Question 20
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.)