2007 AMC 12B Problems/Problem 14
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Problem 14
Point is inside equilateral
. Points
,
, and
are the feet of the perpendiculars from
to
,
, and
, respectively. Given that
,
, and
, what is
?
Solution
Drawing ,
, and
,
is split into three smaller triangles. The altitudes of these triangles are given in the problem as
,
, and
.
Summing the areas of each of these triangles and equating it to the area of the entire triangle, we get:
where
is the length of a side