This is Common Core State Support Video for Grades 9 through 12; standard is G.SRT.2. The standard reads: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Now one of the wonders of technology is that we can take a figure such as a pentagon and create another figure that’s congruent to it. But what is also possible with technology is to take that figure and do a similarity transformation where we simply can enlarge it or shrink it. Now there’s also software—you could probably find a website—that will do this and actually also have the measures of each of the sides so that your students can compare the matching pairs of sides to see if they’re in the same ratio. Now you might also have a situation where you can use the software to take two figures and use the transformations of a rotation and a translation to see if they do match up as far as the angles being congruent. And so, if we were to do this here, rotate first and then once we’ve done our rotation, we can do a slide and match up the corresponding pairs of angles and show that they are equal.
This standard seems to focus on triangles. So if we were to take that same idea and use technology to again take a couple of triangles, and we would first do a rotation. And once we have the rotation set, then we do our slide and do what we did a while ago and again, just match up our pairs of angles and show that the matching, the corresponding pairs of angles are equal, and our corresponding pairs of sides should be proportional. What students can do there is just simply do measurements where, for example, they would measure this side and this side and get that ratio, and then make sure that the ratio of this side and this side are the same and the ratio of this side to this side are the same.
Now if we’re dealing with similarity transformations that deal with being from a given point, then that’s another situation. But I’ve already set this up where, let’s say, our scale factor is two. So that means that the distances are going to be twice as far for the new triangle as they are for the original. And also, since it’s a positive two, that means that our image, our new triangle, is going to be on the same side of that point. So I have already gone through and set this up. I won’t go through all the steps that are involved, but of course, we would have measured the distance from here to here, and that distance is half the distance of the distance from here to here, because again, our scale factor was two. So this distance from here to here is going to be twice what this is, and then the same thing applies to the others. We take those three points and connect them. So now we have a new triangle, and these should be similar. If you notice, in this context, this triangle here and this triangle here are related in that all you have to do is a slide, because they already have the same orientation. And by doing the slide, I can match up the pairs of angles and see that they are congruent.
If we take another scenario where our constant this time, our scale factor, is a negative two, it’s pretty similar to a while ago where it’s going to be twice the distance from this given point, and our new image is going to be (the matching pairs of sides), will be twice as large. The difference is though, that this time, because of the negative, our new triangle is going to be on the opposite side of that point. So again, I’ve already set this up. This distance here, if we were to measure it, whatever that is, let’s say that was a four. Then this distance from here over here has to be twice that, which in this case would be a four—I mean an eight, if this would have been a four. So we do that to all three, and so here’s our new triangle. Now notice that this time they’re not in the same orientation. So what we’re going to have to do is take this triangle and get it oriented in such a way to where we can follow this up with a translation. So now we can take this triangle, slide it over because we want to make sure that the matching pairs of angles are equal. And so by doing this, we can do this informally to again show that the matching pairs of angles are equal.
If we wanted to be a little bit more exact, have a little bit more proof for the students, we can use the idea of parallel lines. In this case, I’ve drawn three of them and two transversals. Notice that I have a triangle formed here and a triangle formed there. Now this time, we can prove the equivalence of the angles. For example, this angle here and this angle here are congruent because they’re vertical. And then, let’s say this angle here would be congruent to this angle here because they’re alternate interior angles. And so, we’ve already proven the similarity because we’ve shown that two pairs of angles are equal, and that’s all you need to show that two triangles are similar. But if we wanted to take this a step further, we could take this triangle and then again, we need to rotate it around to where the orientation would be the same as the other one. And so, once we’ve done that, then again, we can slide and verify. But we already know that the matching pairs of angles are equal because of the alternate interior angles and the vertical angles that we had earlier.
We could also involve the idea of graphing. Let’s say, we start off with this original triangle, and we construct one where the sides are twice the lengths of the corresponding sides of the original. And then, we figure out where our line of reflection should be, and so then, we reflect this triangle across this line. So this is our new image. Then what we can do is do a translation, just slide it over. And again, we can show that the matching pairs of angles are equal, but the main thing here though is that you can also, through the graph and the grid, show that the matching pairs of sides are in the same ratio. So for example, we can compare this distance here to this overall distance here, and it does turn out to be twice as much because this is four and this is eight. Then we can do the same thing with the others. This is three here, that’s six there, so the ratio does hold. It’s a 2 to 1 ratio.
So again, there are several ways that you can approach similarity transformations, especially as it relates to triangles.