2020 AMC 12A Problems/Problem 12
Contents
Problem
Line in the coordinate plane has equation
. This line is rotated
counterclockwise about the point
to obtain line
. What is the
-coordinate of the
-intercept of line
Solution
The slope of the line is . We must transform it by
.
creates an isosceles right triangle since the sum of the angles of the triangle must be
and one angle is
which means the last leg angle must also be
.
In the isosceles right triangle, the two legs are congruent. We can, therefore, construct an isosceles right triangle with a line of slope on graph paper. That line with
slope starts at
and will go to
, the vector
.
Construct another line from to
, the vector
. This is
and equal to the original line segment. The difference between the two vectors is
, which is the slope
, and that is the slope of line
.
Furthermore, the equation passes straight through
since
, which means that any rotations about
would contain
. We can create a line of slope
through
. The
-intercept is therefore
~lopkiloinm
Solution 2
Since the slope of the line is , and the angle we are rotating around is x, then
Hence, the slope of the rotated line is . Since we know the line intersects the point
, then we know the line is
. Set
to find the x-intercept, and so
~Solution by IronicNinja
Solution 3
Let be
and
be
and
respectively. Since the slope of the line is
we know that
Segments
and
represent the before and after of rotating
by 45 counterclockwise. Thus,
and
by tangent addition formula. Since
is 5 and the sidelength of the square is 20 the answer is
Solution 4 (Cheap)
Using the protractor you brought, carefully graph the equation and rotate the given line counter-clockwise about the point
. Scaling everything down by a factor of 5 makes this process easier.
It should then become fairly obvious that the x intercept is (only use this as a last resort).
~Silverdragon
Solution 5 (Linear Algebra)
First note that the given line goes through with a slope of
. This means that
is on the line. Now consider translating the graph so that
goes to the origin, then
becomes
. We now rotate the line
about the origin using a rotation matrix. This maps
to
The line through the origin and
has slope
. Translating this line so that the origin is mapped to
, we find that the equation for the new line is
, meaning that the
-intercept is
.
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 12 Problems and Solutions |
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