Video Transcript
First, translate the given triangle two right and two down, and then rotate this image 180 degrees about the origin. Which of the following sets of coordinates will be the vertices of the image? Is it one, zero; one, one; and negative two, one? Is it (B) zero, negative one; negative one, negative one; and negative one, two? Is it (C) zero, one; one, one; and one, two? Is it (D) zero, one; one, one; and one, negative two? Or is it (E) negative one, zero; one, one, and two, one?
So we’re told to apply two transformations to our triangle 𝐴. They are a translation followed by a rotation. Now usually, we’d look to do this on the diagram, but we’ve not been given a lot of space. And so we have two options. We could redraw our diagram, giving us a little bit more space and showing the four quadrants of our diagram a little more clearly. But alternatively, we do have a couple of tricks. Let’s look at what those tricks might be.
Let’s begin by identifying the coordinates of the vertices of our triangle 𝐴. We have a coordinate here. And that has the coordinate negative three, four. We have one here, which has the coordinates negative three, one. And then our third vertex is at negative two, one. The first instruction says to translate, which is slide, the triangle two right and two down. And we can do this vertex by vertex. But in fact, we can actually consider what happens to one vertex and then apply the pattern to the other two.
Let’s take the vertex at negative three, four. We want to slide this one, two units to the right and then one, two units down. And when we do, we end up with a vertex with coordinates negative one, two. So what has happened to each part of this coordinate? Well, negative three has had two added to it and four has had two subtracted from it. And so we can say that to perform this translation, we’re going to map the coordinate 𝑥, 𝑦 onto the coordinate 𝑥 plus two, 𝑦 minus two. We’re going to add two to the 𝑥-parts and subtract two from the 𝑦-parts.
We’ve already seen that this maps negative three, four onto the point with coordinates negative one, two. And so let’s now look at the coordinate negative three, one. This will be mapped onto the point with coordinates negative three plus two, one minus two, which is negative one, negative one. Similarly, the point with coordinates negative two, one will be mapped onto the point negative two plus two, one minus two, which has coordinates zero, negative one. And so we found the vertices of the image of 𝐴 after the translation. It has vertices at negative one, two; negative one, negative one; and zero, negative one.
And what about the rotation? We’re rotating it 180 degrees about the origin. Well, a rotation by 180 degrees, in either direction, of course, since it’s half a turn about the origin or zero, zero, maps a point with coordinates 𝑎, 𝑏 onto a point with coordinates negative 𝑎, negative 𝑏. Essentially, we simply change the sign of both the 𝑥- and 𝑦-coordinates. And so if we consider the point with coordinates negative one, two, we see that changing the sign of each part gives us a point with coordinates one negative, two.
Similarly, let’s look at negative one, negative one. This will be mapped onto the point with coordinates one, one. And finally, if we rotate the point with coordinates zero, negative one 180 degrees about the origin, we get the point with coordinates zero, one. And so the two transformations map our triangle onto the triangle with vertices one, negative two; one, one; and zero, one. So which sets of coordinates does this apply to? Well, we can see the correct answer here is (D). It’s zero, one; one, one; and one, negative two.
