Video Transcript
Starting with the triangle 𝐴 three, seven; 𝐵 four, one; and 𝐶 eight, seven, apply the transformations: (1) reflect in the 𝑦-axis, (2) reflect in the 𝑥-axis, and (3) translate by three units right and three units up. What are the images of the vertices?
We have three individual transformations to perform. And whilst we might be able to work out the new vertices in our head, it makes much more sense to sketch this out. We begin with a standard pair of 𝑥𝑦-axes. We’ll start by plotting the point with coordinates three, seven. Remember, we go along the corridor, up the stairs, so the point three, seven is here. Let’s call that 𝐴. Our next point has coordinates four, one. Once again, along the corridor, up the stairs, and that brings us to the point four, one here. The third vertex in our triangle has coordinates eight, seven. On our coordinate axes, that’s here. So, we’ve got our triangle 𝐴𝐵𝐶.
The first instruction is to reflect this triangle in the 𝑦-axis. Now, of course, the 𝑦-axis is this vertical line running down the center of our diagram. We need to reflect every vertex on our triangle in this line. And so we need to ensure that the vertices of the image of our triangle are the exact same distance away from the line but out the other side. For example, the perpendicular distance of vertex 𝐴 from the mirror line is three units. And so the image of 𝐴 is three units away from the mirror line as shown. Similarly, the perpendicular distance of vertex 𝐵 from the mirror line is four units. This means the image of 𝐵, which we can call 𝐵 prime, is four units away from the line but on the other side. If we repeat a similar process with vertex 𝐶, we end up with 𝐶 prime and the completed triangle as shown.
The next instruction tells us to reflect this shape in the 𝑥-axis. Now, of course, the 𝑥-axis is the horizontal line in our diagram. Once again, we can measure the perpendicular distances from each of our vertices to the 𝑥-axis. When we reflect our reflected triangle in the 𝑥-axis, we end up with one down here as shown. Now, note that an alternative way to perform these first two steps is to simply rotate by 180 degrees about the origin. A reflection in the 𝑦-axis followed by a reflection in the 𝑥-axis or vice versa is the same as a single rotation 180 degrees about the origin.
Now, the very last instruction we’ve been given is to translate by three units right and three units up. We might, alternatively, sometimes see that written in vector form. As we did with the reflections, we’re going to, once again, do this vertex by vertex. Let’s start with vertex 𝐴. We move one, two, three units right then one, two, three units up, giving us 𝐴 triple prime to be at the point zero, negative four. Let’s do this again with 𝐶 prime prime. We move one, two, three units to the right then one, two, three units up. 𝐶 triple prime lies at the point with coordinates negative five, negative four.
The last point we need to translate is 𝐵. We move one, two, three units right then one, two, three units up. 𝐵 triple prime therefore lies at the point negative one, two. And our triangle is as shown. And so we have the coordinates of the images of our vertices. The image of 𝐴 is at zero, negative four; the image of 𝐵 is at negative one, two; and the image of 𝐶 is negative five, negative four. Note also that it would be very usual just to label the final vertices. In this case, we would say 𝐴 prime is zero, negative four; 𝐵 prime is negative one, two; and 𝐶 prime is negative five, negative four.
