GCSE CCEA Maths: Coordinate Geometry | 坐标几何 考点精讲

📚 GCSE CCEA Maths: Coordinate Geometry | 坐标几何 考点精讲

Coordinate geometry, also known as analytic geometry, is a key topic in GCSE CCEA Mathematics. It bridges algebra and geometry by using a coordinate plane to describe points, lines, and shapes. Mastering formulas for distance, midpoint, gradient, and the equation of a straight line is essential for success in both the calculator and non-calculator papers. This revision guide covers every important concept with clear explanations and worked examples.

坐标几何（解析几何）是GCSE CCEA数学中的核心考点，它利用坐标平面将代数与几何联系起来，描述点、线和图形。掌握距离、中点、斜率和直线方程的公式对于在计算器和非计算器试卷中取得好成绩至关重要。本复习指南将覆盖每个重要概念，并配合清晰的解释和例题。

1. Cartesian Coordinates and Plotting Points | 笛卡尔坐标与描点

The Cartesian coordinate system consists of two perpendicular number lines: the x‑axis (horizontal) and the y‑axis (vertical), intersecting at the origin (0,0). Each point is represented by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance. Points are plotted by moving along the x‑axis first, then parallel to the y‑axis.

笛卡尔坐标系由两条互相垂直的数轴组成：x轴（水平）和y轴（垂直），相交于原点(0,0)。每个点用一个有序对(x, y)表示，其中x是到原点的水平距离，y是垂直距离。描点时先沿x轴移动，再平行于y轴移动。

It is crucial to understand the four quadrants: Quadrant I (+,+), Quadrant II (−,+), Quadrant III (−,−), and Quadrant IV (+,−). Correctly reading coordinates and plotting points is the foundation for all coordinate geometry problems.

理解四个象限非常重要：第一象限(+,+)，第二象限(−,+)，第三象限(−,−)，第四象限(+,−)。正确读取坐标和描点是所有坐标几何问题的基础。

2. Distance Between Two Points | 两点间的距离

The distance d between points A(x₁, y₁) and B(x₂, y₂) is derived from Pythagoras’ theorem. It is given by the formula:

两点A(x₁, y₁)和B(x₂, y₂)之间的距离d由勾股定理导出，公式如下：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Always subtract the coordinates in the same order, square the differences, add them, and take the square root. The distance is always positive. This formula works for any two points on the coordinate plane, regardless of quadrant.

始终以相同顺序相减坐标，将差值平方后相加，再取平方根。距离始终为正。无论点在哪一个象限，该公式都适用。

Example: Find the distance between (2, −3) and (5, 1). d = √[(5−2)² + (1−(−3))²] = √[3² + 4²] = √[9+16] = √25 = 5.

示例：求点(2, −3)与(5, 1)之间的距离。d = √[(5−2)² + (1−(−3))²] = √[3² + 4²] = √[9+16] = √25 = 5。

3. 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:

连接(x₁, y₁)和(x₂, y₂)的线段的中点M通过求x坐标和y坐标的平均值得到：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

The midpoint formula is simply the arithmetic mean of the endpoints. It is often used to find the centre of a line segment or to solve problems involving bisectors.

中点公式就是端点的算术平均值。它常用于求线段的中点或解决涉及平分线的问题。

Example: Midpoint of (4, −2) and (6, 8) is ((4+6)/2, (−2+8)/2) = (5, 3).

示例：(4, −2)和(6, 8)的中点是((4+6)/2, (−2+8)/2) = (5, 3)。

4. Gradient of a Straight Line | 直线的斜率（梯度）

The gradient m measures the steepness and direction of a line. It is defined as the change in y divided by the change in x between two distinct points on the line:

斜率m衡量直线的陡峭程度和方向。它被定义为直线上两个不同点之间y的变化量与x的变化量之比：

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.

斜率为正表示直线从左到右上升；斜率为负表示下降。水平线的斜率为0，而垂直线的斜率无定义（无穷大）。

When given a graph, you can

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