Video: Finding the Vertices of a Triangle’s Image given the Original Is Rotated about the Origin

A triangle has its vertices at the points (2, 1), (3, 2) and (2, 4). The triangle is rotated 90° counterclockwise about the origin. At which of the following coordinates will the image have its vertices? [A] (1, 2), (2, 3), and (2, −4) [B] (−1, 2), (−2, 3), and (−4, 2) [C] (−1, 2), (2, 3), and (2, −4) [D] (2, −1), (3, −2), and (2, −4) [E] (−2, 1), (−2, 3), and (4, −2)

04:22

Video Transcript

A triangle has its vertices at the points two, one; three, two; and two, four. The triangle is rotated 90 degrees counterclockwise about the origin. At which of the following coordinates will the image have its vertices?

And then we’re given five different sets of vertices. The image is the position of the triangle after it’s been rotated. One way to solve this is to remember a rule for this kind of rotation. The rules for rotation for 90 degrees counterclockwise tell us if the preimage is 𝑥, 𝑦, the image will be negative 𝑦, 𝑥. In our starting triangle, in our preimage, we have a vertex at two, one; three, two; and two, four. We could label them 𝐴, 𝐵, and 𝐶. We usually mark the image with a prime dash mark. We call it 𝐴 prime. 𝐴 prime would be negative one, two. We found one vertex of the image. And that eliminates all but options B and C from our answer choice. B and C are the only options which include negative one, two as a vertex.

If we look at vertex 𝐵, its image will be 𝐵 prime: negative two, three. We see this vertex in option B. But it’s still worth checking our final vertex. 𝐶 prime will then be negative four, two. But what if you didn’t remember the rules for rotation? Could you still solve the problem? A second method for solving would involve a graph. Once we label our graph, we can label the points for our preimage. Vertex 𝐴 at two, one; vertex 𝐵 at three, two; and vertex 𝐶 at two, four. When we connect all the vertices, they form a triangle.

Now, we need to think about what a 90-degree counterclockwise rotation about the origin would look like. First, we can draw a line from the origin to vertex 𝐴. A 90-degree counterclockwise rotation about the origin means that the new point needs to form a 90-degree angle with the origin and the original point. You can imagine us putting our finger on the origin and spinning this figure 90 degrees until it arrived at the point negative one, two. And negative one, two would be 𝐴 prime. The same thing is true for vertex 𝐵. And the same thing will happen with point 𝐶 to give us 𝐶 prime. If we connect the vertices, we see the image 𝐴 prime, 𝐵 prime, 𝐶 prime. This is a 90-degree counterclockwise rotation about the origin. And the image has the vertices listed in option B.

If you still weren’t sure, you could plot the points from the other four options. The three points from option A — before we do that, let’s just clear up some space on our graph. Option A has the point one, two; two, three; and two, negative four. The main problem here is that we can immediately tell that the size of the triangle has changed. And in a rotation, the size stays the same. What about option C, negative one, negative two; two, three; and two, negative four? The vertices in option C make a triangle that’s a different size and shape. So it can’t be a rotation.

Option D would look like this. This image is a reflection of our preimage, not a rotation. It’s the same size and the same shape, but it is reflected over the 𝑥-axis instead of rotated 90 degrees counterclockwise. Finally, we’ll consider Option E. And option E doesn’t create a triangle even remotely similar to what we started with. And so, we can confirm the new vertices of the image to be negative one, two; negative two, three; and negative four, two.

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