AMC12 2016 B

AMC12 2016 B · Q6

AMC12 2016 B · Q6. It mainly tests Area & perimeter, Coordinate geometry.

All three vertices of $\triangle ABC$ 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 $BC$?

$\triangle ABC$ 的三个顶点都在抛物线 $y=x^2$ 上，点 $A$ 在原点，且 $BC$ 平行于 $x$ 轴。该三角形的面积为 $64$。求线段 $BC$ 的长度。

(A) 4 4

(B) 6 6

(D) 10 10

(E) 16 16

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let the vertex of the triangle that lies in the first quadrant be $(x,x^2)$. Then the base of the triangle is $2x$ and the height is $x^2$, so $\frac{1}{2}\cdot 2x\cdot x^2=64$. Thus $x^3=64$, $x=4$, and $BC=2x=8$.

答案（C）：设位于第一象限的三角形顶点为 $(x,x^2)$。则三角形的底为 $2x$，高为 $x^2$，所以 $\frac{1}{2}\cdot 2x\cdot x^2=64$。因此 $x^3=64$，$x=4$，并且 $BC=2x=8$。

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