In the given figure, triangle 𝐴 has been transformed to triangle 𝐴 dash by reflecting first in the 𝑦-axis and then reflecting in the 𝑥-axis. What single transformation would have mapped 𝐴 to 𝐴 dash? Is it (A) a rotation about the origin of 90 degrees? Is it (B) a rotation about the origin of 270 degrees? Is it (C) a rotation about the origin of 180 degrees, (D) a reflection in the 𝑦-axis, or (E) a reflection in the line 𝑦 equals 𝑥?
Now, we’ve been told that triangle 𝐴 has been transformed to 𝐴 dash by two reflections, one in the 𝑦-axis and then the next one in the 𝑥-axis. Now, actually, we have a diagram of what’s happened, so we don’t really need to worry too much about these two reflections. Instead, we’re going to look at how we can map 𝐴 directly onto 𝐴 dash.
There are four transformations we need to consider. Those are rotations, reflections, translations, and enlargements. When we rotate a shape, we turn it. So I’ve highlighted the letter “t” as a reminder. When we reflect a shape, we flip it. Let’s highlight “fl” for flip. When we translate to shape, we slide it. We highlight the letters “sl” for slide. Finally, when we enlarge a shape, we change its size. We can make it larger or smaller. So we highlight the word “large” inside the word enlargements to remind us of this.
And we see that we can instantly disregard one of these transformations. Shape 𝐴 and 𝐴 dash are actually the exact same size; they’re congruent. So we can disregard enlargements. Let’s now consider translations. When we translate a shape, we said we slide it. The exact orientation of the shape, though, remains unchanged. We can see that our shape is in a different orientation. It looks to be sort of upside down. And so we’re going to disregard translations.
That leaves us with two options. We have rotations and reflections. Now, when we rotate a shape, we turn it around some point. And when we reflect a shape, we reflect it or flip it in a mirror line. Now, in fact, if we look at shape 𝐴, we see it has been turned. We can, therefore, disregard reflections. And it must have been rotated.
Now, we decide the center of rotation and the angle about which it is rotated. One method is to use tracing paper and trial and error. But in fact, we can do this by observation. If we compare 𝐴 and 𝐴 dash, 𝐴 dash looks to be upside down. That tells us it must have completed half a turn. That’s an angle of 180 degrees. Now, its center must be the origin. If we draw a horizontal and vertical line from the origin to one of the vertices on our original shape, we see this rotates 180 degrees about the origin onto our new vertex. And so the answer must be (C). It’s a rotation about the origin of 180 degrees.