2006 iTest Problems/Problem 27
Problem
Line passes through
and into the interior of the equilateral triangle
.
and
are the orthogonal projections of
and
onto
respectively. If
and
, then the area of
can be expressed as
, where
and
are positive integers and
is not divisible by the square of any prime. Determine
.
Solution
Let be the intercept of
and
. By the Vertical Angle Theorem,
. Also, since both
and
are perpendicular to
,
. Thus,
by AA Similarity. Since
,
and
.
Let be the side length of the triangle, so
. By the Pythagorean Theorem,
. Also,
, so by the Law of Cosines,
.
By using the Pythagorean Theorem again, we have
Thus, the area of the triangle is
, so
.
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 26 |
Followed by: Problem 28 | |
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