Mock AIME 1 2006-2007 Problems/Problem 4
(Redirected from Mock AIME 1 2006-2007/Problem 4)
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 as our answer.