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 as seen in the
figure. Describe the single transformation
that would map 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime. Describe the single transformation
that would map 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 congruent?
In this question, we’re asked about
transformations. We can recall that the four
different transformations are translation, reflection, rotation, and dilation. So let’s look at the first question
where we establish which transformation takes this triangle 𝐴𝐵𝐶 on the left to
the triangle 𝐴 prime 𝐵 prime 𝐶 prime. If we look at our list of the four
transformations, considering dilation, that’s the transformation that usually
changes the size of an object. We can see that our two triangles
here are the same size. So therefore, we can rule out
dilation. When a translation has been
performed, this will keep the object in the same orientation. As our triangles are not the same
orientation here, we can rule out translation.
This leaves us with reflection or
rotation. It does look as though the shapes
could be turned and it could be a rotation. But if we observe the letters and
considering 𝐴 and 𝐶, if we were to have rotated this shape, then 𝐶 prime would be
left of 𝐴 prime. Therefore, the transformation here
is not a rotation. It must be a reflection. We can see that, indeed, our two
shapes do look like a mirror image of each other. In order to fully describe a
transformation, we couldn’t just, for example, say that it’s a reflection. We’d have to also say the line of
reflection.
If we draw a line between each
vertex and its image, for example, between 𝐴 and 𝐴 prime, we can see that this
line would be two units across and two units up. The halfway point along this line
would be one unit across to the right and one unit up. Between the vertices 𝐶 and 𝐶
prime, we can see that it’s five units to the right and five units up. So halfway along would be two and a
half units to the right and two and a half units up. Joining these halfway points would
give us the line of reflection. This has very helpfully given to us
as the line 𝐷𝐸. Putting this into a statement, we
would write that this is a reflection in the line 𝐷𝐸. And that’s our answer for the first
part of the question.
In the second part of this
question, we’re asked to describe the single transformation that would map 𝐴 prime
𝐵 prime 𝐶 prime to 𝐴 double prime 𝐵 double prime 𝐶 double prime. Let’s consider our four
transformation options. We can rule out dilation again
because these triangles haven’t changed size. And as the triangles are a
different way up or a different orientation, we can rule out translation. We can also rule out reflection as
these two triangles are not a mirror image of each other. If we take this transformation as a
rotation, there are a few things we’ll need to establish.
Firstly, the center of rotation,
that’s the point about which the rotation is performed. Next, we’ll also need to find the
angle and the direction of the rotation. If we take a look at the vertex, 𝐵
prime, we could see that if we turned it in this direction, it would give us 𝐵
double prime here. This looks as though it could be a
90-degree angle. We’d need to consider where the
center of rotation would be however. Let’s try working with this point
that’s labeled 𝐹. In this case, a line from 𝐵 prime
to this point 𝐹 and then from 𝐹 to 𝐵 double prime would create a 90-degree
angle.
If we do the same and check out
vertex 𝐴 prime and 𝐴 double prime. In this case, the line between 𝐴
prime and 𝐹 and 𝐹 and 𝐴 double prime would also create a right angle of 90
degrees. We can therefore say that the
center of rotation is the point 𝐹 and the angle is 90 degrees. The direction of the rotation would
be clockwise. We could therefore give our answer
as a complete statement describing this transformation as a rotation of 90 degrees
clockwise about point 𝐹. It would also have been correct to
give this as a rotation of 270 degrees counterclockwise about point 𝐹.
Let’s look at the final part of
this question, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double
prime congruent? We can recall that congruent means
exactly the same shape and size. All the pairs of corresponding
sides and angles in the object and its image would stay the same. The first transformation reflected
𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. The angles and the side lengths
stayed the same, which means that these two triangles are congruent. Then, we rotated 𝐴 prime 𝐵 prime
𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. Even though the shape was rotated
or turned, all the angles and sides stayed the same, which means that these two
triangles are congruent.
Therefore, all three triangles are
congruent. So our answer for the final part of
this question is yes. Triangles 𝐴𝐵𝐶 and 𝐴 double
prime 𝐵 double prime 𝐶 double prime are congruent.
