Video Transcript
The triangle 𝐴𝐵𝐶 has been transformed onto triangle 𝐴 prime 𝐵 prime 𝐶 prime, which has then been transformed onto triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Describe the single transformation that maps 𝐴𝐵𝐶 onto its image 𝐴 prime 𝐵 prime 𝐶 prime. Describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. And the third part of this question says, hence, are triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime similar?
So we’ll begin by answering the first part of this question. We want to describe the single transformation that maps triangle 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. Triangle 𝐴𝐵𝐶 is this one, and its image is defined by the vertices 𝐴 prime 𝐵 prime 𝐶 prime. Watch out for this word “single”. A common mistake is to try and list two transformations, for example, a rotation followed by a transformation. We will only get the relevant marks for this question if we describe one transformation fully.
So how do we map triangle 𝐴𝐵𝐶 onto its image 𝐴 prime 𝐵 prime 𝐶 prime? We recall that there are four transformations we’re interested in. Those are reflections, rotations, enlargements, and translations. And so let’s look at our two triangles. They appear to be the same size, so for now we can disregard an enlargement. Remember, an enlargement makes a shape bigger or smaller. We can say, however, that triangle 𝐴𝐵𝐶 is in the same orientation. It’s the same way up as its image. Once we’ve disregarded enlargements, the only transformation that ensures that the image is the same orientation as the original shape is a translation. When we translate a shape, we slide it along the coordinate plane.
So how far have we slid our shape? Let’s pick a vertex. Let’s pick vertex 𝐴 on the original, which is 𝐴 prime on the image. We see that we have moved one, two, three units to the right and then one unit up. The single transformation that maps 𝐴𝐵𝐶 onto its image is a translation three right and one up.
The second part of this question asks us to describe the transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime. We know 𝐴 prime 𝐵 prime 𝐶 prime is this triangle here. Now, its image is this one. We can see the image 𝐴 double prime 𝐵 double prime 𝐶 double prime is much larger than the first shape 𝐴 prime 𝐵 prime 𝐶 prime. And this means the shape has been enlarged or dilated. When we consider an enlargement or a dilation, we need to describe two things.
Firstly, we’re interested in the scale factor. That’s the single value that we multiply each dimension by to get to the new shape. If we compare line segments 𝐴 prime 𝐵 prime and 𝐴 double prime 𝐵 double prime, we can see that to get from 𝐴 prime 𝐵 prime to its image, we multiply by two. And so our scale factor must be two. Note that a quick way to calculate the scale factor if it’s not instantly obvious is to divide one of the dimensions on the new shape by its corresponding dimension on the original. The last thing we need to find is the center of enlargement.
To do this, we begin by joining the corresponding vertices on our shapes. Let’s join the vertices 𝐴 prime and 𝐴 double prime. We can do the same with 𝐵 prime and 𝐵 double prime. Now, we don’t necessarily need to perform this last step. But for safety, let’s check 𝐶 prime and 𝐶 double prime. We know that all three of these lines, or they’re sometimes called rays, meet at a point. In fact, they meet at the point zero, zero or the origin. And so the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto its image is a dilation from the origin by a scale factor of two.
We’ll now consider the third and final part of this question. This asks us to decide whether triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime and 𝐶 double prime are similar. In fact, it says hence, so we’re going to use what we’ve already done. Before we do though, we recall that shapes are similar if their angles are the same. They might be enlargements of one another. So we’ll begin by considering 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. These shapes are actually the exact same size, so we can say that they’re congruent. When we translate a shape, it does not change its size at all. In general, a shape that has been translated will be congruent to its original.
Then we compare 𝐴 prime 𝐵 prime 𝐶 prime with 𝐴 double prime 𝐵 double prime 𝐶 double prime. Now we said that two shapes that are similar can be considered to be enlargements or dilations of one another. This means that 𝐴 prime 𝐵 prime 𝐶 prime and its image 𝐴 double prime 𝐵 double prime 𝐶 double prime must be similar. But since 𝐴𝐵𝐶 and its image are congruent, this in turn must mean that 𝐴𝐵𝐶 and the enlarged shape 𝐴 double prime 𝐵 double prime 𝐶 double prime are similar. And so the answer is yes. The triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime are similar.
