Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 4"
(apostrophe) |
|||
Line 1: | Line 1: | ||
− | <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 15:19, 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.