📚 GCSE OCR Maths: Coordinate Geometry Key Points | GCSE OCR 数学：坐标几何考点精讲
Coordinate geometry, also known as analytic geometry, is a vital topic in GCSE OCR Mathematics. It combines algebra and geometry to solve problems involving points, lines, and shapes on a coordinate plane. Understanding this topic thoroughly will help you tackle a wide range of exam questions confidently.
坐标几何，又称解析几何，是 GCSE OCR 数学中的一个重要专题。它将代数和几何结合起来，解决关于坐标平面上的点、直线和图形的问题。透彻掌握这一部分内容，将帮助你自信地应对各类考试题目。
1. The Cartesian Coordinate System | 笛卡尔坐标系
A Cartesian coordinate system uses two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), to locate points in a plane. The point where they intersect is called the origin, labelled O, with coordinates (0, 0).
笛卡尔坐标系使用两条垂直的数轴：x 轴（水平）和 y 轴（垂直），用来在平面中定位点。两轴的交点称为原点，记作 O，坐标为 (0, 0)。
Any point is represented by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance. Positive x values move right, negative left; positive y up, negative down.
任意点用一个有序实数对 (x, y) 表示，其中 x 是距原点的水平距离，y 是垂直距离。x 为正向右，负向左；y 为正向上，负向下。
2. Plotting Points and Reading Coordinates | 描点与读取坐标
To plot a point, start at the origin. Move along the x-axis according to the x-coordinate, then move vertically according to the y-coordinate. For example, point A(3, −2) means move 3 units right and 2 units down.
要描点，从原点出发。根据 x 坐标沿 x 轴移动，然后根据 y 坐标垂直移动。例如，点 A(3, −2) 表示向右 3 个单位，向下 2 个单位。
When reading coordinates from a graph, always check the scale on both axes and record the x-value first, then the y-value.
从图中读取坐标时，一定要检查两轴的刻度，并先记录 x 值，再记录 y 值。
3. Gradient: Slope of a Line | 斜率：直线的倾斜度
The gradient (or slope) measures how steep a line is. For two points (x₁, y₁) and (x₂, y₂), the gradient m is calculated by the ratio of vertical change to horizontal change:
m = (y₂ − y₁) / (x₂ − x₁)
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率 m 等于纵坐标差与横坐标差之比：m = (y₂ − y₁) / (x₂ − x₁)。
A positive gradient means the line rises from left to right; a negative gradient means it falls. A horizontal line has gradient 0, while a vertical line has an undefined (infinite) gradient.
斜率为正，直线从左向右上升；斜率为负，直线从左向右下降。水平线的斜率为 0，垂直线的斜率无定义（无穷大）。
4. The Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c
m is the gradient and c is the y-intercept (the point where the line crosses the y-axis).
直线的一般方程为 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴的交点）。
For example, the line y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3).
例如，直线 y = 2x + 3 的斜率为 2，与 y 轴交于点 (0, 3)。
5. Finding Intercepts | 求截距
etting x = 0 in the equation. The x-intercept is found by setting y = 0 and solving for x.
y 轴截距通过令 x = 0 代入方程求得；x 轴截距则令 y = 0 并解出 x 求得。
−½ x + 4, the y-intercept is (0, 4); the x-intercept: 0 = −½ x + 4 → x = 8, giving (8, 0).
对于直线 y = −½ x + 4，y 截距为 (0, 4)；求 x 截距：0 = −½ x + 4，解得 x = 8，因此交点为 (8, 0)。
6. Distance Between Two Points | 两点间的距离
The distance d between points (x₁, y₁) and (x₂, y₂) is given by:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
两点 (x₁, y₁) 和 (x₂, y₂) 间的距离 d 等于根号下 (x₂ − x₁)² 与 (y₂ − y₁)² 之和。这个公式源自勾股定理。
Always subtract the coordinates in the same order, square the differences, add them, and then take the square root.
始终用相同顺序相减，平方后相加，再开平方根。
7. Midpoint of a Line Segment | 线段的中点
The midpoint M of a segment joining (x₁, y₁) and (x₂, y₂) is found by averaging the x-coordinates and the y-coordinates:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点 M 为两点坐标的平均值：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。
This is extremely useful for finding the centre of a line segment or checking if a point lies halfway between two others.
这在求线段中心或检查某点是否位于另外两点中点时非常实用。
8. Parallel Lines and Their Gradients | 平行线及其斜率
nt. If line 1 has gradient m₁ and line 2 has gradient m₂, then m₁ = m₂ for parallel lines. Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
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