2013 AMC 12A Problems/Problem 23
Problem
is a square of side length
. Point
is on
such that
. The square region bounded by
is rotated
counterclockwise with center
, sweeping out a region whose area is
, where
,
, and
are positive integers and
. What is
?
Solution
We first note that diagonal is of length
. It must be that
divides the diagonal into two segments in the ratio
to
. It is not difficult to visualize that when the square is rotated, the initial and final squares overlap in a rectangular region of dimensions
by
. The area of the overall region (of the initial and final squares) is therefore twice the area of the original square minus the overlap, or
.
The area also includes circular segments. Two are quarter-circles centered at
of radii
(the segment bounded by
and
) and
(that bounded by
and
). Assuming
is the bottom-left vertex and
is the bottom-right one, it is clear that the third segment is formed as
swings out to the right of the original square [recall that the square is rotated counterclockwise], while the fourth is formed when
overshoots the final square's left edge. To find these areas, consider the perpendicular from
to
. Call the point of intersection
. From the previous paragraph, it is clear that
and
. This means
, and
swings back inside edge
at a point
unit above
(since it left the edge
unit below). The triangle of the circular sector is therefore an equilateral triangle of side length
, and so the angle of the segment is
. Imagining the process in reverse, it is clear that the situation is the same with point
.
The area of the segments can be found by subtracting the area of the triangle from that of the sector; it follows that the two quarter-segments have areas and
. The other two segments both have area
.
The total area is therefore
Since ,
, and
, the answer is
.
Video Solution by Richard Rusczyk
https://www.youtube.com/watch?v=-3d31c-7xm8
~dolphin7
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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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