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2016 AMC 12B Problem 6

Problem 6 of 25EasierGeometry

All three vertices of △ABC\triangle ABC lie on the parabola defined by y=x2,y=x^2, with AA at the origin and BC‾\overline{BC} parallel to the xx-axis. The area of the triangle is 64.64. What is the length of BC?BC?

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Solution

Let the vertex in the first quadrant be (x,x2).(x,x^2). By symmetry the base is BC=2xBC=2x and the height is x2,x^2, so 12⋅2x⋅x2=x3=64.\tfrac12\cdot2x\cdot x^2=x^3=64. Thus x=4x=4 and BC=2x=8.BC=2x=8. Thus, the correct answer is C.

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Concepts: parabola · triangle area · coordinate geometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.