Video Transcript
A shape 𝐹 has been reflected in the line 𝑦 equals 𝑥 and then rotated 270 degrees clockwise about the origin to 𝐹 prime. Would 𝐹 and 𝐹 prime be congruent?
Let’s begin by recalling what we mean by this word congruent. Two shapes are congruent if they are identical. They have to have the same-sized angles and the same-length sides. They can, however, be in any orientation. And so let’s think about the shape that we have. That’s shape 𝐹.
We begin by reflecting shape 𝐹. Let’s see what that might look like. Let’s begin by imagining shape 𝐹 is the right-angled triangle shown. We’re going to reflect it in the line 𝑦 equals 𝑥. When we reflect a shape, we flip it in the mirror line. It should be exactly the same distance from the mirror line just on the other side. And so if we take this vertex of the triangle, it will end up here. This vertex will end up here, and this vertex will end up over here. Now, if you’re struggling to see this, tracing paper can be a really nice resource to help. And so the first image of 𝐹, the reflection in the line 𝑦 equals 𝑥, is this right triangle shown.
Now we can actually see that reflecting the shape 𝐹 doesn’t actually change the size of the shape. It just changes its orientation and it in fact flips it, which is absolutely fine. And so we can begin by saying that shape 𝐹 is actually congruent with its image after a reflection. And so let’s take the image of 𝐹. And we’re going to rotate it 270 degrees clockwise about the origin to 𝐹 prime. Remember, the origin is the point zero, zero. It’s where the axes meet.
270 degrees is three-quarters of a turn. And so we’re going to end up in this quadrant here. We can see that one of the vertices of our triangle will end up here, another will end up here. And of course, you can use tracing paper to do this. And the third vertex will end up here. And so here we have shape 𝐹 prime.
So let’s compare 𝐹 prime with the image of 𝐹 after the reflection. Once again, the size of the shape hasn’t actually changed, just its orientation and its position on our axes. And so the image of shape 𝐹 after its reflection and the shape 𝐹 prime are congruent. And so we’ve shown that 𝐹 is congruent with its image after a reflection and then that image itself is congruent with 𝐹 prime. And so 𝐹 and 𝐹 prime must themselves also be congruent, meaning that the answer is yes; 𝐹 and 𝐹 prime are congruent.
Now, notice that had we instead started by performing both transformations in a row, we might have noticed that 𝐹 and 𝐹 prime can be transformed onto one another with just one transformation, that is, a reflection in the 𝑦-axis. And we know that when we reflect a shape, its image is congruent to the original. So we could’ve performed it this way. Either way, we find that 𝐹 and 𝐹 prime are indeed congruent.
