Coordinate Geometry of Straight Lines | 直线坐标几何

📚 Coordinate Geometry of Straight Lines | 直线坐标几何

The straight line is one of the most fundamental objects in IGCSE Mathematics. Mastering coordinate geometry of straight lines allows you to describe, predict, and analyse relationships between two variables, and it forms the backbone of many exam questions involving graphs, functions, and problem-solving.

直线是 IGCSE 数学中最基础的对象之一。掌握直线坐标几何，使你能够描述、预测并分析两个变量之间的关系，同时也是考试中许多涉及图像、函数与综合应用题的基石。

1. The Cartesian Plane | 笛卡尔平面

The Cartesian plane is defined by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin, O, with coordinates (0, 0). Every point on the plane is written as an ordered pair (x, y), where x is the horizontal position and y is the vertical position.

笛卡尔平面由两条互相垂直的数轴构成：水平方向的 x 轴和垂直方向的 y 轴。它们的交点称为原点 O，坐标为 (0, 0)。平面上每个点都表示为有序数对 (x, y)，其中 x 表示水平位置，y 表示垂直位置。

The x-coordinate is read from the x-axis; it is positive to the right of the origin and negative to the left.

x 坐标从 x 轴读取；原点右侧为正，左侧为负。
The y-coordinate is read from the y-axis; it is positive above the origin and negative below.

y 坐标从 y 轴读取；原点上方为正，下方为负。
The plane is divided into four quadrants, numbered anticlockwise from the top-right.

平面被分为四个象限，从右上角开始按逆时针方向编号。

2. Gradient of a Line | 直线的斜率

The gradient measures how steep a line is. It is defined as the change in y divided by the change in x as you move along the line.

斜率衡量直线倾斜的程度。其定义为沿直线移动时，y 的变化量除以 x 的变化量。

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line slopes upwards from left to right.

斜率为正表示直线从左向右上升。
A negative gradient means the line slopes downwards from left to right.

斜率为负表示直线从左向右下降。
A zero gradient means the line is horizontal.

斜率为零表示直线是水平的。
A vertical line has an undefined gradient because the change in x is zero.

垂直直线的斜率未定义，因为 x 的变化量为零。

3. Finding the Gradient from Two Points | 由两点求斜率

Given two points on a line, (x₁, y₁) and (x₂, y₂), you can find the gradient by substituting into the formula. The order of subtraction must be consistent between the numerator and the denominator.

已知直线上两点 (x₁, y₁) 和 (x₂, y₂)，代入公式即可求出斜率。分子与分母中减法的顺序必须保持一致。

Example: Find the gradient of the line joining A(1, 2) and B(4, 8).

例：求连接 A(1, 2) 与 B(4, 8) 的直线的斜率。

m = (8 − 2) / (4 − 1) = 6 / 3 = 2

Try to choose (x₂, y₂) as the point with the larger x-coordinate. However, as long as you subtract both coordinates in the same order, the result will be identical.

通常选择 x 坐标较大的点作为 (x₂, y₂)。但只要分子分母按同一顺序相减，结果完全相同。

4. Equation of a Line: y = mx + c | 直线方程 y = mx + c

The most useful form of the equation of a straight line is y = mx + c. Here, m is the gradient and c is the y-intercept, the point where the line crosses the y-axis.

直线方程最常用的形式是 y = mx + c。其中 m 为斜率，c 为 y 截距，即直线与 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)。

The y-intercept is always of the form (0, c), since x = 0 on the y-axis.

y 截距总是形如 (0, c)，因为在 y 轴上 x = 0。
If the equation is not in this form, rearrange it to solve for y before reading off m and c.

若方程不为此形式，应先移项整理为解出 y 的形式，再读出 m 和 c。
For example, 2x + 3y = 6 can be rearranged to y = −(2/3)x + 2.

例如，2x + 3y = 6 可整理为 y = −(2/3)x + 2。

5. Finding the Equation from Two Points | 由两点求直线方程

To find the equation of a line passing through two points, first calculate the gradient m. Then substitute the coordinates of one known point into y = mx + c to solve for c.

要求经过两点的直线方程，先计算斜率 m；然后将一个已知点的坐标代入 y = mx + c，解出 c。

Example: Find the equation of the line through (2, 5) and (4, 9).

例：求经过 (2, 5) 和 (4, 9) 的直线方程。

m = (9 − 5) / (4 − 2) = 4 / 2 = 2

Substitute (2, 5): 5 = 2(2) + c, so 5 = 4 + c and c = 1.

代入 (2, 5)：5 = 2(2) + c，即 5 = 4 + c，所以 c = 1。

y = 2x + 1

Always verify your answer by checking that both original points satisfy the equation.

务必验证答案：检验两个原始点是否都满足该方程。

6. Sketching Straight Lines | 绘制直线草图

There are two convenient strategies for sketching a straight line without plotting many points.

绘制直线草图的两种便捷策略是：不必连接大量点。

Method 1: Use the gradient and y-intercept. Start at (0, c), then move “up/down m units and right 1 unit” repeatedly.

方法一：利用斜率和 y 截距。从 (0, c) 开始，反复“向上/向下移动 m 个单位，再向右移动 1 个单位”。
Method 2: Find the x-intercept and y-intercept. Set y = 0 to find where the line crosses the x-axis, then join the two intercepts.

方法二：求 x 截距和 y 截距。令 y = 0 求与 x 轴的交点，然后连接两个截距。

When sketching, always label the axes, the coordinates of any intercepts, and the equation of the line.

绘图时，务必标明坐标轴、所有截距的坐标以及直线方程。

7. Parallel Lines | 平行线

Two lines are parallel if and only if they have the same gradient. Because they rise at the same rate, they never intersect.

两条直线平行，当且仅当它们的斜率相同。由于它们上升的速率相同，因此永远不会相交。

m₁ = m₂

To find a line parallel to a given line, keep the gradient the same and change the y-intercept c.

要求与已知直线平行的直线，只需保持斜率相同，改变 y 截距 c 即可。

Example: y = 3x + 5 and y = 3x − 2 are parallel, because both have gradient 3.

例：y = 3x + 5 与 y = 3x − 2 平行，因为二者斜率都为 3。

8. Perpendicular Lines | 垂直线

Two lines are perpendicular if the product of their gradients is −1. In other words, one gradient is the negative reciprocal of the other.

两条直线垂直，当且仅当它们的斜率之积为 −1。换句话说，一条直线的斜率是另一条斜率的负倒数。

m₁ × m₂ = −1, so m₂ = −1 / m₁

Example: A line with gradient 2 is perpendicular to a line with gradient −1/2.

例：斜率为 2 的直线垂直于斜率为 −1/2 的直线。

A horizontal line and a vertical line are also perpendicular, but neither has a finite gradient, so the product rule does not apply.

水平线与垂直线也互相垂直，但二者斜率均有限值，因此乘积法则不适用。
When finding a perpendicular line through a specific point, first take the negative reciprocal of m, then substitute the point to find c.

求过特定点的垂线时，先取 m 的负倒数，再代入该点求 c。

9. Midpoint of a Line Segment | 线段中点

The midpoint of a segment joining (x₁, y₁) and (x₂, y₂) is the average of the two x-coordinates and the average of the two y-coordinates.

连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点，是两个 x 坐标的平均值和两个 y 坐标的平均值。

Midpoint = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )

Example: The midpoint of (2, 3) and (6, 7) is ((2 + 6)/2, (3 + 7)/2) = (4, 5).

例：(2, 3) 与 (6, 7) 的中点为 ((2 + 6)/2, (3 + 7)/2) = (4, 5)。

If you know the midpoint and one endpoint, you can work backwards to find the missing endpoint by doubling the midpoint and subtracting the known end.

已知中点和一个端点时，可反向求另一端点：将中点坐标乘以 2，再减去已知端点坐标。

10. Length of a Line Segment | 线段长度

The distance between two points (x₁, y₁) and (x₂, y₂) is found using the Pythagorean theorem. The horizontal difference is (x₂ − x₁) and the vertical difference is (y₂ − y₁).

两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离用勾股定理求。水平差为 (x₂ − x₁)，垂直差为 (y₂ − y₁)。

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Example: Find the distance between (1, 2) and (4, 6).

例：求点 (1, 2) 与 (4, 6) 之间的距离。

d = √[(4 − 1)² + (6 − 2)²] = √(9 + 16) = √25 = 5

This formula is especially useful in problems involving isosceles triangles, perimeter, and coordinate proofs.

该公式在等腰三角形、周长以及坐标证明问题中特别有用。

11. Applications and Problem-Solving | 应用与解题

Coordinate geometry often appears in combined problems with other topics. Common applications include finding the type of a triangle, showing three points are collinear, and solving intersection problems.

坐标几何常与其他知识结合考查。常见应用包括判断三角形类型、证明三点共线，以及求解交点问题。

To show three points are collinear, find the gradient between two pairs of points; if the gradients are equal and share a common point, the points lie on one line.

证明三点共线：求两组点之间的斜率；若斜率相等且共用一点，则三点在同一直线上。
To show a triangle is right-angled, calculate the three side lengths and check whether the square of the longest side equals the sum of the squares of the other two.

证明三角形为直角三角形：计算三条边长，检验最长边的平方是否等于另外两边平方之和。
To find the intersection of two lines, solve their equations simultaneously.

求两条直线的交点：联立解它们的方程组。

12. Common Exam Pitfalls | 常见考试易错点

Students often lose marks in coordinate geometry questions through avoidable arithmetic and sign errors. Review the following reminders before your exam.

学生在坐标几何题中常因可避免的运算和符号错误而失分。考试前请复习以下提醒。

Always subtract x-coordinates in the denominator and y-coordinates in the numerator in the same order.

分子中 y 坐标相减与分母中 x 坐标相减的顺序必须一致。
Do not confuse the x-intercept with the y-intercept. On the x-axis, y = 0; on the y-axis, x = 0.

不要混淆 x 截距与 y 截距。在 x 轴上 y = 0；在 y 轴上 x = 0。
The gradient of a line is the same at every point on that line; you only ever need two points to find it.

直线上的斜率处处相同；求斜率只需任意两个点。
When a line is written as ax + by = c, rearrange to y = mx + c form before identifying m and the y-intercept.

直线写成 ax + by = c 时，先整理为 y = mx + c 形式，再确定 m 和 y 截距。
Remember that perpendicular gradients multiply to −1, not 1.

记住垂直直线的斜率相乘等于 −1，而不是 1。
When using the distance formula, square each difference separately before adding; do not combine the terms prematurely.

使用距离公式时，先将每个差值分别平方后再相加；不要提前合并各项。

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Coordinate Geometry of Straight Lines | 直线坐标几何