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

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

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Video Transcript

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

The first part of this question asks us to compare quadrilateral 𝐴𝐡𝐢𝐷 with its image 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime. And so let’s begin by identifying these two quadrilaterals. 𝐴𝐡𝐢𝐷 is this small rectangle in the first quadrant of our diagram. Its image, 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime, is the other small rectangle shown. And so we need to describe the single β€” remember, that means one β€” transformation that maps 𝐴𝐡𝐢𝐷 onto its image. And so we recall that there are four we’re interested in. We have rotations, reflections, translations, and dilations. We can immediately disregard dilations. When we dilate a shape, we make it bigger or smaller. 𝐴𝐡𝐢𝐷 and its image are clearly the same-size shape, and so we’re going to choose from the remaining three.

We’ll need to be really careful here. At first glance, it might look like 𝐴𝐡𝐢𝐷 maps onto its image by a translation. But when we translate a shape, we slide it. And it will end up in the exact same orientation. If we look carefully, we notice that the orientation of our vertices has changed. And so we can actually disregard translation. And so we have to choose from reflections and rotations.

Now, when we reflect a shape, we flip it in a mirror line, whereas when we rotate a shape, we turn it. If we compare vertex 𝐴 with 𝐴 prime and 𝐡 with 𝐡 prime, we should see that each vertex and its image is the exact same distance from the 𝑦-axis but on the other side. In fact, the same can be said for 𝐢 and its image and 𝐷 and its image. So, is this a rotation or a reflection? Well, by comparing each of the individual vertices, we can see that this shape must have been reflected in the 𝑦-axis. Of course, when we reflect a shape in a mirror line, each vertex will be the exact same distance from that mirror line but on the other side. And we saw that but for each vertex in turn. And so the single transformation that maps 𝐴𝐡𝐢𝐷 onto its image is a reflection in the 𝑦-axis.

We’re now asked to describe the transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime onto 𝐴 double prime 𝐡 double prime 𝐢 double prime 𝐷 double prime. This time that’s this rectangle onto this rectangle. We should immediately spot that one rectangle is much larger than the other. The transformation that makes a shape larger or smaller is a dilation, which is sometimes called an enlargement. But we need to describe two elements of this dilation. We need to describe the scale factor and the point about which it’s been enlarged or dilated.

And so we recall that to find the scale factor, we divide one of the dimensions on the new shape by the corresponding dimension on the older shape. Let’s take the larger shape to be the new shape, and we’ll consider the length of 𝐢 double prime to 𝐷 double prime. It’s three units. The length of the corresponding dimension on the old shape, that’s 𝐢 prime 𝐷 prime, is just one unit. And so the scale factor here must be three divided by one, which is simply three.

But what about the center? To find the center, we draw rays, that’s straight lines, through corresponding vertices. So we’ll begin by joining 𝐴 double prime to 𝐴 prime. We then do the same for 𝐡 double prime and 𝐡 prime. And we could stop here, but it’s always sensible to keep going. We join 𝐢 double prime to 𝐢 prime. And finally, we join 𝐷 double prime to 𝐷 prime. The point at which these rays meet is the center of enlargement or dilation. This point has coordinates negative two, zero. And so we can say that the single transformation that maps 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime onto its image is a dilation from the point negative two, zero by a scale factor of three.

The final part of this question says, β€œHence, are 𝐴𝐡𝐢𝐷 and 𝐴 double prime 𝐡 double prime 𝐢 double prime 𝐷 double prime similar?” The word hence means we need to use what we’ve already done. And if two shapes are similar, one will be a dilation or enlargement of the other. Their angles will be exactly the same. So we really need to ask ourselves, are 𝐴𝐡𝐢𝐷 and 𝐴 double prime 𝐡 double prime 𝐢 double prime 𝐷 double prime dilations of one another? To get from 𝐴𝐡𝐢𝐷 onto its image, we started by reflecting. We saw that reflecting doesn’t change the size of a shape, just its orientation.

And so we can say that 𝐴𝐡𝐢𝐷 and its image are congruent. They’re exactly the same size. But then we took this image and we dilated or enlarged it. So we can say that 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime and its image are similar. And so, if 𝐴𝐡𝐢𝐷 is congruent to 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime, which is in turn similar to 𝐴 double prime 𝐡 double prime 𝐢 double prime 𝐷 double prime, we’re able to say that 𝐴𝐡𝐢𝐷 and 𝐴 double prime 𝐡 double prime 𝐢 double prime 𝐷 double prime must themselves be similar. And so the answer is yes.

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