Difference between revisions of "2020 AMC 12A Problems/Problem 12"
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~Solution by IronicNinja | ~Solution by IronicNinja | ||
+ | |||
+ | ==Solution 3 (Cheap)== | ||
+ | |||
+ | Using the protractor you brought, carefully graph the equation and rotate the given line <math>45^{\circ}</math> counter-clockwise about the line. Scaling everything down by a factor of 5 makes this process easier. | ||
+ | |||
+ | It should then become fairly obvious that the x intercept is <math>x=\boxed{15}</math> (only use this as a last resort). | ||
+ | -Silverdragon | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2020|ab=A|num-b=11|num-a=13}} | {{AMC12 box|year=2020|ab=A|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 03:22, 2 February 2020
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 (Cheap)
Using the protractor you brought, carefully graph the equation and rotate the given line counter-clockwise about the line. 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
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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