Question

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2,$ with $A$ at the origin and $\overline{BC}$ parallel to the $x$ -axis. The area of the triangle is $64.$ What is the length of $BC?$

$\ 4$

$\ 6$

$\ 8$

$\ 10$

$\ 16$

Solution:

Let the points be $A=(0,0),$ $B=(x_1,y_1),$ $C=(x_2,y_2).$

Then, since $BC$ is parallel with the $x$ -axis, we know $y_1=y_2,$ which we will let be $y.$ Then, $x_1^2 = y=x_2^2,$ so $x_1^2=x_2^2 .$ This implies that either $x_1=-x_2$ or $x_1 = x_2.$ The second option cannot happen since that would set two points as the same, which would create an area of $0.$ As such, let $x= -x_1=x_2.$ Then $y=x^2$ also.

Then, the points are $A=(0,0),$ $B=(-x,x^2),$ $C=(x,x^2).$ With a base of $BC,$ the length is $2x$ and the height is $x^2.$ This would make the area $x^3 = 64.$ Therefore, $x=4,$ so $BC = 2\cdot 4 = 8.$

Thus, the correct answer is C .

2016 AMC 10B Problem 9

Solution:

Problem 9 in Other Years