Mock AIME 6 2006-2007 Problems/Problem 9
Problem
is a triangle with integer side lengths. Extend
beyond
to point
such that
. Similarly, extend
beyond
to point
such that
and
beyond
to point
such that
. If triangles
,
, and
all have the same area, what is the minimum possible area of triangle
?
Solution
Let "a", "b", and "c", be the lengths of sides ,
and
respectively.
Let "h_a", "h_b", and "h_c", be the heights of from sides
,
and
respectively.
Since the areas of triangles ,
, and
are equal, then,
Therefore,
and
Since the area of is half any base times it's height, then:
Therefore,
and
Since "a", "b", and "c", and is a prime number, then the minimum integer value that
can have in order for
and
to also be integer is
Therefore ,
, and
minimum possible area of triangle using Heron's formula is
is:
, where
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.