Difference between revisions of "2008 AMC 12B Problems/Problem 17"
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<math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 20 </math> | <math> \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 19\qquad\textbf{(E)}\ 20 </math> | ||
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Revision as of 17:05, 29 December 2009
Let the coordinates of be
and the coordinates of
be
. Since the line
is parallel to the
-axis, the coordinates of
must be
.
Then the slope of line
is
.
The slope of line
is
.
Supposing ,
is perpendicular to
and, it follows, to the
-axis, making
a segment of the line x=m. But that would mean that the coordinates of
are
, contradicting the given that points
and
are distinct. So
is not
. By a similar logic, neither is
.
This means that and
is perpendicular to
. So the slope of
is the negative reciprocal of the slope of
, yielding
.
Because is the length of the altitude of triangle
from
, and
is the length of
, the area of
. Since
,
.
Substituting,
, whose digits sum to
.