Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 9"
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==Solution== | ==Solution== | ||
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Let <math>a</math>, <math>b</math>, and <math>c</math>, be the lengths of sides <math>AB</math>, <math>BC</math> and <math>CA</math> respectively. | Let <math>a</math>, <math>b</math>, and <math>c</math>, be the lengths of sides <math>AB</math>, <math>BC</math> and <math>CA</math> respectively. |
Revision as of 13:46, 26 November 2023
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 ,
, and
, be the lengths of sides
,
and
respectively.
Let ,
, and
, 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 ,
, and
, are integers, 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.