AMC10 2017 A

AMC10 2017 A · Q12

AMC10 2017 A · Q12. It mainly tests Linear inequalities, Graphs (coordinate plane).

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，且第三个量不大于这个公共值。以下哪项是对 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 \(3 = x + 2 \ge y - 4\). This is equivalent to \(x = 1\) and \(y \le 7\), and its graph is the downward-pointing ray with endpoint (1, 7). Similarly, if the two larger quantities are the first and third, then \(3 = y - 4 \ge x + 2\). This is equivalent to \(y = 7\) and \(x \le 1\), and its graph is the leftward-pointing ray with endpoint (1, 7). Finally, if the two larger quantities are the second and third, then \(x+2 = y-4 \ge 3\). This is equivalent to \(y = x + 6\) and \(x \ge 1\), and its graph is the ray with endpoint (1, 7) that points upward and to the right. Thus the graph 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 \(3 = x + 2 \ge y - 4\)。这等价于 \(x = 1\) 且 \(y \le 7\)，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 \(3 = y - 4 \ge x + 2\)。这等价于 \(y = 7\) 且 \(x \le 1\)，其图像是以 (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 \(x+2 = y-4 \ge 3\)。这等价于 \(y = x + 6\) 且 \(x \ge 1\)，其图像是以 (1, 7) 为端点、向上向右的射线。因此图像是由以公共端点 (1, 7) 的三条射线组成。

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