Straight-Line Graphs and Coordinate Geometry | 直线图像与坐标几何

📚 Straight-Line Graphs and Coordinate Geometry | 直线图像与坐标几何

Straight-line graphs appear throughout IGCSE Mathematics, from coordinate geometry to simultaneous equations and real-life linear modelling. Understanding how to find gradient, distance, midpoint and equation of a line gives you a powerful toolkit for both paper-based and calculator questions.

直线图像贯穿 IGCSE 数学的许多内容，从坐标几何到联立方程以及现实生活中的线性模型。掌握如何求斜率、距离、中点和直线方程，可以为你解答笔试和计算器题目提供强大的工具。

1. The Cartesian Plane and Coordinates | 笛卡尔平面与坐标

The Cartesian plane is formed by two perpendicular number lines called the x-axis (horizontal) and the y-axis (vertical). They meet at the origin O, which has coordinates (0, 0).

笛卡尔平面由两条相互垂直的数轴组成，分别是水平的 x 轴和竖直的 y 轴。它们相交于原点 O，原点坐标为 (0, 0)。

Every point is written as an ordered pair (x, y). The x-coordinate is always given first and tells you how far to move left or right; the y-coordinate tells you how far to move up or down.

每个点都写成有序数对 (x, y)。x 坐标总是在前，表示向左或向右移动的距离；y 坐标表示向上或向下移动的距离。

• First quadrant: x > 0, y > 0
• Second quadrant: x < 0, y > 0
• Third quadrant: x < 0, y < 0
• Fourth quadrant: x > 0, y < 0

第一象限：x > 0, y > 0；第二象限：x < 0, y > 0；第三象限：x < 0, y < 0；第四象限：x > 0, y < 0。记住象限中坐标的正负号可以帮助你快速判断点的位置。

2. Distance Between Two Points | 两点之间的距离

To find the distance between two points A(x₁, y₁) and B(x₂, y₂), use Pythagoras’ theorem. The horizontal difference is (x₂ – x₁), the vertical difference is (y₂ – y₁), and the distance is the hypotenuse.

要求两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离，可以使用毕达哥拉斯定理。水平差为 (x₂ – x₁)，竖直差为 (y₂ – y₁)，距离就是直角三角形的斜边。

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

先求两个差值的平方，相加后再开平方。例如 A(1, 2) 与 B(4, 6) 之间的距离为 √[(4 – 1)² + (6 – 2)²] = √(9 + 16) = 5。

A common error is to subtract coordinates in the wrong order, but the squared differences make the direction irrelevant. The formula works for any two points as long as you are consistent.

一个常见错误是坐标相减顺序不一致，但由于差值会被平方，方向不影响结果。只要你保持一致，这个公式适用于任意两点。

3. Midpoint of a Segment | 线段中点

The midpoint of a line segment is simply the average of the x-coordinates and the average of the y-coordinates. If the endpoints are (x₁, y₁) and (x₂, y₂), the midpoint M is:

线段的中点就是两个 x 坐标的平均值和两个 y 坐标的平均值。如果端点为 (x₁, y₁) 和 (x₂, y₂

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