📚 Coordinate Geometry: Key Revision Points | 坐标几何：考点精讲
Coordinate geometry, also known as Cartesian geometry, is a key topic in the Edexcel GCSE Mathematics syllabus. It connects algebra and geometry, allowing you to describe points, lines, and shapes on a graph using equations. Mastering coordinate geometry is essential for solving problems involving straight lines, gradients, distances, midpoints, and intersections. In this revision guide, we will cover all the crucial concepts and formulas you need to know, with step-by-step explanations and exam tips.
坐标几何，也称为笛卡尔几何，是 Edexcel GCSE 数学大纲中的一个重要主题。它将代数与几何联系起来，使你能够用方程描述图上的点、直线和图形。掌握坐标几何对于解决涉及直线、斜率、距离、中点和交点的问题至关重要。在本复习指南中，我们将涵盖你需要了解的所有关键概念和公式，并配有逐步的解释和考试技巧。
1. Cartesian Coordinates and Points | 笛卡尔坐标与点
The Cartesian plane consists of a horizontal x-axis and a vertical y-axis, intersecting at the origin O(0,0). Every point is written as an ordered pair (x, y), where the x-coordinate comes first, then the y-coordinate.
笛卡尔平面由水平的 x 轴和垂直的 y 轴构成，两轴相交于原点 O(0,0)。每个点都用有序数对 (x, y) 表示，先写 x 坐标，再写 y 坐标。
Positive x values lie to the right of the origin, negative to the left. Positive y values lie above the origin, negative below. The axes divide the plane into four quadrants, labelled I to IV anticlockwise starting from the upper right.
正的 x 值位于原点右侧，负的位于左侧。正的 y 值位于原点上侧，负的位于下侧。坐标轴将平面分为四个象限，从右上角开始按逆时针方向依次标记为第 I 至第 IV 象限。
When plotting a point such as (3, -2), move 3 units right and 2 units down. Always read coordinates along the x-direction first.
绘制点如 (3, -2) 时，先向右移动 3 个单位，再向下移动 2 个单位。读取坐标时始终先沿 x 方向。
2. Midpoint of a Line Segment | 线段的中点
The midpoint of a line segment joining two points (x₁, y₁) and (x₂, y₂) is found by averaging the x-coordinates and averaging the y-coordinates. This gives the point exactly halfway between them.
连接两点 (x₁, y₁) 和 (x₂, y₂) 的线段中点，通过分别对 x 坐标和 y 坐标取平均值来求得。所得点正好位于两点正中间。
Midpoint M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )
For example, the midpoint of (2, 5) and (8, 11) is ((2+8)/2, (5+11)/2) = (5, 8). This works even with negative coordinates; between (-3, 4) and (1, -2) the midpoint is (-1, 1).
例如，(2, 5) 和 (8, 11) 的中点是 ((2+8)/2, (5+11)/2) = (5, 8)。即使有负坐标，该方法依然成立：(-3, 4) 和 (1, -2) 的中点是 (-1, 1)。
3. Distance Between Two Points | 两点间的距离
The distance d between two points (x₁, y₁) and (x₂, y₂) follows directly from Pythagoras’ theorem. The horizontal and vertical differences form a right-angled triangle, so d = √[(x₂ – x₁)² + (y₂ – y₁)²].
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离 d 直接由勾股定理导出。水平差与垂直差构成直角三角形，因此 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Calculate the differences first, square them, sum them, and then take the square root. For instance, the distance from (1, 2) to (4, 6) is √[(4−1)² + (6−2)²] = √[3² + 4²] = √25 = 5. Always leave your answer in surd form unless asked otherwise.
先计算差值，再平方、求和，最后取平方根。例如，从 (1, 2) 到 (4, 6) 的距离为 √[(4−1)² + (6−2)²] = √[3² + 4²] = √25 = 5。除非另有要求，答案应保留根号形式。
4. Gradient of a Straight Line | 直线的斜率
The gradient (or slope) m of a line measures its steepness. It is defined as the change in y divided by the change in x between any two distinct points on the line.
直线的斜率（坡度）m 衡量其倾斜程度，定义为线上任意两个不同点之间 y 的变化量除以 x 的变化量。
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 zero gradient (m = 0), while a vertical line has an undefined gradient because division by zero occurs.
正斜率表示直线从左向右上升；负斜率表示下降。水平线的斜率为零 (m = 0)，而垂直线的斜率未定义，因为会出现除以零的情况。
Find the gradient of the line through (2, 7) and (6, 3): m = (3 − 7)/(6 − 2) = −4/4 = −1. The line slopes downward.
求经过 (2, 7) 和 (6, 3) 的直线斜率：m = (3 − 7)/(6 − 2) = −4/4 = −1。该直线向下倾斜。
5. Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c
The most common form for a straight line is y = mx + c. Here, m represents the gradient and c is the y-intercept — the point where the line crosses the y-axis (0, c).
直线最常见的形式是 y = mx + c。其中 m 表示斜率，c 是 y 轴截距 —— 即直线与 y 轴的交点 (0, c)。
If a line has gradient 2 and crosses the y-axis at −3, its equation is y = 2x − 3. To identify m and c from an equation, simply compare it to y = mx + c: in y = 5 − 0.5x, rewrite as y = −0.5x + 5, so m = −0.5, c = 5.
若直线斜率为 2 且与 y 轴交于 −3，则方程为 y = 2x − 3。从方程中识别 m 和 c 只需与 y = mx + c 形式对照：如 y = 5 − 0.5x，改写为 y = −0.5x + 5，因此 m = −0.5, c = 5。
Always ensure the equation is in the form ‘y = …’ when identifying m and c. This form also makes it easy to sketch the line using the intercept and gradient.
识别 m 和 c 时务必确保方程为 ‘y = …’ 的形式。这种形式也便于利用截距和斜率画出直线草图。
6. Using the General Form ax + by + c = 0 | 一般式 ax + by + c = 0 的使用
Straight lines can also be written as ax + by + c = 0, where a, b, and c are integers. This is called the general form. To find the gradient and y-intercept, rearrange into y = mx + c.
直线方程也可写成 ax + by + c = 0，其中 a, b 和 c 为整数，这称为一般式。要求出斜率和 y 轴截距，可将其化为 y = mx + c 的形式。
y = −(a/b)x − (c/b), so gradient m = −a/b, y-intercept = −c/b
For 3x + 2y − 6 = 0, bring terms: 2y = −3x + 6, then y = −1.5x + 3. The gradient is −3/2 and the line crosses the y-axis at 3. The x-intercept (y=0) can be found similarly.
以 3x + 2y − 6 = 0 为例，移项得 2y = −3x + 6，然后 y = −1.5x + 3。斜率为 −3/2，与 y 轴交于 3。x 轴截距（令 y=0）也可类似求出。
In the Edexcel exam you may be asked to convert between forms or compare gradients using the general equation. Knowing m = −a/b saves time.
在 Edexcel 考试中，可能会要求你转换方程形式或利用一般式比较斜率。记住 m = −a/b 可节省时间。
7. Finding the Equation from Two Points or a Point and Gradient | 由两点或点斜式求直线方程
When you know a point (x₁, y₁) on the line and the gradient m, you can use the point-slope formula: y − y₁ = m(x − x₁). Then rearrange into y = mx + c or ax + by + c = 0.
当已知直线上一点 (x₁, y₁) 和斜率 m 时，可使用点斜式：y − y₁ = m(x − x₁)，然后整理成 y = mx + c 或 ax + by + c = 0。
y − y₁ = m(x − x₁)
If two points are given, first compute the gradient m = (y₂ − y₁)/(x₂ − x₁), then substitute one point into the formula. For example, line through (1, 4) and (3, 10): m = (10−4)/(3−1) = 3. Using (1,4): y − 4 = 3(x − 1), which simplifies to y = 3x + 1.
若给定两点，先计算斜率 m = (y₂ − y₁)/(x₂ − x₁)，再代入其中一个点。例如，过 (1, 4) 和 (3, 10) 的直线：m = (10−4)/(3−1) = 3。使用 (1,4) 得 y − 4 = 3(x − 1)，化简为 y = 3x + 1。
Always check your final equation by plugging in both original points. This guarantees you haven’t made an arithmetic mistake.
务必用两个原始点代入最终方程进行检验，这样可以确保没有算术错误。
8. Parallel and Perpendicular Lines | 平行与垂直直线
Parallel lines have the same gradient, so if line L₁ has gradient m₁, any parallel line also has gradient m₁. Perpendicular lines meet at right angles; their gradients multiply to −1.
平行线具有相同的斜率，因此若直线 L₁ 的斜率为 m₁，则任何平行线的斜率也是 m₁。垂直直线相交成直角，它们的斜率乘积为 −1。
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1, or m₂ = −1/m₁
If a line has equation y = 3x + 5, the gradient is 3. A line parallel to it also has gradient 3. A line perpendicular to it has gradient −1/3. This is called the negative reciprocal.
若直线方程为 y = 3x + 5，斜率为 3。与之平行的直线斜率也为 3。与之垂直的直线斜率为 −1/3，即负倒数。
Be careful: when a line is vertical (gradient undefined), its perpendicular is horizontal (gradient 0). You won’t usually be asked to use the reciprocal rule there.
注意：若直线为垂直线（斜率未定义），其垂直线为水平线（斜率为 0），此时通常不涉及倒数规则。
9. Drawing Straight Line Graphs | 绘制直线图
To draw a straight line, you only need two points — a third point can serve as a check. The easiest approach is to use the y-intercept and then use the gradient to find another point.
绘制一条直线只需两个点即可，第三个点可用于检验。最简单的方法是先标出 y 轴截距，然后利用斜率找到另一个点。
For y = 2x − 3, the y-intercept is (0, −3). The gradient 2 means ‘rise over run’ = 2/1, so from (0, −3) move 1 unit right and 2 units up to reach (1, −1). Plot both and draw a line through them.
对于 y = 2x − 3，y 轴截距为 (0, −3)。斜率 2 表示 ‘纵升/横距’ = 2/1，因此从 (0, −3) 向右移动 1 个单位、向上移动 2 个单位到达 (1, −1)。标出两点并过它们画线。
You can also find intercepts by setting y=0 for the x-intercept and x=0 for the y-intercept. Connecting (0, c) and (−c/m, 0) is a reliable method when m ≠ 0.
你也可以通过令 y=0 求 x 截距、令 x=0 求 y 截距来画图。当 m ≠ 0 时，连接 (0, c) 和 (−c/m, 0) 是一种可靠的方法。
10. Intersections of Lines and Axes (Intercepts) | 直线与坐标轴的交点(截距)
The intercepts are the points where the line crosses the axes. To find the y-intercept, set x = 0 and solve for y. To find the x-intercept, set y = 0 and solve for x.
截距是直线与坐标轴相交的点。求 y 轴截距时，令 x = 0 解出 y。求 x 轴截距时，令 y = 0 解出 x。Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
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