2007 AMC 12B Problems/Problem 22
Problem 22
Two particles move along the edges of equilateral in the direction
starting simultaneously and moving at the same speed. One starts at
, and the other starts at the midpoint of
. The midpoint of the line segment joining the two particles traces out a path that encloses a region
. What is the ratio of the area of
to the area of
?
Solution
First, notice that each of the midpoints of ,
, and
are on the locus.
Suppose after some time the particles have each been displaced by a short distance
, to new positions
and
respectively. Consider
and drop a perpendicular from
to hit
at
. Then,
and
. From here, we can use properties of a
triangle to determine the lengths
and
as monomials in
. Thus, the locus of the midpoint will be linear between each of the three special points mentioned above. It follows that the locus consists of the only triangle with those three points as vertices. (A cheaper way to find the shape of the region is to look at the answer choices: if it were any sort of conic section then the ratio would not generally be rational.)
Comparing inradii between this "midpoint" triangle and the original triangle, the area contained by
must be
of the total area.
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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