2020 AIME II Problems/Problem 4
Problem
Triangles and
lie in the coordinate plane with vertices
,
,
,
,
,
. A rotation of
degrees clockwise around the point
where
, will transform
to
. Find
.
Solution
After sketching, it is clear a rotation is done about
. Looking between
and
,
and
. Solving gives
. Thus
.
~mn28407
Solution 2 (Official MAA)
Because the rotation sends the vertical segment to the horizontal segment
, the angle of rotation is
degrees clockwise. For any point
not at the origin, the line segments from
to
and from
to
are perpendicular and are the same length. Thus a
clockwise rotation around the point
sends the point
to the point
. This has the solution
. The requested sum is
.
Solution 3
A degree rotation is obvious. Let's look at
and
. They are very close to each other. Let's join
and
with a line. Then construct a perpendicular bisector to
with the midpoint being
which is at
. We also draw a point
on the perpendicular bisector such that
is
. That point
is the same distance to
as
is to
but it is on a line perpendicular to
Therefore
is at
or
. The sum is
.
Video Solution
~IceMatrix
See Also=
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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