Question Video: Using Transformations to Determine Similarity Mathematics • 8th Grade

Video Transcript

The triangle 𝐴𝐵𝐶 has been transformed to triangle 𝐴 prime 𝐵 prime 𝐶 prime which has then been transformed to triangle 𝐴 double prime 𝐵 double prime 𝐶 double prime. Describe the single transformation that maps 𝐴𝐵𝐶 to 𝐴 prime 𝐵 prime 𝐶 prime.

And there are two further parts to this question which we’ll consider in a moment. Let’s begin by identifying triangle 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime. 𝐴𝐵𝐶 is made up of the vertices 𝐴, 𝐵, and 𝐶 as it sounds. Then 𝐴 prime 𝐵 prime 𝐶 prime is this triangle directly above it. So, what single transformation maps the smaller triangle onto the larger triangle? Well, there’s a little bit of a hint in how we worded that. The triangle has changed in size. And so we know that the transformation is a dilation, sometimes called an enlargement. Remember, that simply makes the shape bigger or smaller.

There are, however, two further things that we need to find, and those are the center of the dilation and the scale factor. We find the center of the dilation by drawing rays; those are straight lines that pass through corresponding pairs of vertices. So for instance, we’ll join 𝐴 prime and 𝐴. We’ll join 𝐵 prime and 𝐵. And finally, we’ll join 𝐶 prime and 𝐶. The point where these lines intersect is the center of the dilation. We can see that our lines intersect at the origin or the point zero, zero. So, that’s our center, and we have a dilation from the origin.

But what’s the scale factor? We can calculate this by dividing a length on the new shape by the corresponding length on the original shape. So, for instance, let’s take the line segment 𝐴 prime 𝐵 prime and divide it by the length of the line segment 𝐴𝐵. 𝐴 prime 𝐵 prime is nine units in length, and 𝐴𝐵 is three units. And so, the scale factor of dilation, or enlargement here, is nine divided by three, which is just three. And so the single transformation that maps 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime is a dilation from the origin by a scale factor of three.

The second part of this question asks us to describe the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime.

And so, that’s the larger of the triangles we just looked at and this triangle down below the 𝑥-axis. So, let’s compare these triangles. We might spot that each triangle is the same distance away from the 𝑥-axis, but on opposite sides. And in fact, the triangle appears to have been flipped over this line. The mathematical word for flipping a shape is reflecting it. And since each vertex on the original shape has the same perpendicular distance away from the 𝑥-axis, as the vertices on the image, then we could say the 𝑥-axis is our mirror line. And this means that the single transformation that maps 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime is simply a reflection in the 𝑥-axis.

The third and final part to this question says: Hence, are the triangles 𝐴𝐵𝐶 and 𝐴 double prime 𝐵 double prime 𝐶 double prime similar?

Firstly, the word hence means we need to use what we’ve already done. But what does the word similar mean? When two shapes are similar, they must have the exact same set of angles. Often this will look like one is a dilation or an enlargement of the original. And so, let’s see what’s happened when we mapped 𝐴𝐵𝐶 onto 𝐴 prime 𝐵 prime 𝐶 prime. This was a dilation, as we saw. Since 𝐴 prime 𝐵 prime 𝐶 prime is a dilation or an enlargement of 𝐴𝐵𝐶, then by definition, their angles have to be equal. And so these two triangles are similar.

Then, when we mapped 𝐴 prime 𝐵 prime 𝐶 prime onto 𝐴 double prime 𝐵 double prime 𝐶 double prime, we reflected the shape. And so the actual size of the triangle didn’t change. This means that since the orientation is the only difference here, 𝐴 prime 𝐵 prime 𝐶 prime and 𝐴 double prime 𝐵 double prime 𝐶 double prime themselves are congruent. And so, if 𝐴 double prime 𝐵 double prime 𝐶 double prime is the same size, it’s congruent to 𝐴 prime 𝐵 prime 𝐶 prime, but 𝐴 prime 𝐵 prime 𝐶 prime is in turn an enlargement or a dilation, it’s similar to 𝐴𝐵𝐶, then triangle 𝐴𝐵𝐶 must be similar to 𝐴 double prime 𝐵 double prime 𝐶 double prime. And so the answer is yes.