Video: Using Transformations to Determine Similarity

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′, which has then been transformed onto triangle 𝐴″𝐡″𝐢″. Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′. Describe the single transformation that maps 𝐴′𝐡′𝐢′ onto 𝐴″𝐡″𝐢″. Hence, are triangles 𝐴𝐡𝐢 and 𝐴″𝐡″𝐢″ similar?

06:38

Video Transcript

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴 prime 𝐡 prime 𝐢 prime, which has then been transformed onto triangle 𝐴 double prime 𝐡 double prime 𝐢 double prime. Describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. Describe the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. Hence, are triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime similar?

We’ll begin with the first part of this question. That asks us to describe the single transformation that maps 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. 𝐴𝐡𝐢 is this small triangle at the center of our drawing. Then, 𝐴 prime 𝐡 prime 𝐢 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 β€œsingle.” We’re going to describe exactly one transformation rather than a series of these.

So, let’s 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 β€œfl” in this word remind us that we flip the shape. When we translate, the β€œsl” reminds us to slide the shape. And finally, the dilation, sometimes called an enlargement, makes a shape larger, that’s the l, or smaller.

So, let’s see what’s happened to transform 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime. We should quite quickly notice that the image 𝐴 prime 𝐡 prime 𝐢 prime is larger than the original shape. That’s an indication to us that 𝐴𝐡𝐢 has been dilated. That’s not enough, though. We’re 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, we’re going to divide the length of the line segment 𝐴 prime 𝐡 prime by the length of the line segment 𝐴𝐡. Line segment 𝐴 prime 𝐡 prime is 12 units long, whereas the line segment 𝐴𝐡 is four. So, the scale factor of enlargement or dilation, which I’ve 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 𝐴 prime and 𝐴. Similarly, we’ll join vertex 𝐢 prime and 𝐢. And then, we join the vertices 𝐡 prime and 𝐡. The point at which these rays meet is the center of enlargement or dilation. And we can see that happens at point 𝐷. And so, the single transformation that maps triangle 𝐴𝐡𝐢 onto 𝐴 prime 𝐡 prime 𝐢 prime is a dilation from point 𝐷 by a scale factor of three.

We now move on to question two. And that says, describe the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime. We’ve already identified that 𝐴 prime 𝐡 prime 𝐢 prime is the enlargement of 𝐴𝐡𝐢. And 𝐴 double prime 𝐡 double prime 𝐢 double prime is this triangle here. So, how have we got from 𝐴 prime 𝐡 prime 𝐢 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, we’ll disregard the translation. So, we have two options. We have a reflection, that’s a flip of the shape, and a rotation, that’s a turn.

Well, in fact, we see that if we turn or rotate 𝐴 prime 𝐡 prime 𝐢 prime, we end up in the same orientation as 𝐴 double prime 𝐡 double prime 𝐢 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 there’s only really one point about which the shape can have rotated. That’s the point 𝐷.

Then, we’re going to join one of our vertices to this point. We’ll begin by joining 𝐴 prime to point 𝐷. And then, we’re going to join the rotated vertex, that’s 𝐴 double prime, to point 𝐷. We can now see that this line has rotated 180 degrees. And we can, therefore, say that the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime is a rotation by 180 degrees about point 𝐷.

Finally, we’re asked, hence are triangles 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime similar? The word β€œhence” indicates to us that we need to use what we’ve just done. And so, what we’re 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 𝐴 prime 𝐡 prime 𝐢 prime is a dilation of 𝐴𝐡𝐢. So, we can certainly say that 𝐴𝐡𝐢 and 𝐴 prime 𝐡 prime 𝐢 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 𝐴 prime 𝐡 prime 𝐢 prime and 𝐴 double prime 𝐡 double prime 𝐢 double prime must be congruent. They’re actually exactly the same. And so, if 𝐴𝐡𝐢 is similar to 𝐴 prime 𝐡 prime 𝐢 prime, but actually 𝐴 prime 𝐡 prime 𝐢 prime is congruent to 𝐴 double prime 𝐡 double prime 𝐢 double prime, then this in turn means that 𝐴𝐡𝐢 and 𝐴 double prime 𝐡 double prime 𝐢 double prime must in fact be similar. And so, the answer is yes.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.