### Video Transcript

The triangle π΄π΅πΆ has been transformed onto triangle π΄ prime π΅ prime πΆ prime, which has then been transformed onto triangle π΄ double prime π΅ double prime πΆ double prime. Describe the single transformation that maps π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime. Describe the single transformation that maps π΄ prime π΅ prime πΆ prime onto π΄ double prime π΅ double prime πΆ double prime. Hence, are triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime similar?

Weβll begin with the first part of this question. That asks us to describe the single transformation that maps π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime. π΄π΅πΆ is this small triangle at the center of our drawing. Then, π΄ prime π΅ prime πΆ prime is the larger one that sits around it. Now, we need to be really careful when describing the transformation. The keyword in this question is the word βsingle.β Weβre going to describe exactly one transformation rather than a series of these.

So, letβs recall the transformations that we need to know. These are rotations, reflections, translations, and dilations. We recall that a rotation, the key is the letter T here, turns the shape. We have reflections, and the βflβ in this word remind us that we flip the shape. When we translate, the βslβ reminds us to slide the shape. And finally, the dilation, sometimes called an enlargement, makes a shape larger, thatβs the l, or smaller.

So, letβs see whatβs happened to transform π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime. We should quite quickly notice that the image π΄ prime π΅ prime πΆ prime is larger than the original shape. Thatβs an indication to us that π΄π΅πΆ has been dilated. Thatβs not enough, though. Weβre going to need to describe two more things. We need to give a center of enlargement or dilation, and we need to give a scale factor. To find a scale factor for enlargement, we divide a dimension on the new shape by the corresponding dimension on the old shape.

And so, to find the scale factor of our dilation or enlargement, weβre going to divide the length of the line segment π΄ prime π΅ prime by the length of the line segment π΄π΅. Line segment π΄ prime π΅ prime is 12 units long, whereas the line segment π΄π΅ is four. So, the scale factor of enlargement or dilation, which Iβve shortened to s.f., is 12 divided by four, which is equal to three.

So, we have the type of transformation and the scale factor. We need to decide where the center of enlargement lies. To do this, we join each corresponding vertex by a ray. So, for example, we can join vertex π΄ prime and π΄. Similarly, weβll join vertex πΆ prime and πΆ. And then, we join the vertices π΅ prime and π΅. The point at which these rays meet is the center of enlargement or dilation. And we can see that happens at point π·. And so, the single transformation that maps triangle π΄π΅πΆ onto π΄ prime π΅ prime πΆ prime is a dilation from point π· by a scale factor of three.

We now move on to question two. And that says, describe the single transformation that maps π΄ prime π΅ prime πΆ prime onto π΄ double prime π΅ double prime πΆ double prime. Weβve already identified that π΄ prime π΅ prime πΆ prime is the enlargement of π΄π΅πΆ. And π΄ double prime π΅ double prime πΆ double prime is this triangle here. So, how have we got from π΄ prime π΅ prime πΆ prime onto its image?

It appears that these triangles are the same size. And so we can disregard the dilation at this point. A translation involves a slide of the shape. When we slide the shape, they end up in the same orientation. And these ones are clearly not in the same orientation; one appears to be upside down. So, weβll disregard the translation. So, we have two options. We have a reflection, thatβs a flip of the shape, and a rotation, thatβs a turn.

Well, in fact, we see that if we turn or rotate π΄ prime π΅ prime πΆ prime, we end up in the same orientation as π΄ double prime π΅ double prime πΆ double prime. And so, the shape has been rotated. There are two more things we need to decide. We need to decide the angle of rotation and the point about which the shape is rotated. If we clear the annotations off of our diagram, we see thereβs only really one point about which the shape can have rotated. Thatβs the point π·.

Then, weβre going to join one of our vertices to this point. Weβll begin by joining π΄ prime to point π·. And then, weβre going to join the rotated vertex, thatβs π΄ double prime, to point π·. We can now see that this line has rotated 180 degrees. And we can, therefore, say that the single transformation that maps π΄ prime π΅ prime πΆ prime onto π΄ double prime π΅ double prime πΆ double prime is a rotation by 180 degrees about point π·.

Finally, weβre asked, hence are triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime similar? The word βhenceβ indicates to us that we need to use what weβve just done. And so, what weβre really asking is if we start off with a shape, then dilate it, and then rotate it, will we end up with two shapes that are similar?

Well, for two shapes to be similar, one must be an enlargement or a dilation of the other. We initially showed that π΄ prime π΅ prime πΆ prime is a dilation of π΄π΅πΆ. So, we can certainly say that π΄π΅πΆ and π΄ prime π΅ prime πΆ prime are similar.

But what about when we rotate a shape? Well, when we rotate a shape, it does change the orientation, but it otherwise does not change the size of that shape. And we can, therefore, say that π΄ prime π΅ prime πΆ prime and π΄ double prime π΅ double prime πΆ double prime must be congruent. Theyβre actually exactly the same. And so, if π΄π΅πΆ is similar to π΄ prime π΅ prime πΆ prime, but actually π΄ prime π΅ prime πΆ prime is congruent to π΄ double prime π΅ double prime πΆ double prime, then this in turn means that π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime must in fact be similar. And so, the answer is yes.