Question

设 $S$ 是坐标平面中的点 $(x,y)$ 的集合，满足三个量 $3,x+2$ 、 $y-4$ 中有两个相等，而第三个量不大于这个公共值。下面哪一项正确描述了 $S$ ？

Let $S$ be a 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 $S?$

一个点

a single point

两条相交直线

two intersecting lines

三条直线，且两两交于三个不同点

three lines whose pairwise intersections are three distinct points

一个三角形

a triangle

有共同端点的三条射线

three rays with a common endpoint

答案：E

知识点：坐标几何分类讨论

难度评级：1540

解答：

分情况讨论哪两个量相等。若 $3 = x + 2$ ，则 $x = 1$ 。这还给出这是一条以 $(1, 7)$ 为端点、沿负 $y$ 方向延伸的射线。 $y - 4 \leq 3$ $y \leq 7.$

类似地，若 $3 = y - 4$ ，则 $y = 7$ ，并且这也是一条以 $(1, 7)$ 为端点、但沿负 $x$ 方向延伸的射线。 $x + 2 \leq 3$ $x \leq 1.$

最后，若 $x + 2 = y - 4$ ，则直线为 $y = x + 6$ 。此外有且注意，由等式可知满足其中一个条件时，另一个也必然满足。 $3 \leq x + 2$ $x \geq 1$ $3 \leq y - 4$ $y \geq 7.$

当 $y = 7$ 时， $x = 1$ 。其余点在线上，满足 $y \gt 7$ 且 $x \gt 1$ 。

这描述了另一条从 $(1, 7)$ 出发、沿第三个方向延伸的射线。

三种情形都得到从 $(1, 7)$ 出发、方向各不相同的射线。

所以正确答案是 E。

Let us case on which of the values are equal. If $3 = x + 2,$ then $x = 1.$ This also tells us that $y - 4 \leq 3$ $y \leq 7.$ This describes a ray starting at $(1, 7)$ and extending in the negative $y$ direction.

Similarly, if $3 = y - 4,$ then $y = 7$ and $x + 2 \leq 3$ $x \leq 1.$ This also describes a ray starting at $(1, 7)$ but instead extending in the negative $x$ direction.

Finally, if $x + 2 = y - 4,$ then we have the line $y = x + 6.$ Furthermore, we have that $3 \leq x + 2$ $x \geq 1$ and $3 \leq y - 4$ $y \geq 7.$ Note that if one of these conditions is met, the other is also necessarily true due to the equation of the line.

If $y = 7,$ then $x = 1.$ The other points are along the line, where $y \gt 7$ and $x \gt 1.$

This describes another ray that starts at $(1, 7)$ and goes off in some third direction.

All three cases result in rays originating from $(1, 7)$ that all go in different directions.

Thus, E is the correct answer.

2017 AMC 10A 第 12 题

答案：E

难度评级：1540

解答：

其他年份的第 12 题