Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 4"
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− | <math>\triangle ABC</math> has all of | + | <math>\triangle ABC</math> has all of its [[vertex| vertices]] on the [[parabola]] <math>y=x^{2}</math>. The slopes of <math>AB</math> and <math>BC</math> are <math>10</math> and <math>-9</math>, respectively. If the <math>x</math>-coordinate of the triangle's centroid is <math>1</math>, find the area of <math>\triangle ABC</math>. |
Revision as of 14:19, 4 September 2006
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 shoestring method (or your chosen alternative) to calculate the area of the triangle we get 665 as our answer.