AMC12 2017 A

AMC12 2017 A · Q9

AMC12 2017 A · Q9. It mainly tests Linear inequalities, Transformations.

Let S be the set of points $(x, y)$ in the coordinate plane such that two of the three quantities 3, $x + 2$, and $y -4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of S?

设 S 为坐标平面中满足以下条件的点集 $(x, y)$：三个量 3、$x + 2$ 和 $y -4$ 中有两个相等，且第三个量不超过这个公共值。以下哪项是 S 的正确描述？

(A) a single point 单个点

(B) two intersecting lines 两条相交直线

(D) a triangle 一个三角形

(E) three rays with a common endpoint 三条有公共端点的射线

Answer

Correct choice: (E)

正确答案：（E）

Solution

Suppose that the two larger quantities are the first and the second. Then $3 = x + 2 \geq y -4$. This is equivalent to $x = 1$ and $y \leq 7$, and its graph is the downward-pointing ray with endpoint $(1, 7)$. Similarly, if the two larger quantities are the first and third, then $3 = y -4 \geq x + 2$. This is equivalent to $y = 7$ and $x \leq 1$, and its graph is the leftward-pointing ray with endpoint $(1, 7)$. Finally, if the two larger quantities are the second and third, then $x + 2 = y - 4 \geq 3$. This is equivalent to $y = x + 6$ and $x \geq 1$, and its graph is the upward-pointing ray with endpoint $(1, 7)$. Thus S consists of three rays with common endpoint $(1, 7)$.

假设两个较大的量是第一个和第二个。那么 $3 = x + 2 \geq y -4$。这等价于 $x = 1$ 和 $y \leq 7$，其图像是以 $(1, 7)$ 为端点的向下射线。类似地，如果两个较大的量是第一个和第三个，则 $3 = y -4 \geq x + 2$，等价于 $y = 7$ 和 $x \leq 1$，图像是以 $(1, 7)$ 为端点的向左射线。最后，如果两个较大的量是第二个和第三个，则 $x + 2 = y - 4 \geq 3$，等价于 $y = x + 6$ 和 $x \geq 1$，图像是以 $(1, 7)$ 为端点的向上射线。因此 S 由三条以公共端点 $(1, 7)$ 的射线组成。

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