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2019 CMIMC
13
13
Part of
2019 CMIMC
Problems
(1)
2019 T13: A Peculiar Laser
Source:
1/27/2019
Points
A
A
A
,
B
B
B
, and
C
C
C
lie in the plane such that
A
B
=
13
AB=13
A
B
=
13
,
B
C
=
14
BC=14
BC
=
14
, and
C
A
=
15
CA=15
C
A
=
15
. A peculiar laser is fired from
A
A
A
perpendicular to
B
C
‾
\overline{BC}
BC
. After bouncing off
B
C
BC
BC
, it travels in a direction perpendicular to
C
A
CA
C
A
. When it hits
C
A
CA
C
A
, it travels in a direction perpendicular to
A
B
AB
A
B
, and after hitting
A
B
AB
A
B
its new direction is perpendicular to
B
C
BC
BC
again. If this process is continued indefinitely, the laser path will eventually approach some finite polygonal shape
T
∞
T_\infty
T
∞
. What is the ratio of the perimeter of
T
∞
T_\infty
T
∞
to the perimeter of
△
A
B
C
\triangle ABC
△
A
BC
?
2019
team