📚 Coordinate Geometry for CIE IGCSE Mathematics | CIE IGCSE 数学：坐标几何考点精讲
Coordinate geometry, also known as analytic geometry, links algebra with geometry through the use of a coordinate system. In CIE IGCSE Mathematics, this topic focuses on the Cartesian plane, allowing you to describe points, lines, and shapes using equations. Mastering the key formulas for distance, midpoint, gradient, and the equation of a straight line is essential for success in both the Core and Extended papers. This revision guide breaks down every crucial concept with clear explanations and worked examples.
坐标几何，又称解析几何，通过坐标系将代数与几何联系起来。在 CIE IGCSE 数学中，该主题侧重于直角坐标平面，使你能够用方程描述点、线和形状。掌握距离、中点、斜率和直线方程的关键公式对核心和扩展试卷的成功至关重要。本复习指南通过清晰的解释与例题，逐一剖析每个重要概念。
1. The Cartesian Plane and Coordinates | 直角坐标系与坐标
The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They intersect at the origin, E(0,0). Any point is written as an ordered pair (x, y), where x represents the horizontal displacement and y the vertical displacement from the origin. The plane is divided into four quadrants, labelled I, II, III, and IV in an anticlockwise direction starting from the top right.
直角坐标平面由两条垂直的数轴构成：x 轴（水平）和 y 轴（垂直）。它们相交于原点 E(0,0)。任意点用有序数对 (x, y) 表示，其中 x 代表相对于原点的水平位移，y 代表垂直位移。平面被分为四个象限，从右上角开始按逆时针方向标记为 I、II、III 和 IV。
For example, point A(3, 4) lies in quadrant I, while point B(−2, 5) lies in quadrant II. Knowing the signs of coordinates helps you quickly locate a point without plotting.
例如，点 A(3, 4) 位于第一象限，而点 B(−2, 5) 位于第二象限。知道坐标的符号有助于你无需描点就能快速定位。
2. Distance Between Two Points | 两点间距离
The distance d between two points (x₁, y₁) and (x₂, y₂) is found using Pythagoras' theorem. The horizontal difference is (x₂ − x₁) and the vertical difference is (y₂ − y₁). The formula is:
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离 d 可利用勾股定理求得。水平差为 (x₂ − x₁)，垂直差为 (y₂ − y₁)。公式如下：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
istance between P(1, 2) and Q(4, 6). Substitute: d = √[(4 − 1)² + (6 − 2)²] = √[3² + 4²] = √9 + 16 = √25 ys check that you subtract coordinates in the same order. The distance is always positive.
例题：求 P(1, 2) 与 Q(4, 6) 之间的距离。代入公式：d = √[(4 − 1)² + (6 − 2)²] = √[3² + 4²] = √9 + 16 = √25 = 5 单位。务必按相同顺序相减坐标。距离始终为正值。
3. Midpoint of a Line Segment | 线段的中点
The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) is simply the average of the x-coordinates and the average of the y-coordinates. The formula is:
连接 (x₁, y₁) 与 (x₂, y₂) 的线段中点 M 就是 x 坐标的平均值与 y 坐标的平均值。公式如下：
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
(2, 2). Midpoints are especially useful in geometry proofs and when working with symmetry.
例题：求 A(−3, 5) 与 B(7, −1) 的中点。x 坐标为 (−3 + 7)/2 = 4/2 = 2。y 坐标为 (5 + (−1))/2 = 4/2 = 2。因此 M = (2, 2)。中点在几何证明和处理对称性问题时特别有用。
4. Gradient of a Straight Line | 直线的斜率
The gradient (or slope) m of a line measures its steepness. For two points (x₁, y₁) and (x₂, y₂), the gradient is the ratio of the vertical change to the horizontal change:
直线的斜率 m 衡量其倾斜程度。对于两点 (x₁, y₁) 与 (x₂, y₂)，斜率等于垂直变化与水平变化之比：
m = (y₂ − y₁) / (x₂ − x₁)
ne has an undefined gradient (division by zero). For example, the line through (1, 2) and (3, 8) has m = (8 − 2)/(3 − 1) = 6/2 = 3, a steep upward slope.
斜率为正表示直线从左向右上升；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（分母为零）。例如，经过 (1, 2) 和 (3, 8) 的直线斜率为 m = (8 − 2)/(3 − 1) = 6/2 = 3，表示一条较陡的上升线。
5. Parallel and Perpendicular Lines | 平行线与垂直线
lel because both have gradient 2. Two lines are perpendicular if the product of their gradients is −1, or one gradient is the negative reciprocal of the other:
两条直线平行当且仅当它们的斜率相等：m₁ = m₂。例如，y = 2x + 3 与 y = 2x − 5 平行，因为两者的斜率均为 2。两条直线垂直，如果它们的斜率乘积为 −1，或者说一个斜率为另一个斜率的负倒数：
m₁ × m₂ = −1
If a line has gradient 3, a line perpendicular to it has gradient −1/3. Special care is needed: a vertical line (undefined gradient) is perpendicular to a horizontal line (gradient 0), but the product formula does not apply directly. Always check for these cases separately.
若一条直线的斜率为 3，则与之垂直的直线斜率为 −1/3。需特别注意：垂直线（斜率无定义）垂直于水平线（斜率为 0），此时乘积公式并不直接适用。应单独检验这类情况。
6. Equation of a Straight Line | 直线方程
t and c is the y-intercept (the point where the line crosses the y-axis). Another vital form is the point-slope form, which is extremely helpful when you know the gradient and one point on the line:
直线方程可有多种有用的表达形式。最常见的是斜截式：y = mx + c，其中 m 为斜率，c 为 y 轴截距（直线与 y 轴交点的纵坐标）。另一种重要形式是点斜式，当你已知斜率及直线上一点时，该形式极为有用：
y − y₁ = m(x − x₁)
2x + y − 5 = 0. Being able to switch between forms is a key skill in coordinate geometry.
一般式为 ax + by + c = 0，其中 a、b、c 为整数且 a 通常为正数。例如，y = −2x + 5 可整理为 2x + y − 5 = 0。能够灵活转换不同形式是坐标几何中的关键技能。
7. Finding the Equation of a Line | 求直线方程
using the gradient formula, then substitute one point into y − y₁ = m(x − x₁) to obtain the equation. Always express the final answer in the required form, often y = mx + c or ax + by + c = 0.
要求直线方程，通常需要给定两个点或一个点与斜率。若给定两点，先用斜率公式计算 m，然后将其中一个点代入 y − y₁ = m(x − x₁) 得到方程。最后答案应根据要求表达为所需形式，通常为 y = mx + c 或 ax + by + c = 0。
(x − 2) → y − 3 = 2x − 4 → y = 2x − 1. Check with B: when x=6, y = 2(6) − 1 = 11. Correct.
例题：求经过 A(2, 3) 和 B(6, 11) 的直线方程。斜率 m = (11 − 3)/(6 − 2) = 8/4 = 2。代入点 A(2, 3)：y − 3 = 2(x − 2) → y − 3 = 2x − 4 → y = 2x − 1。用点 B 验证：当 x=6 时，y = 2(6) − 1 = 11，正确。
8. Intersection of Two Lines | 两条直线的交点
ou can use substitution or elimination. For instance, to find where y = 2x + 1 and y = −x + 7 meet, set 2x + 1 = −x + 7 → 3x = 6 → x = 2. Then y = 2(2) + 1 = 5. The intersection is (2, 5).
两条直线的交点可通过联立求解它们的方程得到，由此求出满足两个方程的 (x, y) 值。可使用代入法或消元法。例如，求 y = 2x + 1 与 y = −x + 7 的交点，令 2x + 1 = −x + 7 → 3x = 6 → x = 2。然后 y = 2(2) + 1 = 5。交点为 (2, 5)。
If the lines are parallel, there is no intersection (or they are coincident). If the equations are given in general form, you can also solve using algebraic manipulation. Finding intersections is a common requirement in area and linear programming problems.
若两直线平行，则没有交点（或重合）。若方程以一般式给出，也可通过代数方法求解。求交点是面积问题和线性规划问题中的常见要求。
9. Coordinate Geometry in Proofs | 坐标几何证明
points A, B, C are collinear (lie on the same straight line), prove that gradient AB = gradient BC. To prove a triangle is right-angled, show that two sides are perpendicular by checking if the product of their gradients equals −1. Midpoints and distance can verify properties of special quadrilaterals.
坐标几何可用于证明几何性质，如共线性、共点性以及四边形的种类。要证明三点 A、B、C 共线（位于同一直线上），可证明梯度 AB = 梯度 BC。要证明一个三角形是直角三角形，可通过检验两条边的斜率乘积是否为 −1 来说明两边垂直。中点和距离可用于验证特殊四边形的性质。
ce gradients are equal, the points lie on the same line. Always present your reasoning clearly.
例题：证明 A(1, 1)、B(3, 4)、C(5, 7) 共线。mAB = (4 − 1)/(3 − 1) = 3/2；mBC = (7 − 4)/(5 − 3) = 3/2。由于斜率相等，这些点位于同一直线上。务必清晰地呈现推理过程。
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