Video: Identifying the Type of Geometric Transformation of a Triangle given Its Vertices before and after the Transformation

A triangle has vertices at the points (1, 1), (7, 1), and (1, 2). The image of this triangle has vertices at the points (−1, 1), (−1, 7), and (−2, 1). Which transformation has taken place?

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Video Transcript

A triangle has vertices at the points one, one; seven, one; and one, two. The image of this triangle has vertices at the points negative one, one; negative one, seven; and negative two, one. Which transformation has taken place?

There are a couple of ways to answer this question. The first is to sketch this out on a coordinate grid. Let’s put each vertex in turn. The first vertex is the vertex one, one, which is here. We have a vertex at seven, one, which is here, and one at one, two, which is here. And so, we have a right triangle located in the first quadrant. Let’s now plot the image of the triangle. The first vertex has coordinates negative one, one, which is here. The next has coordinates negative one, seven, which is here. And the last has coordinates negative two, one, which is here. And so, we now have the original triangle and its image. So, we need to decide which transformation has taken place.

We can choose from reflections where the fl reminds us we flip a shape across the mirror line. We have rotations where the t reminds us that we turn the shape about a center for a given angle and in a given direction. When we dilate a shape, the l reminds us we make it larger or smaller. And we could translate a shape. And when we translate it, we slide it. So, are there any of these that we can instantly disregard?

Well, yes. Firstly, the shapes are the same size, and so we can disregard dilations. We can dilate a shape and end up with a shape that’s the same size if we use a negative scale factor, in fact, a scale factor of negative one. But the orientation for that isn’t quite right. Similarly, when we translate a shape, we slide it. This means its orientation doesn’t change, just its position on the grid. And so, we’re disregarding translations, leaving us with reflections or rotations.

When we reflect a shape, we flip it in a mirror line. Now there’s no way to draw a mirror line on our diagram so that when we flip it or reflect the original shape, we end up with the new one. And so, that leaves us with rotation. Now, actually, it does indeed look like this shape has been rotated. It’s been turned. If we look carefully, we see it’s been turned by 90 degrees in a counterclockwise direction. In fact, if we were to describe the full transformation, we could say this is a rotation by 90 degrees in a counterclockwise direction about the origin or center zero, zero. But we’re just being asked which transformation has taken place, and so the answer is rotation.