### Video Transcript

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

The first part of this question asks us to compare quadrilateral π΄π΅πΆπ· with its image π΄ prime π΅ prime πΆ prime π· prime. And so letβs begin by identifying these two quadrilaterals. π΄π΅πΆπ· is this small rectangle in the first quadrant of our diagram. Its image, π΄ prime π΅ prime πΆ prime π· prime, is the other small rectangle shown. And so we need to describe the single β remember, that means one β transformation that maps π΄π΅πΆπ· onto its image. And so we recall that there are four weβre 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. π΄π΅πΆπ· and its image are clearly the same-size shape, and so weβre going to choose from the remaining three.

Weβll need to be really careful here. At first glance, it might look like π΄π΅πΆπ· 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 π΄ with π΄ prime and π΅ with π΅ prime, we should see that each vertex and its image is the exact same distance from the π¦-axis but on the other side. In fact, the same can be said for πΆ and its image and π· 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 π¦-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 π΄π΅πΆπ· onto its image is a reflection in the π¦-axis.

Weβre now asked to describe the transformation that maps π΄ prime π΅ prime πΆ prime π· prime onto π΄ double prime π΅ double prime πΆ double prime π· double prime. This time thatβs 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 itβs 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. Letβs take the larger shape to be the new shape, and weβll consider the length of πΆ double prime to π· double prime. Itβs three units. The length of the corresponding dimension on the old shape, thatβs πΆ prime π· 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, thatβs straight lines, through corresponding vertices. So weβll begin by joining π΄ double prime to π΄ prime. We then do the same for π΅ double prime and π΅ prime. And we could stop here, but itβs always sensible to keep going. We join πΆ double prime to πΆ prime. And finally, we join π· double prime to π· 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 π΄ prime π΅ prime πΆ prime π· 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, βHence, are π΄π΅πΆπ· and π΄ double prime π΅ double prime πΆ double prime π· double prime similar?β The word hence means we need to use what weβve 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 π΄π΅πΆπ· and π΄ double prime π΅ double prime πΆ double prime π· double prime dilations of one another? To get from π΄π΅πΆπ· onto its image, we started by reflecting. We saw that reflecting doesnβt change the size of a shape, just its orientation.

And so we can say that π΄π΅πΆπ· and its image are congruent. Theyβre exactly the same size. But then we took this image and we dilated or enlarged it. So we can say that π΄ prime π΅ prime πΆ prime π· prime and its image are similar. And so, if π΄π΅πΆπ· is congruent to π΄ prime π΅ prime πΆ prime π· prime, which is in turn similar to π΄ double prime π΅ double prime πΆ double prime π· double prime, weβre able to say that π΄π΅πΆπ· and π΄ double prime π΅ double prime πΆ double prime π· double prime must themselves be similar. And so the answer is yes.