2020 AMC 12A Problem 20

Problem 20 of 25HarderGeometry

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

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. Let

r

be the

90^\circ

rotation and

s

reflection across the

x

-axis. The five allowed maps are

r,r^2,r^3,s,r^2s;

the missing nonidentity maps are the diagonal reflections

rs,r^3s.

In an ordered triple, the third map is forced by the first two and is allowed precisely when their product is one of the five. Exactly

5

ordered pairs have product identity: each first map is paired with its inverse. A diagonal reflection requires one rotation and one axis reflection;

r

r^3

may be paired on either side with

s

r^2s,

giving

2\cdot2\cdot2=8

pairs. The remaining

25-5-8=12

ordered pairs give valid sequences. Thus, A is the correct answer.