📚 IGCSE Maths Lesson 15-9: Coordinate Geometry and Straight-Line Graphs | IGCSE 数学第15-9课：坐标几何与直线图像
In IGCSE Mathematics, coordinate geometry connects algebra and geometry by describing points and lines on a plane using ordered pairs. This lesson (15-9) focuses on the essential skills of finding distances, midpoints, gradients, and equations of straight lines. These tools often appear in both Paper 2 and Paper 4, especially in problem-solving with intersections and perpendicular bisectors.
在 IGCSE 数学中，坐标几何通过有序数对在平面上描述点和线，将代数与几何联系起来。本课（15-9）重点讲授求距离、中点、斜率和直线方程的核心技能。这些工具常出现在试卷二和试卷四中，尤其是涉及交点与垂直平分线的问题。
1. The Cartesian Plane and Coordinates | 笛卡尔平面与坐标
The Cartesian plane consists of a horizontal x-axis and a vertical y-axis intersecting at the origin (0, 0). A point is written as (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
笛卡尔平面由水平 x 轴和垂直 y 轴组成，两条轴在原点 (0, 0) 处相交。一个点写作 (x, y)，其中 x 是离原点的水平距离，y 是垂直距离。
The plane is divided into four quadrants. In the first quadrant both coordinates are positive; in the second x is negative and y positive; in the third both are negative; in the fourth x is positive and y negative.
该平面分为四个象限。第一象限两个坐标都为正；第二象限 x 为负、y 为正；第三象限两个坐标都为负；第四象限 x 为正、y 为负。
2. Plotting Points and Reading Coordinates | 描点与读取坐标
To plot a point such as (3, −2), start at the origin, move 3 units right along the x-axis, then move 2 units down parallel to the y-axis. To read coordinates from a graph, project the point perpendicularly onto both axes.
要描出点 (3, −2)，从原点出发，沿 x 轴向右移动 3 个单位，然后平行于 y 轴向下移动 2 个单位。要从图中读取坐标，将点垂直投影到两条轴上。
This skill is the foundation for drawing straight lines and identifying intersections. Always use brackets and a comma in coordinate form.
这项技能是绘制直线和识别交点的基础。坐标形式始终使用括号和逗号。
3. Distance Between Two Points | 两点之间的距离
The distance d between points (x₁, y₁) and (x₂, y₂) comes from Pythagoras' theorem:
点 (x₁, y₁) 与 (x₂, y₂) 之间的距离 d 来自勾股定理：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
It does not matter which point is labelled as 1 or 2, because squaring removes the sign of each difference.
哪个点标为 1 或 2 并不重要，因为平方会消去每个差值的符号。
For example, the distance between (1, 2) and (4, 6) is √[(4−1)²+(6−2)²] = √9+16 = √25 = 5.
例如，(1, 2) 与 (4, 6) 之间的距离为 √[(4−1)²+(6−2)²] = √9+16 = √25 = 5。
4. Midpoint of a Line Segment | 线段的中点
The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) is found by averaging the x-values and y-values separately.
联结 (x₁, y₁) 与 (x₂, y₂) 的线段中点 M 通过分别对 x 值和 y 值取平均求得。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
This midpoint is the centre of the segment and is often used to simplify symmetry or to construct a perpendicular bisector.
这个中点是线段的中心，常用于简化对称性或构造垂直平分线。
For example, the midpoint of (2, 4) and (8, 10)
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