Describe the single transformation that would map triangle 𝐴𝐵𝐶 onto triangle 𝐴 prime 𝐵 prime 𝐶 prime in the given figure. Option (A) a rotation of 90 degrees clockwise about 𝐷. Option (B) a rotation of 90 degrees counterclockwise about 𝐷. Option (C) a rotation of 90 degrees clockwise about 𝐸. Option (D) a rotation of 270 degrees clockwise about 𝐸. Or option (E) a rotation of 270 degrees counterclockwise about 𝐷.
In this question, we’re considering how the triangle 𝐴𝐵𝐶 is transformed to give us triangle 𝐴 prime 𝐵 prime 𝐶 prime. The four transformation options are translation, reflection, rotation, or dilation. We can consider that, in the case of a dilation, this would expand or contract the object. As we can see that our two triangles are the same size, we can rule out dilation. In a translation, the object and the image after translation would be in the same orientation. That means the same way up. We can see that our two triangles here are in a different orientation. So we can rule out translation.
In a reflection, the object and its image would appear to be a mirror image of each other. As we can see that they’re not a mirror image here, then we can rule out reflection, which leaves us with rotation. When we describe a rotation, we need to give several pieces of information. We need to state the center of rotation. That’s the point or coordinate about which the shapes are rotated. We also need to give the angle of rotation and the direction. Sometimes, it can be difficult to find the center of rotation. But if we were to use tracing paper and trace over triangle 𝐴𝐵𝐶, what we’re looking for is a point about which we could turn this shape to get to 𝐴 prime 𝐵 prime 𝐶 prime.
Let’s say that we move our tracing paper in the counterclockwise direction. As there’s no overlap between the object and the image, then we know that the center of rotation will be outside the shape. In fact, that the center of rotation would lie somewhere in this area. We can check if the center of rotation is at 𝐷 by seeing if we can find an angle that would rotate each vertex to its image. We can see that the angle created between 𝐴 and 𝐷 and 𝐴 prime and 𝐷 would be a 90-degree angle. Drawing a line from 𝐵 to 𝐷 and then from 𝐷 to 𝐵 prime would also create a 90-degree angle. So it looks as though we’ve find the angle and the center of rotation. But we can check with the final vertex.
Between 𝐶 and 𝐶 prime, there would also be a right angle of 90 degrees. And therefore, we’ve found that the center of rotation is 𝐷, the angle is 90 degrees, and the direction we turned it through was counterclockwise. We can put these pieces of information into statement form that this is a rotation of 90 degrees counterclockwise about 𝐷. However, there is also one other way we could’ve described this rotation. And that is by changing the direction of the rotation.
So here, the direction would have been clockwise and the angle this time would have been 270 degrees. Even though we’ve changed the angle and the direction, the center of rotation would be the same. So we could’ve described this as a rotation of 270 degrees clockwise about 𝐷. Although both of these statements are equally valid, there’s only one that’s given in the answer options. And that’s the one given in option (B). This is a rotation of 90 degrees counterclockwise about 𝐷.