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.)

Answer: A

Concepts:transformationsystematic listing

Difficulty rating: 1950

AMC 8AMC 10AMC 12AIME

Video solution:

Click to load, then click again to play

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.

Among two rotations, $6$ ordered pairs have a nonidentity rotation as their product. A rotation and a reflection have an allowed axis-reflection as their product exactly when the rotation is $R^2$, giving $4$ ordered pairs. Finally, the two different axis-reflections can occur in either order, giving $2$ more pairs. Altogether there are $6+4+2=12$ valid sequences. Thus, A is the correct answer.