Question Video: Identifying the Types of Transformations Applied on a Given Figure from the Graph Mathematics • 8th Grade

In the given figure, what combination of transformations would map circle 𝐴 onto circle 𝐵?

04:09

Video Transcript

In the given figure, what combination of transformations would map circle 𝐴 onto circle 𝐵? Is it (A) a translation of four left and six down, followed by a dilation of scale factor one-half? Is it (B) a translation of six left and four down, followed by a dilation of scale factor two-thirds? (C) A translation of four left and six down, followed by a dilation of scale factor two-thirds. (D) A translation of six right and four up, followed by a dilation of scale factor three over two. Or is it (E) a translation of four left and six up, followed by a dilation of scale factor three-quarters?

So, in this question, we’re looking to describe a combination of transformations. And that’s great because we can include more than one transformation when we’re describing this. We’re looking to describe the transformations that map circle 𝐴 onto circle 𝐵. Well, 𝐴 is the larger circle, and 𝐵 is the smaller. And so, that’s a little bit of a hint as to what transformation we might consider.

The four transformations, sometimes called similarity transformations, that we’re going to look at are rotations, reflections, dilations, and translations. When we rotate a shape, we turn it. The use of the letter “t ” here is a reminder of that. When we reflect it, the “fl ” reminds us we flip the shape in a mirror line. Dilation, which is sometimes called enlargement, is where we make a shape larger, that’s the “l, ” or sometimes smaller. And when we translate a shape, the “sl ” reminds us that we slide it. And when we do so, we slide it left or right and up or down.

We said we were mapping the larger circle onto the smaller circle. So, there’s an indication to us that we’re definitely going to be dilating the shape. But also we notice that the center of the circle 𝐴, which is currently at the point with coordinates six, zero, maps onto the center of circle 𝐵 with coordinates zero, negative four. So, how do we achieve that? Well, we’re going to need to slide the entire circle, and so we’re going to be translating it as well.

Let’s deal with the translation first. We said the translation slides the center of circle 𝐴 onto the center of circle 𝐵. Note that we’ve chosen the center of the circle rather than a point on its circumference because it’s the only constant when we dilate it. And so, how many units left or right are we going to need to slide the center? Well, the 𝑥-coordinate moves from six to zero. So, we’re going to be moving this six units to the left. Then, the 𝑦-coordinate goes from zero to negative four, so we slide it four down. And we, therefore, have a translation of six left and four down.

So, that’s the translation part, but what about the dilation? Well, the shape has gotten smaller. So we’re going to have a fractional scale factor between zero and one. And we might recall that we can calculate a scale factor by dividing a dimension on the new shape by a corresponding dimension on the original shape. So, let’s consider the radius of our circle. I’ve drawn that in on circle 𝐴. And here it is on circle 𝐵. Now, remember, the radius is simply the length of the line that joins the center of the circle to its circumference. So it doesn’t matter which radius we choose.

On shape 𝐵, the radius is two units, whereas on shape 𝐴, it’s three units. Since we’re mapping 𝐴 onto 𝐵, we take the radius of circle 𝐵 and we divide that by the radius of circle 𝐴. So, the scale factor must be two-thirds. And so, we’re ready to fully describe the combination of transformations.

We have a translation of six units left and four down, followed by a dilation or an enlargement of scale factor two-thirds. Comparing that to the answers given here, we see the correct answer is (B), a translation of six left and four down, followed by a dilation of scale factor two-thirds.

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