2016 AMC 10B Problem 9

Problem 9 of 25EasierGeometry

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?

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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,

x_1^2=x_2^2 .

This implies that either

x_1=-x_2

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,

BC = 2\cdot 4 = 8.

Thus, the correct answer is C .