📚 Coordinate Geometry | 坐标几何
Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe and analyse geometric figures on a plane. In the IB Mathematics: Analysis and Approaches (AA) and Applications and Interpretation (AI) courses, the Core syllabus requires you to master the fundamentals of points, lines, distances, midpoints, gradients and intersections. These tools form the backbone of problem‑solving in both pure and applied contexts, from calculating the perpendicular bisector of a segment to finding areas of triangles using coordinates.
坐标几何，又称解析几何，是数学的一个分支，它利用代数方程在平面上描述和分析几何图形。在 IB 数学：分析与方法 (AA) 以及应用与解释 (AI) 课程中，核心大纲要求你掌握点、直线、距离、中点、斜率和交点等基础知识。这些工具构成了解决纯数学及应用数学问题的基础，从计算线段的垂直平分线，到利用坐标求三角形面积，都离不开坐标几何。
1. The Cartesian Plane | 笛卡尔平面
The Cartesian plane consists of two perpendicular number lines, the x‑axis (horizontal) and the y‑axis (vertical), intersecting at the origin O(0, 0). Any point P is identified by an ordered pair (x, y), where x is the horizontal displacement and y the vertical displacement from the origin. The plane is divided into four quadrants, numbered counter‑clockwise from the top right: Quadrant I (x > 0, y > 0), Quadrant II (x < 0, y > 0), Quadrant III (x < 0, y < 0) and Quadrant IV (x > 0, y < 0). Understanding the sign of coordinates in each quadrant is essential for sketching graphs and solving geometric problems.
笛卡尔平面由两条互相垂直的数轴组成：水平的 x 轴和竖直的 y 轴，它们相交于原点 O(0, 0)。任意一点 P 用一个有序对 (x, y) 来表示，其中 x 表示从原点出发的水平位移，y 表示竖直位移。整个平面被划分为四个象限，从右上方开始逆时针编号：第一象限 (x > 0, y > 0)、第二象限 (x < 0, y > 0)、第三象限 (x < 0, y < 0) 和第四象限 (x > 0, y < 0)。理解各象限中坐标的符号对于绘制图形和解决几何问题至关重要。
You will often need to plot points, read coordinates from graphs and determine the quadrant in which a point lies. A point on the x‑axis has a y‑coordinate of 0, and a point on the y‑axis has an x‑coordinate of 0. In IB questions, being able to visualise a point’s location helps when applying formulas such as distance and midpoint.
你经常需要描点、从图形中读取坐标，并判断一个点位于哪个象限。x 轴上的点纵坐标为 0，y 轴上的点横坐标为 0。在 IB 考题中，能够直观想象点的位置有助于你应用距离公式和中点公式。
2. Distance Between Two Points | 两点间的距离
l differences are Δx = x₂ − x₁ and Δy = y₂ − y₁, and the distance is given by the formula:
若 A(x₁, y₁) 和 B(x₂, y₂) 是笛卡尔平面上的两点，则距离 AB 由勾股定理推导而来。水平差和竖直差分别为 Δx = x₂ − x₁ 和 Δy = y₂ − y₁，距离公式如下：
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This formula works regardless of the quadrants of the points. The order of subtraction does not matter because the differences are squared. Always take the positive square root to obtain a length.
无论点位于哪个象限，该公式均适用。相减的顺序并不重要，因为差值要平方。始终取正的平方根，以得到长度。
en P(3, 1) and Q(−1, 4). Here, Δx = −1 − 3 = −4, Δy = 4 − 1 = 3. Then PQ = √[(−4)² + 3²] = √16 + 9 = √25 s, you may have to use the distance formula to prove that a triangle is isosceles or right‑gled, or to find the radius of a circle given its centre and a point on the circumference.
例如，求 P(3, 1) 和 Q(−1, 4) 之间的距离。这里 Δx = −1 − 3 = −4，Δy = 4 − 1 = 3。则 PQ = √[(−4)² + 3²] = √16 + 9 = √25 = 5 个单位。在 IB 考试中，你可能需要运用距离公式证明某个三角形是等腰三角形或直角三角形，或者已知圆心和圆周上一点来求半径。
3. Midpoint of a Segment | 线段的中点
The midpoint M of the line segment joining A(x₁, y₁) and B(x₂, y₂) is found by averaging the x‑coordinates and the y‑coordinates:
连接 A(x₁, y₁) 和 B(x₂, y₂) 的线段的中点 M 可以通过求横坐标和纵坐标的平均值得到：
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
This point lies exactly halfway between A and B. The midpoint formula is frequently used in IB problems involving symmetry, the centre of a circle (as the midpoint of a diameter) and the construction of perpendicular bisectors.
该点恰好位于 A 和 B 的正中间。中点公式在 IB 题目中频繁出现，涉及对称性、圆的圆心（作为直径的中点）以及垂直平分线的构造等问题。
he negative sign when averaging. When solving an unknown endpoint problem, you can set up two separate equations using the midpoint formula.
例如，若 A(2, −3) 和 B(8, 5)，则中点为 ((2+8)/2, (−3+5)/2) = (5, 1)。求平均值时务必注意负号。在求解未知端点的问题时，可以利用中点公式列出两个独立的方程。
4. Gradient (Slope) of a Line | 直线的斜率
The gradient m of the line passing through (x₁, y₁) and (x₂, y₂) measures its steepness and direction. It is defined as the ratio of the vertical change to the horizontal change:
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率 m 衡量了该直线的倾斜程度和方向。它被定义为竖直变化量与水平变化量之比：
m = (y₂ − y₁) / (x₂ − x₁)
ertical and its gradient is undefined. A positive gradient indicates the line rises from left to right, a negative gradient means it falls, a zero gradient gives a horizontal line, and an undefined gradient corresponds to a vertical line.
条件是 x₂ ≠ x₁；如果 x₂ = x₁，则直线是竖直的，斜率未定义。正斜率表示直线从左到右上升，负斜率表示直线下降，斜率为零时得到水平直线，斜率未定义时对应竖直直线。
arallel (m₁ = m₂) or perpendicular (m₁ × m₂ = −1), and for writing the equation of a line.
你必须能够准确地计算斜率，包括涉及分数和负数的情况。在 IB 中，斜率对于判断直线是否平行 (m₁ = m₂) 或垂直 (m₁ × m₂ = −1) 以及写出直线方程至关重要。
5. Equations of a Straight Line | 直线方程
A straight line can be expressed in several forms. The most common is the gradient–intercept form:
一条直线可以用多种形式表示。最常见的是斜截式：
y = mx + c
where m is the gradient and c is the y‑intercept (the value of y when x = 0). Another widely used form is the point–nt form, which is particularly useful when you know one point (x₁, y₁) and the gradient m:
其中 m 为斜率，c 为 y 轴截距（当 x = 0 时的 y 值）。另一种广泛使用的形式是点斜式，当你已知一点 (x₁, y₁) 和斜率 m 时，该形式尤为有用：
y − y₁ = m(x − x₁)
essing final answers in some IB questions. All forms can be rearranged through algebraic manipulation. Mastery of converting between forms is tested, for example, taking a line from a diagram and writing its equation in the requested format.
还有一般式 ax + by + d = 0，某些 IB 题目要求以此形式给出最终答案。所有形式都可以通过代数变换相互转换。考试中会考查你在形式间转换的能力，例如，从图形中读取一条直线，并用题目要求的格式写出其方程。
To find the equation of the line through (2, 5) with gradient −3, apply the point– r 3x + y − 11 = 0 in general form. Always check your work by substituting the given point.
要求出经过点 (2, 5) 且斜率为 −3 的直线方程，使用点斜式：y − 5 = −3(x − 2)。化简得 y = −3x + 11，或写成一般式 3x + y − 11 = 0。务必代入给定的点进行验算。
6. Parallel and Perpendicular Lines | 平行与垂直直线
1. This condition also tells us that one gradient is the negative reciprocal of the other.
两条互异直线平行，当且仅当它们的斜率相等：m₁ = m₂。平行线永不相交。两条直线垂直，当且仅当它们的斜率之积等于 −1：m₁ × m₂ = −1。这个条件也说明其中一条直线的斜率是另一条的负倒数。
al line (undefined gradient), but the product rule cannot be applied directly because multiplication by an undefined quantity is not valid. In IB problems, you are often asked to find the equation of a line parallel or perpendicular to a given line and passing through a specific point.
对于水平线和竖直线要特别留意。水平线 (m = 0) 与竖直线（斜率未定义）垂直，但不能直接使用乘积法则，因为未定义的值无法进行乘法运算。在 IB 题目中，经常要求你求出一条与给定直线平行或垂直，并且经过特定点的直线方程。
ar to L₁ must have gradient −½; passing through (3, 0), its equation is y − 0 = −½(x − 3).
例如，直线 L₁ 的方程为 y = 2x − 1。与 L₁ 平行的直线 L₂ 必须具有斜率 2。若 L₂ 经过点 (3, 0)，则其方程为 y − 0 = 2(x − 3) → y = 2x − 6。与 L₁ 垂直的直线 L₃ 必须具有斜率 −½；经过点 (3, 0)，其方程为 y − 0 = −½(x − 3)。
7. Perpendicular Bisector | 垂直平分线
The perpendicular bisector of a line segment AB is the line that passes through the midpoint of AB and is perpendicular to AB. It is the locus of points equidistant from A and B. Finding its equation is a common IB Core topic that combines the midpoint, gradient and perpendicular line concepts in a single problem.
线段 AB 的垂直平分线，是通过 AB 的中点且垂直于 AB 的直线。它是到 A 和 B 距离相等的点的轨迹。求垂直平分线的方程是 IB 核心课程的常见题型，它将中点、斜率和垂直直线的概念整合在一个问题中。
Steps to determine the perpendicular bisector:
1. Find the midpoint M of AB.
2. Calculate the gradient mAB of AB.
3. Determine the perpendicular gradient mperp = −1 / mAB (if mAB = 0, the bisector is vertical, and if AB is vertical, the bisector is horizontal).
4. Use the point–gradient form with point M and gradient mperp to write the equation.
求垂直平分线的步骤：
1. 求出 AB 的中点 M。
2. 计算 AB 的斜率 mAB。
3. 确定垂直斜率 mperp = −1 / mAB（若 mAB = 0，则平分线为竖直线；若 AB 是竖直线，则平分线为水平线）。
4. 利用点 M 和斜率 mperp，通过点斜式写出方程。
As an example, for A(1, 2) and B(5, 8), the midpoint is (3, 5). The gradient mAB simplifies to 2x + 3y = 21. Such questions often require the final answer in general form.
举例来说，对于 A(1, 2) 和 B(5, 8)，中点为 (3, 5)。斜率 mAB = (8−2)/(5−1) = 6/4 = 1.5。垂直斜率为 −2/3。垂直平分线方程为 y − 5 = −2/3 (x − 3)，化简得 2x + 3y = 21。这类题目通常要求最终答案以一般式表示。
8. Intersection of Lines | 直线的交点
The point of intersection of two lines is found by solving their equations simultaneously. Two non‑parallel lines always intersect at a single point. In coordinate geometry, the solution corresponds to the graphical intersection, and the algebraic methods (substitution or elimination) are used.
两条直线的交点可通过联立它们的方程求解得出。两条不平行且不重合的直线总相交于唯一一点。在坐标几何中，该解对应图形上的交点，代数方法（代入法或消元法）被用来求解。
t is (1.8, 4.6). If the equations are both in general form, elimination may be more efficient. IB questions sometimes ask you to verify that a point lies on both lines or to find the intersection as part of a larger problem, e.g., with triangles.
对于直线 L₁: y = 2x + 1 和 L₂: 3x + y = 10，将 L₁ 中的 y 代入 L₂：3x + (2x + 1) = 10 → 5x + 1 = 10 → x = 9/5 = 1.8。则 y = 2(1.8) + 1 = 4.6。交点为 (1.8, 4.6)。当两个方程都是普通式时，消元法可能更高效。IB 题目有时会要求验证某点是否同时在两条直线上，或者作为更复杂题目（例如涉及三角形）的一部分来求交点。
9. Finding the Area of a Triangle from Coordinates | 由坐标求三角形的面积
Given the vertices of a triangle A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), the area can be calculated without finding side lengths by using the determinant (or shoelace) formula:
已知三角形的顶点 A(x₁, y₁)、B(x₂, y₂) 和 C(x₃, y₃)，无需计算边长，即可使用行列式（或鞋带）公式求出面积：
Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |
This formula is often arranged in a cyclical pattern to avoid sign errors. The absolute value ensures the area is positive. In IB Core, this formula is particularly useful when the triangle does not have a base or height conveniently aligned with the axes.
该公式通常按循环模式排列，以避免符号错误。绝对值确保了面积为正。在 IB 核心课程中，当三角形的底边或高没有恰好与坐标轴对齐时，该公式尤为有用。
For example, triangle with vertices (0, 0), (4, 2) and (1, 5): Area = ½ | 0(2−5) + 4(5−0) + 1(0−2) | = ½ | 0 + 20 − 2 | lar distance from the third vertex to that base, but the shoelace method is often quicker.
例如，顶点为 (0, 0), (4, 2) 和 (1, 5) 的三角形：面积 = ½ | 0(2−5) + 4(5−0) + 1(0−2) | = ½ | 0 + 20 − 2 | = 9 平方单位。你也应能运用距离公式求出底边长度，再求第三个顶点到该底边的垂直距离，但鞋带法往往更快。
10. Applications to Quadrilaterals and Geometric Proofs | 在四边形与几何证明中的应用
Coordinate geometry is a powerful tool for proving properties of quadrilaterals. By placing a shape on a coordinate system (often with one vertex at the origin and sides along axes), you can use algebraic conditions to prove that a quadrilateral is a parallelogram, rectangle, rhombus or square. For instance, a quadrilateral is a parallelogram if both pairs of opposite sides are parallel (equal gradients) or if the diagonals bisect each other (same midpoint).
坐标几何是证明四边形性质的强有力工具。通过将图形放置在坐标系中（通常使一个顶点位于原点，且边与坐标轴对齐），你可以利用代数条件来证明某个四边形是平行四边形、矩形、菱形或正方形。例如，若两组对边互相平行（斜率相等），或者对角线互相平分（中点相同），那么这个四边形就是平行四边形。
Using gradients and distances, you can check:
- Parallelogram: both pairs of opposite sides parallel.
- Rectangle: parallelogram with one right angle (adjacent sides perpendicular).
- Rhombus: parallelogram with all sides equal.
- Square: both a rectangle and a rhombus.
利用斜率和距离，你可以验证：
- 平行四边形：两组对边平行。
- 矩形：有一个直角的平行四边形（邻边垂直）。
- 菱形：四边相等的平行四边形。
- 正方形：既是矩形又是菱形。
You may also need to prove collinearity (three points lying on the same line) by showing that the gradients between pairs are equal, or that the area of the triangle formed by them is zero. Another common task is to determine the coordinates of a vertex of a shape when other conditions are given, which involves solving equations derived from the geometric properties.
你可能还需要通过证明两点间斜率相等，或证明三点构成的三角形面积为零，来证明它们共线（三点在同一直线上）。另一个常见题型是：在给定其他条件的情况下，确定图形某个顶点的坐标，这需要求解由几何性质导出的方程。
11. Coordinate Geometry in the IB Exam – Tips and Common Pitfalls | IB考试中的坐标几何 – 技巧与常见误区
In the IB Diploma Mathematics exam, coordinate geometry often appears in both Paper 1 (non‑calculator) and Paper 2 (calculator). You must be able to apply formulas quickly and accurately, showing full working for algebraic manipulation. Here are some key exam tips:
- Always sketch a quick diagram, even a rough one. It helps you to avoid sign errors and to see whether your final answer makes sense.
- When solving for an unknown coordinate, treat the x and y components separately and set up equations using the distance, midpoint or gradient formulas.
- Be careful when a gradient is zero or undefined. Remember the special perpendicular cases.
- Leave answers in exact simplified form unless a decimal is specified. This often means leaving roots or fractions.
- For perpendicular bisector and intersection problems, verify by substituting coordinates back into both equations.
在 IB 文凭数学考试中，坐标几何经常同时出现在卷一（不可用计算器）和卷二（可用计算器）里。你必须能快速准确地应用公式，并详细展示代数演算过程。以下是一些关键的考试技巧：
- 始终画一个快速图示，即使很粗略。这有助于你避免符号错误，并判断最终答案是否合理。
- 在求解未知坐标时，分别处理横坐标和纵坐标，利用距离、中点或斜率公式建立方程。
- 当斜率为零或未定义时要格外小心，牢记垂直关系的特例。
- 除非题目明确要求，否则答案要保留精确的简化形式，这通常意味着保留根号或分数。
- 对于垂直平分线和交点问题，把坐标代回两个方程进行验证。
A common mistake is to misapply the gradient formula by subtracting coordinates in the wrong order. Another is forgetting to take the absolute value in the area formula. In perpendicular bisector questions, students sometimes use the original gradient instead of the negative reciprocal. Repeated practice with past paper questions will make these processes automatic.
一个常见错误是弄错斜率公式中坐标的相减顺序。另一个常见错误是忘记在面积公式中使用绝对值。在垂直平分线问题中，学生有时会误用原直线的斜率，而不是它的负倒数。反复练习历年真题会使这些步骤变得自然而然。
12. Summary and Key Formulae Review | 总结与核心公式回顾
Coordinate geometry is a wonderfully visual and algebraic topic that ties together many strands of mathematics. The fundamental relationships to remember are presented in the table below.
坐标几何是一个奇妙地结合了几何直观与代数运算的主题，它将数学中的许多分支联系在了一起。需要牢记的基本关系如下表所示。
| Concept / 概念 | Formula / 公式 |
|---|---|
| Distance / 距离 | d = √[(x₂ − x₁)² + (y₂ − y₁)²] |
| Midpoint / 中点 | M = ( (x₁+x₂)/2 , (y₁+y₂)/2 ) |
| Gradient / 斜率 | m = (y₂ − y₁) / (x₂ − x₁) |
| Line (gradient‑intercept) / 直线(斜截式) | y = mx + c |
| Line (point‑gradient) / 直线(点斜式) | y − y₁ = m(x − x₁) |
| Parallel lines / 平行线 | m₁ = m₂ |
| Perpendicular lines / 垂直线 | m₁ × m₂ = −1 |
| Area of triangle / 三角形面积 | ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) | |
Mastering these tools will allow you to approach any Core coordinate geometry problem with confidence. Remember that each formula stems from a simple geometric idea, and understanding that idea is more valuable than memorising symbols. As you practice, focus on linking the algebraic steps back to their geometric meaning, and you will find coordinate geometry not only manageable but genuinely rewarding.
掌握这些工具将使你能够自信地应对核心课程中任何坐标几何问题。请记住，每个公式都源于一个简单的几何概念，而理解那个概念比记忆符号更有价值。在练习时，注重将代数步骤与几何意义相联系，你会发现坐标几何不仅容易掌握，而且真正充满乐趣。
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