Video: Finding the Coordinates of the Vertices of a Triangle after Reflection

List the images of the vertices of the triangle shown, after reflection in the dashed line.

04:12

Video Transcript

List the images of the vertices of the triangle shown, after reflection in the dashed line.

So what we’re looking to here is one of our transformations. And the transformation we’re looking to do is a reflection. And for any reflection, what we need is a mirror line. And we’ve been told in the question that the reflection is gonna be in the dashed line. So the dashed line is gonna be our mirror line. And I’ve shown that here. So what we’re looking to do in the question is show where the images of the vertices of our triangle going to be.

So what are the vertices of our triangle? Well, I’ve shown them here in orange. So I’ve circled our vertices. And we’ve got three of them. So I’ve also shown their coordinates. And I’ve done this by going along the 𝑥-axis first and then up the 𝑦-axis, so along the corridors or up the stairs, sometimes known as. So our vertices are two, one; five, four; and 10, one. Now, as we have a horizontal mirror line, then each of our vertices on our image or our reflected image are going to be the same distance vertically from the mirror line as those in the original triangle.

So if we start with this vertex here, we can see that it’s six units below the mirror line. So therefore, the corresponding vertex on our reflected image is going to be six units above the mirror line. It’s worth noting that they’re both vertically below and above. Okay, now, let’s move on to the next vertex. Well, this vertex is three units below the mirror line for original triangle. So therefore, for the image, it’s gonna be three units above the mirror line. So now, what we can do is move on to the final vertex. And we can see that the original vertex of the original triangle is six units directly below, so vertically below our mirror line. So that means that the vertex of the image is gonna be six units vertically above the mirror line.

Okay, great, so now we’ve got the vertices of the image or our reflected image of our triangle. So if we join them up, we can take a look and say, “Yes, this is a reflection of the original triangle.” And what we can do to double-check is have a look at some of the dimensions. So we’ve got a base of eight. That’s on both of them. Great. And we’ve got a perpendicular height of three. So, yes, we’ve checked. They’ve got the same dimensions. So we have got a reflection of the original triangle. Because it is worth noting if you’re reflecting the shape, it should not change the size because that would be an enlargement.

So now, we need to find the coordinates for our vertices. And the first one is gonna be two, 13 because it’s two on the 𝑥-axis, 13 on the 𝑦-axis. And we can check that to make sure it makes sense because we know that the 𝑥-coordinate should remain the same. And that’s because we’ve got a horizontal mirror line and the 𝑦-coordinate is 12 greater than the original 𝑦-coordinate of the original triangle. Well, let’s think about that. Does that make logical sense? Well, we said that it was six units away from the mirror line. Well, that means we traveled six units from the original triangle to the mirror line, another six units from the mirror line to the reflection vertex. So therefore, six add six is 12. 1 add 12 is 13. So yes, this does look correct.

Then, the next vertex will be five, 10. And again, we check it in the same way. We’ve got five, four as our original vertex. 𝑥 stays the same. Then, we’ve got three units away from mirror Line. So that’s three units up from the original triangle to the mirror line and then another three units up from the mirror line to the reflection. That’s six. Four add six is 10. So, yes, that makes sense. That is our second vertex. Then, in exactly the same way, we can write the third vertex. That’s 10, 13. And we did that again with 10 and then one plus 12.

So great, we’ve checked them all. And we can say that the images of the vertices of the triangle shown after reflection in the dashed line are two, 13; five, 10; and 10, 13.