Difference between revisions of "2021 Fall AMC 10B Problems/Problem 17"
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<math>(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0</math> | <math>(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0</math> | ||
− | == | + | ==Solutions== |
+ | ===Solution 1=== | ||
It is well known that the composition of 2 reflections , one after another, about two lines <math>l</math> and <math>m</math>, respectively, that meet at an angle <math>\theta</math> is a rotation by <math>2\theta</math> around the intersection of <math>l</math> and <math>m</math>. | It is well known that the composition of 2 reflections , one after another, about two lines <math>l</math> and <math>m</math>, respectively, that meet at an angle <math>\theta</math> is a rotation by <math>2\theta</math> around the intersection of <math>l</math> and <math>m</math>. | ||
Revision as of 23:38, 23 November 2021
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
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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