Coordinate Geometry and Straight Line Graphs | 坐标几何与直线图像

📚 Coordinate Geometry and Straight Line Graphs | 坐标几何与直线图像

Coordinate geometry is one of the most important topics in IGCSE Mathematics. It connects algebra with geometry by using coordinates to describe points, lines, distances and gradients on the Cartesian plane. This article covers the core skills you need: plotting points, finding distances, midpoints, gradients, equations of straight lines, and the conditions for parallel and perpendicular lines.

坐标几何是 IGCSE 数学中最重要的主题之一。它通过使用坐标来描述笛卡尔平面上的点、直线、距离和斜率，将代数与几何联系起来。本文涵盖你需要掌握的核心技能：描点、求距离、中点、斜率、直线方程，以及平行线和垂直线的条件。

1. Coordinate Plane and Points | 坐标平面与点

Every point in coordinate geometry is written as an ordered pair (x, y). The first value, x, represents the horizontal position from the origin, and the second value, y, represents the vertical position. The origin is the point (0, 0), where the x-axis and y-axis intersect.

在坐标几何中，每个点都写成有序数对 (x, y)。第一个值 x 表示相对于原点的水平位置，第二个值 y 表示垂直位置。原点是 x 轴与 y 轴相交的点 (0, 0)。

The Cartesian plane is divided into four quadrants. Quadrant I has both coordinates positive, Quadrant II has x negative and y positive, Quadrant III has both negative, and Quadrant IV has x positive and y negative.

笛卡尔平面分为四个象限。第一象限的两个坐标均为正，第二象限 x 为负、y 为正，第三象限两个均为负，第四象限 x 为正、y 为负。

x-axis: horizontal axis | x 轴：水平轴
y-axis: vertical axis | y 轴：垂直轴
Quadrants I, II, III, IV are counted anticlockwise from the top right | 象限 I、II、III、IV 从右上角开始按逆时针计数

Always write coordinates in the form (x, y), not (y, x). Reversing the order is a very common error when reading or plotting points.

务必始终以 (x, y) 的形式书写坐标，而不是 (y, x)。在读取或描点时颠倒顺序是一个非常常见的错误。

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

The distance between two points A(x₁, y₁) and B(x₂, y₂) is found using Pythagoras’ theorem. The difference in x-coordinates and the difference in y-coordinates form the two shorter sides of a right-angled triangle.

两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离可以用毕达哥拉斯定理求得。x 坐标之差和 y 坐标之差构成直角三角形的两条较短边。

d = √((x₂ − x₁)² + (y₂ − y₁)²)

For example, the distance between (1, 2) and (4, 6) is √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5 units.

例如，点 (1, 2) 与点 (4, 6) 之间的距离为 √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5 个单位。

This formula works for any two points, including points with negative coordinates. Be careful with signs when subtracting a negative number.

该公式适用于任意两点，包括带有负坐标的点。当减去负数时，请务必注意符号。

3. Midpoint of a Line Segment | 线段的中点

The midpoint of a line segment is the point exactly halfway between the two endpoints. It is calculated by averaging the x-coordinates and averaging the y-coordinates.

线段的中点是恰好位于两个端点中间的点。它通过分别求 x 坐标的平均值和 y 坐标的平均值来计算。

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

For example, the midpoint of (−2, 3) and (4, −1) is ((−2 + 4)/2, (3 + (−1))/2) = (1, 1).

例如，点 (−2, 3) 与点 (4, −1) 的中点为 ((−2 + 4)/2, (3 + (−1))/2) = (1, 1)。

The midpoint formula is useful when you are given one endpoint and the midpoint, and you need to find the other endpoint. Simply work backwards by doubling the midpoint coordinates and subtracting the known endpoint.

当你已知一个端点和中点，而需要求另一个端点时，中点公式非常有用。只需将中点坐标乘以 2，再减去已知端点即可。

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

The gradient of a straight line measures how steep the line is. It is the ratio of the vertical change to the horizontal change between any two points on the line.

直线的斜率衡量直线的倾斜程度。它是直线上任意两点之间垂直变化与水平变化的比值。

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line rises from left to right. A negative gradient means the line falls from left to right. A horizontal line has gradient 0, and a vertical line has an undefined gradient.

斜率为正表示直线从左到右上升。斜率为负表示直线从左到右下降。水平线的斜率为 0，垂直线的斜率没有定义。

Always subtract the y-coordinates and x-coordinates in the same order. If you reverse one order without reversing the other, the sign of the gradient will be wrong.

务必以相同的顺序减去 y 坐标和 x 坐标。如果只颠倒其中一个顺序而没有颠倒另一个，斜率的符号就会出现错误。

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

The equation of a straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept. The y-intercept is the point where the line crosses the y-axis, so its coordinates are (0, c).

直线方程可以写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距。y 轴截距是直线与 y 轴相交的点，因此其坐标为 (0, c)。

y = mx + c

For example, the line y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3). The line y = −x + 5 has gradient −1 and y-intercept 5.

例如，直线 y = 2x + 3 的斜率为 2，与 y 轴相交于 (0, 3)。直线 y = −x + 5 的斜率为 −1，y 轴截距为 5。

If the equation is not in the exact form y = mx + c, rearrange it first. For example, 2y = 4x + 6 becomes y = 2x + 3 after dividing both sides by 2.

如果方程不是 y = mx + c 的精确形式，请先重新整理。例如，2y = 4x + 6 在两边同时除以 2 后变为 y = 2x + 3。

6. Parallel and Perpendicular Lines | 平行线与垂直线

Two straight lines are parallel if and only if their gradients are equal. This means the lines never meet, no matter how far they are extended.

两条直线平行，当且仅当它们的斜率相等。这意味着无论延长多远，这两条直线都不会相交。

m₁ = m₂ for parallel lines

Two straight lines are perpendicular if and only if the product of their gradients is −1. This also means 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。这也意味着其中一个斜率是另一个斜率的负倒数。

m₁ × m₂ = −1

For example, if line A has gradient 3, then a line parallel to A also has gradient 3, but a line perpendicular to A has gradient −1/3.

例如，如果直线 A 的斜率为 3，那么与 A 平行的直线斜率也为 3，但与 A 垂直的直线斜率为 −1/3。

Remember that vertical lines have undefined gradient. They are perpendicular to horizontal lines, whose gradient is 0, but the product rule m₁ × m₂ = −1 does not apply in that special case.

请记住，垂直线的斜率没有定义。它们与斜率为 0 的水平线垂直，但斜率乘积规则 m₁ × m₂ = −1 在这种特殊情况下不适用。

7. Finding the Equation from Two Points | 已知两点求直线方程

To find the equation of a straight line given two points, first calculate the gradient using m = (y₂ − y₁) / (x₂ − x₁). Then substitute the gradient and one of the points into y = mx + c to solve for c.

已知两点求直线方程时，首先用 m = (y₂ − y₁) / (x₂ − x₁) 计算斜率。然后将斜率和其中一个点代入 y = mx + c 来求出 c。

For example, find the line passing through (1, 4) and (3, 10). First, m = (10 − 4) / (3 − 1) = 6 / 2 = 3. Then use (1, 4): 4 = 3(1) + c, so c = 1. The equation is y = 3x + 1.

例如，求经过点 (1, 4) 和 (3, 10) 的直线。首先，m = (10 − 4) / (3 − 1) = 6 / 2 = 3。然后代入 (1, 4)：4 = 3(1) + c，因此 c = 1。所求方程为 y = 3x + 1。

You can also use the point-gradient form y − y₁ = m(x − x₁), which is especially useful if the y-intercept is not immediately obvious.

你也可以使用点斜式 y − y₁ = m(x − x₁)，当 y 轴截距不是立刻明显时，这种形式尤其有用。

y − y₁ = m(x − x₁)

8. Intercepts and Sketching Straight Lines | 截距与直线草图

The x-intercept is the point where the line crosses the x-axis, so y = 0. To find it, substitute y = 0 into the line equation and solve for x. The y-intercept is where the line crosses the y-axis, so x = 0.

x 轴截距是直线与 x 轴相交的点，因此 y = 0。要求它，将 y = 0 代入直线方程并解出 x。y 轴截距是直线与 y 轴相交的点，因此 x = 0。

For example, for the line y = 2x + 4, the y-intercept is (0, 4). To find the x-intercept, set y = 0: 0 = 2x + 4, so x = −2. The x-intercept is (−2, 0).

例如，对于直线 y = 2x + 4，y 轴截距为 (0, 4)。要求 x 轴截距，令 y = 0：0 = 2x + 4，所以 x = −2。x 轴截距为 (−2, 0)。

To sketch a straight line, plot the y-intercept first. Then use the gradient to find a second point by rising or falling in the vertical direction and moving across in the horizontal direction. Draw a straight line through the two points.

要绘制直线草图，首先标出 y 轴截距。然后利用斜率求出第二个点：先在垂直方向上升或下降，再在水平方向上移动。通过这两个点画一条直线。

9. Real-Life Applications of Linear Graphs | 线性图像的实际应用

Linear equations model many real-life situations, including taxi fares, phone bills, temperature conversion and distance-time graphs for constant speed. In these problems, the gradient often represents a rate, and the y-intercept represents a fixed starting value.

线性方程可以模拟许多现实生活中的情况，包括出租车费、电话费、温度转换以及匀速运动的距离-时间图像。在这些问题中，斜率通常表示变化率，y 轴截距表示固定的初始值。

For example, a taxi company charges a fixed fee of 2 dollars plus 3 dollars per mile. The cost C for a journey of d miles is C = 3d + 2. The gradient 3 is the cost per mile, and the y-intercept 2 is the fixed fee.

例如，一家出租车公司收取 2 美元的固定费用，另加每英里 3 美元。行驶 d 英里的费用 C 为 C = 3d + 2。斜率 3 是每英里的费用，y 轴截距 2 是固定费用。

When interpreting a straight-line graph, always state the meaning of the gradient and the y-intercept in the context of the question, using the correct units.

在解读直线图像时，务必结合题目的情境说明斜率和 y 轴截距的含义，并使用正确的单位。

10. Worked Example: Full Solution | 典型例题：完整解答

Question: Find the equation of the line that passes through the point (2, −1) and is perpendicular to the line y = 3x + 5. Also find the midpoint of the two points A(2, −1) and B(4, 3).

题目：求经过点 (2, −1) 且垂直于直线 y = 3x + 5 的直线方程。同时求两点 A(2, −1) 与 B(4, 3) 的中点。

Step 1: The given line has gradient 3. A perpendicular line has gradient m = −1/3.

步骤 1：给定直线的斜率为 3。垂直直线的斜率为 m = −1/3。

Step 2: Use the point (2, −1) in y = mx + c: −1 = (−1/3)(2) + c. This gives −1 = −2/3 + c, so c = −1 + 2/3 = −1/3.

步骤 2：将点 (2, −1) 代入 y = mx + c：−1 = (−1/3)(2) + c。由此得到 −1 = −2/3 + c，所以 c = −1 + 2/3 = −1/3。

Step 3: The equation is y = −1/3 x − 1/3, which can also be written as 3y = −x − 1 or x + 3y + 1 = 0.

步骤 3：所求方程为 y = −1/3 x − 1/3，也可以写成 3y = −x − 1 或 x + 3y + 1 = 0。

Step 4: For the midpoint of A(2, −1) and B(4, 3), use M = ((2 + 4)/2, (−1 + 3)/2) = (3, 1).

步骤 4：对于 A(2, −1) 与 B(4, 3) 的中点，使用 M = ((2 + 4)/2, (−1 + 3)/2) = (3, 1)。

Final answer: the perpendicular line is x + 3y + 1 = 0, and the midpoint is (3, 1).

最终答案：垂直直线为 x + 3y + 1 = 0，中点为 (3, 1)。

11. Exam Tips and Common Mistakes | 考试技巧与常见错误

In IGCSE exams, coordinate geometry questions often combine several skills: finding a gradient, writing an equation, using parallel or perpendicular conditions, and finding midpoints or distances. Work step by step and show all your working.

在 IGCSE 考试中，坐标几何题通常综合考查多种技能：求斜率、写方程、使用平行或垂直条件，以及求中点或距离。请一步一步作答，并展示所有解题过程。

Always write coordinates in the order (x, y). | 务必按 (x, y) 的顺序书写坐标。
Check the sign of the gradient when using negative coordinates. | 使用负坐标时，请检查斜率的符号。
For perpendicular lines, use the negative reciprocal: m₂ = −1/m₁. | 对于垂直线，请使用负倒数：m₂ = −1/m₁。
Do not confuse distance with midpoint: distance is a length, midpoint is a point. | 不要混淆距离与中点：距离是一个长度，中点是一个点。
If the equation is not in y = mx + c form, rearrange it before identifying m and c. | 如果方程不是 y = mx + c 的形式，请在确定 m 和 c 之前先重新整理。
When sketching, label the intercepts with their coordinates. | 绘制草图时，请用坐标标出截距。

Time management is also important. If a question asks for both an equation and a midpoint, complete the easier midpoint part first to secure those marks.

时间管理也很重要。如果一道题同时要求方程和中点，请先完成较容易的中点部分，以确保拿到这些分数。

12. Summary of Key Formulas | 关键公式总结

The following table summarises the most important formulas for IGCSE coordinate geometry. Learn them thoroughly and practise applying them in mixed questions.

下表总结了 IGCSE 坐标几何中最重要的公式。请熟练掌握，并在综合题中练习应用。

Quantity	Formula
Distance between two points \| 两点间距离	√((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint \| 中点	( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Gradient \| 斜率	(y₂ − y₁) / (x₂ − x₁)
Line equation \| 直线方程	y = mx + c
Parallel lines \| 平行线	m₁ = m₂
Perpendicular lines \| 垂直线	m₁ × m₂ = −1

Practice with past paper questions is the best way to build confidence. Try to identify which formula is needed before starting each calculation, and always check that your final answer makes sense in the context of the question.

练习历年真题是建立信心的最佳方式。在开始每次计算之前，先确定需要哪个公式，并始终检查最终答案在题目情境中是否合理。

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Coordinate Geometry and Straight Line Graphs | 坐标几何与直线图像