Video Transcript
Triangle 𝐴𝐵𝐶 has been
transformed under triangle 𝐴 prime 𝐵 prime 𝐶 prime, which has then been
transformed under triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Describe the single transformation
that maps 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime.
Now, there are two other parts to
this question, and we’ll consider them in a moment. Triangle 𝐴𝐵𝐶 is this one here,
and triangle 𝐴 prime 𝐵 prime 𝐶 prime is defined by those vertices. It’s this triangle here. And so we’re looking to find the
single transformation that maps the smaller triangle onto the larger triangle. And of course, one of the keywords
here is the word single. We need to make sure we’re
describing just one transformation. So what are those
transformations?
Well, we have the reflection. And in a reflection, the “fl”
reminds us that we flip the shape. We reflect it in a mirror line. We have a rotation. The “t” reminds us that we turn the
shape. We rotate it by a given angle in a
given direction, and about a given point. We have dilations, which is
sometimes called enlargements. The “l” reminds us that we make the
shape larger or smaller about a given point, and we need to give a scale factor. And then we have translations. And here the letters “sl” remind us
that this slides the shape in a given direction.
So let’s compare triangle 𝐴𝐵𝐶
with 𝐴 prime 𝐵 prime 𝐶 prime. We notice that one triangle is much
larger than the other. And so we must be dealing with a
dilation. Our job then is to work out the
scale factor of this dilation and its center. And there’s a really neat way to
find the center of dilation or the center of enlargement. And that is by drawing a ray or a
straight line through corresponding pairs of vertices. In other words, let’s join vertex
𝐴 and 𝐴 prime with an extended line.
We’re then going to join vertex 𝐵
and 𝐵 prime with an extended line. And finally, we’ll join vertex 𝐶
and 𝐶 prime and notice how each of these rays meets at point 𝐷. And so we can say that the center
of dilation or the center of enlargement is point 𝐷. So what’s the scale factor?
Well, we find a scale factor by
dividing a length on the new triangle by a corresponding length on the old
triangle. Let’s take the side 𝐴 prime 𝐵
prime. We can see that is six units in
length. The corresponding side on the
original triangle is 𝐴𝐵. And this time, that’s two units in
length. And so the scale factor must be six
over two or six divided by two, which is three. And so we’ve answered the first
part of this question. We can say that the single
transformation that maps 𝐴𝐵𝐶 onto its image 𝐴 prime 𝐵 prime 𝐶 prime is a
dilation from or about point 𝐷 by a scale factor of three. We might also notice that these two
triangles are similar. They are different sizes, but their
angles are exactly the same.
Let’s now move on to part two of
this question.
The second part of the question
says, “Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto
𝐴 double prime 𝐵 double prime 𝐶 double prime.” That’s this third triangle
here. So how do we map from 𝐴 prime 𝐵
prime 𝐶 prime onto this triangle?
Well, we’ll actually begin by
disregarding one of our transformations. We can disregard a translation. And that’s because when we
translate a shape, we slide it. Its orientation remains
unchanged. It’s just its position that
changes. We can also disregard
reflections. When we reflect a shape, we flip it
in a mirror line. And we can see that there’s no way
to do this. And that leaves us with rotation or
dilation.
Now, when we dilate a shape or
enlarge it, we make it bigger or smaller. There is one instance where this
isn’t necessarily true. And that’s when we have a scale
factor of negative one. But what that does is it brings it
out the other side of the center of rotation and in a different orientation. And we can see that hasn’t actually
happened here. So we must have a rotation. And actually, this makes a lot of
sense. We can see that the orientation of
our shape has rotated by some angle. So what angle has this rotated
by?
Well, let’s compare the sides 𝐵
prime 𝐶 prime and 𝐵 double prime 𝐶 double prime. It looks like 𝐵 prime 𝐶 prime has
rotated either by 270 degrees or by 90 degrees. And so we have to choose a
direction. And in fact, the original arrow we
drew was a counterclockwise direction. And so we can say that their shape
must be rotated 90 degrees counterclockwise. We could alternatively write this
as a 90 degrees’ rotation counterclockwise. But we’re not quite finished; we
need to find the center of rotation.
Now, one way to find the center of
rotation is to use some tracing paper and a little bit of trial and error. Alternatively, you can join
corresponding vertices and then find the perpendicular bisectors of these. The point at which the
perpendicular bisectors meet is the center of rotation. The perpendicular bisector of the
line segment 𝐵 prime 𝐵 double prime is shown. And the perpendicular bisector of
the line segment between 𝐶 prime and 𝐶 double prime is also shown. Notice how these meet at point
𝐷. And so that is our center of
rotation. The single transformation then that
maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime
is a 90 degrees’ rotation in a counterclockwise direction about point 𝐷.
Now, in fact, we can notice that
these two triangles are congruent. They’re exactly the same. They have the same angles and the
same-length sides. So with all this information in
mind, let’s look at the third and final part of this question.
Hence, are triangles 𝐴𝐵𝐶 and 𝐴
double prime 𝐵 double prime 𝐶 double prime similar?
So thinking about our
transformations, we’re looking to compare the very first triangle with the very
final triangle. And so let’s think about what we
said. We said that because we were
dealing with a dilation, the triangle 𝐴𝐵𝐶 was similar but not congruent to 𝐴
double prime 𝐵 double prime 𝐶 double prime. We also said that our latter two
triangles, 𝐴 prime 𝐵 prime 𝐶 prime and 𝐴 double prime 𝐵 double prime 𝐶 double
prime, since they were a rotation of one another, are congruent. And so this must mean that since 𝐴
prime 𝐵 prime 𝐶 prime and 𝐴 double prime 𝐵 double prime 𝐶 double prime are
congruent, they’re identical, that the original triangle must also be similar to 𝐴
double prime 𝐵 double prime 𝐶 double prime.
And so the answer to the third part
of this question is yes, 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime
are similar.
