2016 AMC 10B Problems/Problem 9

Problem

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 $BC$ ?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$

Solution

Let the point in the first quadrant be $(a, a^2)$ . Then, the area of the triangle is $\frac{2a\cdot a^2}{2}=a^3$ . Solving the equation $a^3=64$ for $a$ , we get $a=4$ , so $BC=2a=8$ . So the answer is $\boxed{\textbf{(C)} 8}$ .

2016 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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All AMC 10 Problems and Solutions

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2016 AMC 10B Problems/Problem 9

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