2020 AMC 10A Problem 23

Problem 23 of 25HarderGeometry Counting & Probability

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.)

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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.

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