2020 AMC 10A Problem 23

Let T be the triangle in the coordinate plane with vertices (0,0), (4,0), and (0,3). Consider the following five isometries (rigid transformations) of the plane: rotations of 90^{\circ}, 180^{\circ}, and 270^{\circ} counterclockwise around the origin, reflection across the x-axis, and reflection across the y-axis. How many of the 125 sequences of three of these transformations (not necessarily distinct) will return T to its original position? (For example, a 180^{\circ} rotation, followed by a reflection across the x-axis, followed by a reflection across the y-axis will return T to its original position, but a 90^{\circ} rotation, followed by a reflection across the x-axis, followed by another reflection across the x-axis will not return T to its original position.)

\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 25

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.