Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 4"
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Revision as of 14:52, 3 April 2012
has all of its vertices on the parabola 
. The slopes of 
and 
are 
and 
, respectively. If the 
-coordinate of the triangle's centroid is 
, find the area of .
Solution
If a triangle in the Cartesian plane has vertices 
and 
then its centroid has coordinates 
. Let our triangle have vertices 
and 
. Then we have by the centroid condition that 
. From the first slope condition we have 
and from the second slope condition that 
. Then 
, 
and 
, so our three vertices are 
and 
.
Now, using the Shoelace Theorem (or your chosen alternative) to calculate the area of the triangle we get 665 as our answer.
