Question Video: Identifying the Image of a Shape after a Rotation about the Origin Mathematics • 8th Grade

Video Transcript

If triangle 𝐴𝐵𝐶 is rotated by 𝑅 brackets 𝑂, 90 degrees, which triangle would represent its final position?

And we are given four possible answer options for the position of triangle 𝐴𝐵𝐶. In this first, largest graph, we can observe the right triangle 𝐴𝐵𝐶 with all its coordinates in the first quadrant. And we are told that this triangle is rotated. The 𝑂 in the brackets notation means that the rotation is performed about 𝑂, the origin, which means that the rotation has a center of rotation at 𝑂. The value of 90 degrees indicates that the angle of rotation is 90 degrees. And when no direction is specified in a rotation, we use the convention that a positive angle of rotation means a counterclockwise direction, which is what we have here.

All of the coordinates of triangle 𝐴𝐵𝐶 are going to move 90 degrees counterclockwise and will be in the second quadrant. That means we can straightaway tell that the shapes in the graphs in options (A) and (D) are in the wrong quadrant. So let’s see what triangle 𝐴𝐵𝐶 would look like after the correct rotation. We can take each of the points — here, for example, we have point 𝐴 — and draw a line from the point to the center of rotation, which is the point 𝑂. A 90 degrees’ counterclockwise rotation would take point 𝐴 to here. This is the image of point 𝐴, and we can label it with the notation 𝐴 prime. The coordinates of 𝐴 prime are at negative one, one.

We can do the same with point 𝐵, drawing a line from this point to the origin. A 90-degree counterclockwise turn would create the image 𝐵 prime at the coordinates negative one, three. And we can do the same for point 𝐶. The rotation of 90 degrees counterclockwise would produce the point 𝐶 prime at the coordinates negative two, one. We can join the points to create the triangle 𝐴 prime 𝐵 prime 𝐶 prime. This is the answer given in the graph in option (C).

Notice that the graph in option (D) which we already eliminated would show the rotation of triangle 𝐴𝐵𝐶 by a rotation of 90 degrees clockwise. Or using the same notation, we could say that this is the rotation 𝑅 about the point 𝑂 by negative 90 degrees. Options (A) and (B) appear to be reflections instead, with option (A) being a reflection in the 𝑥-axis and option (B) being a reflection in the 𝑦-axis. So answers (A), (B), and (D) are incorrect and only the graph in option (C) is correct.