Mock AIME 1 2006-2007 Problems/Problem 4

$\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$ . The slopes of $AB$ and $BC$ are $10$ and $-9$ , respectively. If the $x$ -coordinate of the triangle's centroid is $1$ , find the area of $\triangle ABC$ .

Solution

If a triangle in the Cartesian plane has vertices $(a_1, a_2), (b_1, b_2)$ and $(c_1, c_2)$ then its centroid has coordinates $\left(\frac{a_1 + b_1 + c_1}{3}, \frac{a_2 + b_2 + c_2}{3}\right)$ . Let our triangle have vertices $A(a, a^2), B(b, b^2)$ and $C(c, c^2)$ . Then we have by the centroid condition that $a + b + c = 3$ . From the first slope condition we have $10 = \frac{b^2 - a^2}{b - a} = b + a$ and from the second slope condition that $-9 = \frac{c^2 - b^2}{c - b} = c + b$ . Then $c = (a + b + c) - (a + b) = -7$ , $b = (b + c) - c = -2$ and $a = (a + b) - b = 12$ , so our three vertices are $(-7, 49), (-2, 4)$ and $(12, 144)$ .