2005 AIME I Problems/Problem 14
Problem
Consider the points and
There is a unique square
such that each of the four points is on a different side of
Let
be the area of
Find the remainder when
is divided by
.
Solution
Solution 1
Consider a point such that
is perpendicular to
,
intersects
, and
. E will be on the same side of the square as point
.
Let the coordinates of be
. Since
is perpendicular to
, and
, we have
and
The coordinates of
are thus
.
Now, since and
are on the same side, we find the slope of the sides going through
and
to be
. Because the other two sides are perpendicular, the slope of the sides going through
and
are now
.
Let be the vertices of the square so that
contains point
,
contains point
, and etc. Since we know the slopes and a point on the line for each side of the square, we use the point slope formula to find the linear equations. Next, we use the equations to find
vertices of the square, then apply the distance formula.
We find the coordinates of to be
and the coordinates of
to be
. Applying the distance formula, the side length of our square is
.
Hence, the area of the square is . The remainder when
is divided by
is
.
Solution 2
Let denote a normal vector of the side containing
. Note that
intersect and hence must be opposite vertices of the square. The lines containing the sides of the square have the form
,
,
, and
. The lines form a square, so the distance between
and the line through
equals the distance between
and the line through
, hence
, or
. We can take
and
. So the side of the square is
, the area is
, and the answer to the problem is
.
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |