Difference between revisions of "2024 AMC 12B Problems/Problem 12"
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Revision as of 02:44, 14 November 2024
Problem
Let be a complex number with real part greater than
and
. In the complex plane, the four points
,
,
, and
are the vertices of a quadrilateral with area
. What is the imaginary part of
?
Diagram
Solution 1 (similar triangles)
By making a rough estimate of where ,
, and
are on the complex plane, we can draw a pretty accurate diagram (like above.)
Here, points ,
, and
lie at the coordinates of
,
, and
respectively, and
is the origin.
We're given , so
and
. This gives us
,
, and
.
Additionally, we know that (since every power of
rotates around the origin by the same angle.) We set these angles equal to
.
This gives us enough info to say that by SAS (since
.)
It follows that as the ratio of side lengths of the two triangles is 2 to 1.
This means or
as we were given
.
Using , we get that
, so
, giving
.
Thus, .
~nm1728
See also
2024 AMC 12B (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 |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.