Coordinate Geometry: Straight Lines | 坐标几何：直线

📚 Coordinate Geometry: Straight Lines | 坐标几何：直线

Coordinate geometry connects algebra and geometry by describing points, lines, and shapes on the Cartesian plane. In IGCSE Maths, you must be able to calculate gradients, midpoints, and distances, and work confidently with the equation of a straight line. This guide covers the key methods, common exam traps, and worked examples to help you master straight-line coordinate geometry.

坐标几何将代数与几何联系起来，在笛卡尔平面上描述点、直线和图形。在 IGCSE 数学中，你必须能够计算斜率、中点和距离，并能熟练运用直线方程。本指南涵盖关键方法、常见考试陷阱和例题，帮助你掌握直线坐标几何。

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

The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection is the origin, O(0, 0). Points are written as ordered pairs (x, y).

笛卡尔平面由两条互相垂直的数轴构成：x 轴（水平）和 y 轴（垂直）。它们的交点是原点 O(0, 0)。点写作有序数对 (x, y)。

Positive x-values are to the right of the origin, while negative x-values are to the left. Positive y-values are above the origin, and negative y-values are below it.

正的 x 值位于原点右侧，负的 x 值位于左侧。正的 y 值位于原点上侧，负的 y 值位于下侧。

Always read the x-coordinate first, then the y-coordinate. For example, the point (3, -2) means 3 units right and 2 units down.

始终先读 x 坐标，再读 y 坐标。例如，点 (3, -2) 表示向右 3 个单位、向下 2 个单位。

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2. Plotting Points and Reading Coordinates | 描点与读取坐标

To plot a point, start at the origin. Move along the x-axis by the x-coordinate, then move parallel to the y-axis by the y-coordinate.

描点时，从原点出发。沿 x 轴移动 x 坐标对应的距离，再沿平行于 y 轴的方向移动 y 坐标对应的距离。

To read coordinates from a graph, draw perpendicular lines from the point to both axes. The x-coordinate is where the vertical line meets the x-axis; the y-coordinate is where the horizontal line meets the y-axis.

从图像中读取坐标时，从该点向两条坐标轴作垂线。x 坐标是垂直线与 x 轴相交处的数值；y 坐标是水平线与 y 轴相交处的数值。

Common errors include swapping x and y, or miscounting the scale when each square is not 1 unit. Check the axis labels carefully.

常见错误包括调换 x 和 y，或者在每个方格不代表 1 个单位时数错刻度。务必仔细检查坐标轴上的刻度标注。

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

Gradient measures how steep a line is. It is the change in y divided by the change in x between two points on the line.

斜率衡量直线的倾斜程度。它是直线上两点之间 y 的变化量除以 x 的变化量。

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.

正斜率表示直线从左到右上升。负斜率表示直线从左到右下降。

Horizontal lines have gradient 0, and vertical lines have undefined gradient because the change in x is zero.

水平线的斜率为 0，垂直线的斜率不存在，因为 x 的变化量为零。

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4. Midpoint of a Line Segment | 线段的中点

The midpoint is the point exactly halfway between two given points. To find it, average the x-coordinates and average the y-coordinates.

中点是两个已知点正中间的点。求中点时，对 x 坐标取平均值，对 y 坐标取平均值。

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

For example, the midpoint of (2, 5) and (8, -1) is ((2 + 8) / 2, (5 + (-1)) / 2) = (5, 2).

例如，(2, 5) 和 (8, -1) 的中点是 ((2 + 8) / 2, (5 + (-1)) / 2) = (5, 2)。

This formula works for any two points, including points with negative coordinates. Add the coordinates first, then divide by 2.

该公式适用于任意两点，包括带负坐标的点。先把坐标相加，再除以 2。

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

The distance between two points comes from Pythagoras’ theorem. It is the square root of the sum of the squared horizontal and vertical differences.

两点间的距离来源于勾股定理。它是水平差和垂直差的平方和的正平方根。

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.

例如，(1, 2) 和 (4, 6) 之间的距离是 √((4 – 1)² + (6 – 2)²) = √(9 + 16) = √25 = 5。

Be careful with negative signs: when you subtract coordinates and square them, the result is always positive.

注意

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