2024 AMC 12B Problems/Problem 12
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 ?
$\textbf{(A) }\frac{3}{4}\qquad \textbf{(B) }1\qquad \textbf{(C) }\frac{4}[3}\qquad \textbf{(D) }\frac{3}{2}\qquad \textbf{(E) }\frac{5}[3}$ (Error compiling LaTeX. Unknown error_msg)
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