请按照如下方式生成视频,用于给初中生讲解题目:
1.读题干和第1问
2.以图上标准解析为参考,来讲解第1问
3.读第2问
4.以图上标准解析为参考,来讲解第2问
5.读第3问
6.以图上标准解析为参考,来讲解第3问
7.总结题目的考查知识点、难度、易错点---**Question Stem:**
如图1, 平面上两条直线 $l_1$、$l_2$相交于点$O$, 对于平面上任意一点$M$, 若点$M$到直线$l_1$的距离为$p$, 到直线$l_2$的距离为$q$, 则称有序数对$(p, q)$为点$M$的“距离坐标”, 例如, 图1中点$O$的“距离坐标”为$(0,0)$, 点$N$的“距离坐标”为$(3.6, 4.2)$。
As shown in Figure 1, two lines $l_1$ and $l_2$ on the plane intersect at point $O$. For any point $M$ on the plane, if the distance from point $M$ to line $l_1$ is $p$ and the distance to line $l_2$ is $q$, then the ordered pair $(p, q)$ is called the "distance coordinate" of point $M$. For example, in Figure 1, the "distance coordinate" of point $O$ is $(0,0)$, and the "distance coordinate" of point $N$ is $(3.6, 4.2)$.
(1) 如图2, 点$A$的“距离坐标”为\_\_\_\_\_\_\_\_, 点$B$的“距离坐标”为\_\_\_\_\_\_\_\_;
As shown in Figure 2, the "distance coordinate" of point A is \_\_\_\_\_\_\_\_, and the "distance coordinate" of point B is \_\_\_\_\_\_\_\_;
(2) 如图3, 点$C$、$D$分别在直线$l_1$、$l_2$上, 则$C$、$D$两个点中, “距离坐标”为$(3,0)$的点是\_\_\_\_\_\_\_\_;
As shown in Figure 3, points C and D are respectively on lines $l_1$ and $l_2$. Among points C and D, the point whose "distance coordinate" is $(3,0)$ is \_\_\_\_\_\_\_\_;
(3) 平面上“距离坐标”为$(0,5)$的点有\_\_\_\_\_\_\_\_个, “距离坐标”为$(5,5)$的点有\_\_\_\_\_\_\_\_个。
There are \_\_\_\_\_\_\_\_ points on the plane with "distance coordinate" $(0,5)$, and there are \_\_\_\_\_\_\_\_ points with "distance coordinate" $(5,5)$.
**Figure Descriptions:**
* **图1 (Figure 1):**
* Type: Geometric diagram with lines and points.
* Main Elements: Two straight lines labeled $l_1$ and $l_2$ intersecting at point $O$. Point $M$ is shown in the upper region. Dashed lines indicate perpendicular distance from $M$ to $l_1$ (labeled $p$) and from $M$ to $l_2$ (labeled $q$). Point $N$ is shown in the lower region. Dashed lines indicate perpendicular distance from $N$ to $l_1$ (labeled 3.6) and from $N$ to $l_2$ (labeled 4.2). Labels: $l_1$, $l_2$, $O$, $M$, $N$, $p$, $q$, 3.6, 4.2, 图1.
* **图2 (Figure 2):**
* Type: Geometric diagram with lines and points.
* Main Elements: Two straight lines labeled $l_1$ and $l_2$ intersecting at point $O$. Point $A$ is shown in the upper-left region. Dashed lines indicate perpendicular distance from $A$ to $l_1$ (length 1.6) and from $A$ to $l_2$ (length 2.5). Point $B$ is shown in the lower-right region. Dashed lines indicate perpendicular distance from $B$ to $l_1$ (length 2.2) and from $B$ to $l_2$ (length 1.5). Right angle symbols are shown at the feet of the perpendiculars. Labels: $l_1$, $l_2$, $O$, $A$, $B$, 1.6, 2.5, 2.2, 1.5, 图2.
* **图3 (Figure 3):**
* Type: Geometric diagram with lines and points.
* Main Elements: Two straight lines labeled $l_1$ and $l_2$ intersecting at point $O$. Point $C$ is shown on line $l_1$. Point $D$ is shown on line $l_2$. A dashed line indicates perpendicular distance from $C$ to $l_2$ (length 3). A dashed line indicates perpendicular distance from $D$ to $l_1$ (length 3). Right angle symbols are shown at the feet of the perpendiculars. Labels: $l_1$, $l_2$, $O$, $C$, $D$, 3, 3, 图3.
* **Supplementary Figure (for part 3 solution):**
* Type: Geometric diagram showing two intersecting lines and lines parallel to them.
* Main Elements: Two straight lines labeled $l_1$ and $l_2$ intersecting. Two dashed lines are drawn parallel to $l_1$, one on each side. The distance between these two dashed lines is indicated as 5. Two dashed lines are drawn parallel to $l_2$, one on each side. The distance between these two dashed lines is indicated as 5. These four dashed lines form a parallelogram. The four vertices of this parallelogram are labeled $A$, $B$, $C$, and $D$. Labels: $l_1$, $l_2$, $A$, $B$, $C$, $D$, 5, 5. Dashed lines are used for the parallel lines.
**Standard Solution Explanation:**
解:
(1) 图形点A到直线$l_1$、$l_2$的距离分别是1.6和2.5, 点B到直线$l_1$、$l_2$的距离分别是2.2和1.5.
答案为: $(1.6, 2.5)$, $(2.2, 1.5)$
(2) “距离坐标”的两个有序数对的第一个数和第二个数分别表示点到直线$l_1$、$l_2$的距离, 所以, 到直线$l_1$的距离是3, 到直线$l_2$的距离是0.
结合已知图形, 可知满足条件的点为D.
答案为: D,
(3) $(0, 5)$代表点到直线$l_1$的距离是0和5, 则所求点在直线$l_1$上, 且到$l_2$的距离为5, 这样的点在直线$l_2$两侧各有一个, 共有2个.
如图, 直线$AB||CD||l_2$且相邻两条直线距离为5, 直线$AD||BC||l_1$, 且相邻两条直线距离为5, $A、B、C、D$四点的“距离坐标”为$(5, 5)$, 共有4个点.
答案为: 2, 4.
**Solution Translation (for clarity, not part of required output):**
Solution:
(1) In the figure, the distances from point A to lines $l_1$ and $l_2$ are 1.6 and 2.5 respectively. The distances from point B to lines $l_1$ and $l_2$ are 2.2 and 1.5 respectively.
Answer: $(1.6, 2.5)$, $(2.2, 1.5)$
(2) The first and second numbers in the "distance coordinate" ordered pair represent the distance from the point to line $l_1$ and $l_2$ respectively. Therefore, the distance to line $l_1$ is 3 and the distance to line $l_2$ is 0.
Combining with the given figure, the point that satisfies the condition is D.
Answer: D,
(3) $(0, 5)$ represents a point whose distance to line $l_1$ is 0 and distance to line $l_2$ is 5. So the required point is on line $l_1$ and is 5 units away from $l_2$. There is one such point on each side of line $l_2$ along $l_1$, totaling 2 points.
As shown in the figure, lines $AB||CD||l_2$ and the distance between adjacent parallel lines is 5. Lines $AD||BC||l_1$, and the distance between adjacent parallel lines is 5. The "distance coordinates" of the four points $A, B, C, D$ are $(5, 5)$, totaling 4 points.
Answer: 2, 4.