, we
could incorrectly describe it as a translation of two units to the right and five
units down. But we can make a note that the
first transformation will be a translation of two units left and five units up.
We can now look at the final set of
three coordinates to find the second transformation. Plotting the coordinates one,
eight; negative two, four; and three, one will give us a triangle that looks like
this. We need to work out the
transformation that takes our triangle in pink to the triangle in orange. Looking at our four transformation
options, we can eliminate dilation as these triangles are the same size. We can also eliminate another
translation as these two triangles are in a different orientation. The triangles are not a reflected
or mirror image of each other. So we can eliminate reflection,
which leaves us with rotation.
In order to describe a rotation, we
need to give the center of rotation, the angle, and the direction. Sometimes, it can be difficult to
find the center of rotation. What weโre really looking for is a
point about which the shapes can pivot or turn. If we were to draw the pink
triangle on tracing paper and then put the point of our pencil on this green ๐ฅ. Then moving our tracing paper in
the direction of the green arrow would take us to the orange triangle. This is the clockwise
direction.
In order to find the angle of
rotation, we look at the angle between each vertex and its image. Here, we can see that we have a
right angle. So, the angle of rotation will be
90 degrees. Putting these pieces of information
together โ where the center of rotation is at one, eight, the angle is 90 degrees,
and the direction is clockwise โ we would have a rotation of 90 degrees clockwise
about one, eight. We could also describe this
transformation as a rotation of 270 degrees counterclockwise about one, eight.
So if we look at our answer
options, the first transformation was a translation of two units left and five units
up. So, we can eliminate options (A)
and (C). Our second transformation is a
rotation of 90 degrees clockwise. So, we can eliminate option (B) as
that has a rotation of 180 degrees. We can see the fully correct answer
is given in option (D). It was translated two units left
and five units up. And then it was rotated 90 degrees
clockwise about the point one, eight.
Now, letโs summarize what we learnt
in this video. We saw that there are four
transformations: translation, reflection, rotation, and dilation. We can combine the transformations
by performing one transformation and then another. And finally, if weโre performing or
describing a series or combination of transformations, we must be careful that this
is done in the correct given order.