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2006 Stanford Mathematics Tournament
23
23
Part of
2006 Stanford Mathematics Tournament
Problems
(1)
Mirrors and Lasers
Source:
3/15/2010
Consider two mirrors placed at a right angle to each other and two points A at
(
x
,
y
)
(x,y)
(
x
,
y
)
and B at
(
a
,
b
)
(a,b)
(
a
,
b
)
. Suppose a person standing at point A shines a laser pointer so that it hits both mirrors and then hits a person standing at point B (as shown in the picture). What is the total distance that the light ray travels, in terms of
a
a
a
,
b
b
b
,
x
x
x
, and
y
y
y
? Assume that
x
x
x
,
y
y
y
,
a
a
a
, and
b
b
b
are positive. [asy]draw((0,4)--(0,0)--(4,0),linewidth(1)); draw((1,3)--(0,2),MidArrow); draw((0,2)--(2,0),MidArrow); draw((2,0)--(3,1),MidArrow); dot((1,3)); dot((3,1)); label("
A
(
x
,
y
)
A (x,y)
A
(
x
,
y
)
", (1,3),NE); label("
B
(
a
,
b
)
B (a,b)
B
(
a
,
b
)
", (3,1),NE);[/asy]