2021 Fall AMC 10B Problems/Problem 17
Contents
[hide]Problem
Distinct lines and
lie in the
-plane. They intersect at the origin. Point
is reflected about line
to point
, and then
is reflected about line
to point
. The equation of line
is
, and the coordinates of
are
. What is the equation of line
Solutions
Solution 1
It is well known that the composition of 2 reflections , one after another, about two lines and
, respectively, that meet at an angle
is a rotation by
around the intersection of
and
.
Now, we note that is a 90 degree rotation clockwise of
about the origin, which is also where
and
intersect. So
is a 45 degree rotation of
about the origin clockwise.
To rotate 90 degrees clockwise, we build a square with adjacent vertices
and
. The other two vertices are at
and
. The center of the square is at
, which is the midpoint of
and
. The line
passes through the origin and the center of the square we built, namely at
and
. Thus the line is
. The answer is (D)
.
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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All AMC 10 Problems and Solutions |
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