# 2016 AMC 10B Problems/Problem 9

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## 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 $[asy]import graph;size(7cm,IgnoreAspect); real f(real x) {return x*x;} draw((0,0)--(4,16)--(-4,16)--cycle,blue); draw(graph(f,-5,5,operator ..),gray); xaxis("x");yaxis("y",-1); label("y=x^2",(4.5,20.25),E); draw((4.2,0)--(4.2,16),Arrows); label("r^2",(4.2,0)--(4.2,16),E); draw((0,17)--(4,17),Arrows); label("r",(0,17)--(4,17),N); [/asy]$ The area of the triangle is $\frac{(2r)(r^2)}{2} = r^3$, so $r^3=64\implies r=4$, giving a total distance across the top of $8$, which is answer $\textbf{(C)}$.

## Video Solution

~savannahsolver

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