Video Transcript
The triangle 𝐴𝐵𝐶 has been
transformed onto triangle 𝐴 prime 𝐵 prime 𝐶 prime which has been transformed onto
triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Describe the single transformation
that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. Describe the single transformation
that maps 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double
prime. Hence, are triangles 𝐴𝐵𝐶 and 𝐴
double prime 𝐵 double prime 𝐶 double prime similar?
In this question, we have a series
or combinations of transformations which begin with triangle 𝐴𝐵𝐶. The first transformation takes us
to the second smaller triangle. And the second transformation takes
us to the larger triangle of 𝐴 double prime 𝐵 double prime 𝐶 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
𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime, we can see that they’re 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. Let’s see if we could describe this
transformation as a rotation. Starting with triangle 𝐴𝐵𝐶, if
we rotated this in a clockwise direction, we could then work out the angles for
which this must be rotated. Between 𝐴 and 𝐴 prime, there’s a
right angle of 90 degrees. Between 𝐶 and 𝐶 prime, we can
also see a 90-degree angle. This will confirm that we have a
rotation of 90 degrees.
Notice that we’ve 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 we’ve discovered — 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.
Let’s look at the second
question. Describe the single transformation
that maps 𝐴 prime 𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double
prime. We need to be careful that we’re
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 — translation, reflection, and rotation — keep the
object and its image the same size. As the triangles here are different
sizes, that means there’s 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 𝐴
double prime 𝐵 double prime and 𝐴 prime 𝐵 prime. We can see that on 𝐴 prime 𝐵
prime, the length is two units, and on the top length of 𝐴 double prime 𝐵 double
prime, this is four units long. So, it looks like we’ll have a
scale factor of two. But it’s always worth checking some
of the lengths on the other sides, just to be sure. The length 𝐵 prime 𝐶 prime is
three units long and the length 𝐵 double prime 𝐶 double prime is six units
long. And as that’s twice as large, then
we’ve confirmed that the scale factor is two.
There’s 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 𝐵
prime and 𝐵 double prime and 𝐴 prime and 𝐴 double prime. We can do the same between 𝐶 prime
and 𝐶 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 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 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 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime, we didn’t change the size of these
triangles, which means that they are congruent. And when we transformed 𝐴 prime 𝐵
prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime, the image here
did get larger. So, these two triangles would not
be congruent.
Each of the lengths in the image of
𝐴 double prime 𝐵 double prime 𝐶 double prime was in proportion to those in the
triangles of 𝐴 prime 𝐵 prime 𝐶 prime. All corresponding pairs of angles
are congruent. So, our triangles 𝐴𝐵𝐶 and 𝐴
double prime 𝐵 double prime 𝐶 double prime are similar. So, our answer for the final part
of this question is: yes, these triangles are similar.
