Video: Determining Which Triangle Is an Original Triangle’s Image after Reflection

Reflect triangle 𝐴 in the 𝑦-axis and then in the 𝑥-axis. Which triangle is its image?

02:52

Video Transcript

Reflect triangle 𝐴 in the 𝑦-axis and then in the 𝑥-axis. Which triangle is its image?

Let’s begin by reflecting triangle 𝐴, that’s this one up here, in the 𝑦-axis, which is this vertical line. We know that when we reflect a shape in a mirror line, the shape will end up exactly the same distance from the mirror line, but on the other side. And it will be sort of flipped. And one way we can find the image after reflection is to do this vertex by vertex.

Let’s begin with this vertex on triangle 𝐴. We measure the perpendicular distance of this vertex from the mirror line, and that’s two units. We now come the exact same distance out of the mirror line on the other side. And when we do, we see that the vertex is on triangle 𝐶. And so, we can probably deduce that the image of triangle 𝐴 after its first reflection is triangle 𝐶. But let’s double-check this with another vertex.

Let’s take this vertex at the top of our triangle. Once again, we measure the perpendicular distance of this vertex from the mirror line, and that’s three units. We come out the other side of the mirror line the exact same distance, and we end up with the second vertex at 𝐶. And in fact, if we did that with the third vertex, we’d end up with the third vertex at 𝐶. And so, the image of triangle 𝐴 after the first reflection is triangle 𝐶.

But then we’re told this triangle is reflected in the 𝑥-axis, and this is this horizontal line. And so once again, we know that the image of 𝐶 after the reflection will be the exact same distance away from the line directly at the other side. And so we measure the perpendicular distance of our first vertex from the mirror line, and it’s two units once again. We carry on out the other side of the mirror line at the same distance, and we end up at this vertex at point two, negative two. We can do the same with our other vertices, and we end up with a vertex at point five, negative three and one at three, negative five. And so, the image of 𝐶 after its reflection is triangle 𝐷.

And so, we could therefore say that when we reflect triangle 𝐴 in the 𝑦-axis and then in the 𝑥-axis, the triangle that is its image is triangle 𝐷. Now, in fact, this wasn’t necessarily the quickest way to achieve this. And this is because if we reflect a shape in one axis and then reflect it in the other, that’s the same as rotating that shape 180 degrees about the origin or the point zero, zero. If we look at 𝐴, if we rotate that 180 degrees about this point, we do indeed get to triangle 𝐷. Either method is perfectly valid as long as we get the answer 𝐷.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.