2022 AMC 12A Problems/Problem 13
Problem
Let be the region in the complex plane consisting of all complex numbers
that can be written as the sum of complex numbers
and
, where
lies on the segment with endpoints
and
, and
has magnitude at most
. What integer is closest to the area of
?
Solution
If is a complex number and
, then the magnitude (length) of
is
. Therefore,
has a magnitude of 5. If
has a magnitude of at most one, that means for each point on the segment given by
, the bounds of the region
could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of
.
$\$ The total area is, therefore,$ (Error compiling LaTeX. Unknown error_msg)5(2) + \pi \approx 10 + 3 = 13$ (A)