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

2016 AMC 12B Problem 6

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All three vertices of

\triangle 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

Let a top vertex be

(x,x^2);

by symmetry the other is

(-x,x^2)

The base is

BC=2x

and the height is

x^2,

\tfrac12(2x)(x^2)=64

Let the vertex in the first quadrant be

(x,x^2).

By symmetry the base is

BC=2x

and the height is

x^2,

\tfrac12\cdot2x\cdot x^2=x^3=64.

Thus

x=4

and

BC=2x=8.

Thus, the correct answer is C.