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.)

Video solution:

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Written solution:

Let $R$ be a $90^\circ$ rotation, so the allowed rotations are $R,R^2,R^3$. Let $X$ and $Y$ be the reflections across the coordinate axes. Once the first two transformations are chosen, the third is forced to be the inverse of their product.

The ordered first-two choices whose forced third transformation is still in the allowed set are $(R,R),(R,R^2),(R^2,R),(R^2,R^3),(R^3,R^2),(R^3,R^3)$, and $(R^2,X),(R^2,Y),(X,R^2),(Y,R^2),(X,Y),(Y,X)$. There are $12$ such sequences. Thus, A is the correct answer.