Which of the following is equivalent to a 25-degree rotation about the origin? Option (A) a negative 25-degree rotation about the origin. Option (B) a 25-degree rotation about the point one, one. Option (C) a negative 155-degree rotation about the origin. Option (D) a 335-degree rotation about the origin. Or option (E) a negative 335-degree rotation about the origin.
In this problem, we are considering a 25-degree rotation about the origin, which is the point zero, zero. Now, we don’t know if this is a point or a line segment or even a polygon that is being rotated. And it doesn’t really matter which. So let’s take any point here in the first quadrant and call it point 𝐴. If we are rotating through a positive angle and we aren’t given a direction, then we use the convention that this is in the counterclockwise direction, so 25 degrees counterclockwise. And after this rotation of 25 degrees counterclockwise, we would obtain the image of 𝐴, which we could call 𝐴 prime. So are there any different rotations which would also produce this point 𝐴 prime at the same coordinates?
Well, because we know that the sum of the angle measures about a point is 360 degrees, we know that a 360-degree rotation of 𝐴 would take us back to point 𝐴. And another 25 degrees would take us to 𝐴 prime. Adding 360 degrees and 25 degrees would give us 385 degrees. So a 385-degree rotation, which would be counterclockwise, about the origin would also take us to the image 𝐴 prime. And indeed, if we added any multiple of 360 degrees to 25 degrees, we would get an equivalent rotation.
But if we look at the given answer options, none of these match the rotation we have found. So let’s recall an important property about rotation, which comes from the property we have already seen, that the angle measures about a point sum to 360 degrees. It is that a rotation of 𝑥 degrees is equivalent to a rotation of 𝑥 plus or minus 360 degrees. So, if we go back to the 25-degree rotation, it’s equivalent to both 25 plus 360 degrees and 25 minus 360 degrees, which is why we had the 385-degree rotation. But now we can also see that negative 335 degrees is equivalent. That’s because a rotation of 335 degrees clockwise or a rotation of negative 335 degrees about the origin would rotate point 𝐴 to the same image 𝐴 prime as the 25-degree rotation, which was counterclockwise.
Therefore, a negative 335-degree rotation about the origin would also be an equivalent rotation. This is the answer that was given in option (E).
None of the other answer options would produce an equivalent rotation. And it is worth noting that although option (B) had an equal angle of 25 degrees, it does not have the same center of rotation at the origin. This would mean that the image produced by this rotation would not be in the same position as the image produced by a 25-degree rotation about the origin. The only correct answer from the given options is a negative 335-degree rotation about the origin.