Question

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.)

$12$

$15$

$17$

$20$

$25$

Solution:

Because $T$ is a scalene right triangle, the only isometry carrying $T$ to itself is the identity, so a sequence works exactly when the three transformations compose to the identity.

The five maps are all of the square's symmetry group except the identity and the two diagonal reflections. In an ordered triple the third map must be the inverse of the first two composed, and it is allowed precisely when the product of the first two is again one of the five.

Of the $25$ ordered pairs, $5$ compose to the identity and $8$ compose to a diagonal reflection. The remaining $25 - 13 = 12$ pairs give valid sequences.

Thus, A is the correct answer.

2020 AMC 12A Problem 20

Solution:

Problem 20 in Other Years