Straight Line Graphs: Gradient, Equations and Midpoints | 直线图像：斜率、方程与中点

📚 Straight Line Graphs: Gradient, Equations and Midpoints | 直线图像：斜率、方程与中点

In IGCSE Mathematics, coordinate geometry brings algebra and geometry together on the xy-plane. You need to be confident finding gradients, midpoints, distances, and equations of straight lines because these skills appear in both Paper 2 and Paper 4, often linked with simultaneous equations and real-world graphs.

在 IGCSE 数学中，坐标几何将代数与几何结合在 xy 平面上。你需要熟练求斜率、中点、距离和直线方程，因为这些技能在试卷二和试卷四中经常出现，并常与联立方程和实际问题图像结合考查。

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1. The Coordinate Plane and Plotting Points | 坐标平面与描点

A point on the plane is written as an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate. The x-axis is horizontal, the y-axis is vertical, and they meet at the origin (0, 0).

平面上的点写成有序数对 (x, y)，其中 x 是水平坐标，y 是垂直坐标。x 轴是水平轴，y 轴是垂直轴，它们相交于原点 (0, 0)。

Plotting points accurately is the first step in drawing any straight-line graph. For example, the point (3, −2) is 3 units to the right of the origin and 2 units down.

准确描点是绘制任何直线图像的第一步。例如，点 (3, −2) 位于原点右侧 3 个单位、下方 2 个单位处。

• The x-coordinate is always written first: (x, y).

  x 坐标总是先写：(x, y)。

• Negative x values are to the left; negative y values are below the origin.

  x 为负时点在左侧；y 为负时点在原点下方。

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2. Gradient Between Two Points | 两点间的斜率

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

斜率度量一条直线的倾斜程度。它是直线上任意两点之间垂直变化与水平变化之比。

For two points (x₁, y₁) and (x₂, y₂), the gradient m is given by:

对于两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m 的公式为：

m = (y₂ − y₁) ÷ (x₂ − x₁)

It makes no difference which point is chosen first, as long as you subtract the y-coordinates and x-coordinates in the same order.

先选哪个点没有关系，只要 y 坐标和 x 坐标按相同的顺序相减即可。

Example: Find the gradient of the line through (2, 5) and (6, 13).

例题：求经过 (2, 5) 和 (6, 13) 的直线的斜率。

m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2

A positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards.

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。

Note: a vertical line has an undefined gradient because the horizontal change is zero. A horizontal line has gradient zero.

注意：竖直线的斜率无定义，因为水平变化为零。水平线的斜率为零。

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3. Midpoint of a Line Segment | 线段的中点

The midpoint is the point exactly halfway between two given points. Its coordinates are the averages of the x-coordinates and y-coordinates.

中点是恰好位于两个给定点中间的点。它的坐标是两个点 x 坐标和 y 坐标的平均值。

For points (x₁, y₁) and (x₂, y₂), the midpoint M is:

对于点 (x₁, y₁) 和 (x₂, y₂)，中点 M 为：

M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Example: The midpoint of (−3, 4) and (7, 10) is:

例题：(−3, 4) 和 (7, 10) 的中点为：

M = ((−3 + 7) ÷ 2, (4 + 10) ÷ 2) = (2, 7)

This formula is often used when a question gives one endpoint and the midpoint, then asks you to find the other endpoint.

当题目给出一个端点和中点，再要求你求另一个端点时，经常会用到这个公式。

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

The distance between two points is found using Pythagoras’ theorem. Treat the horizontal and vertical differences as the two shorter sides of a right-angled triangle.

两点间的距离使用毕达哥拉斯定理求解。将水平差和垂直差看作直角三角形的两条短边。

For points (x₁, y₁) and (x₂, y₂), the distance d is:

对于点 (x₁, y₁) 和 (x₂, y₂)，距离 d 为：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Example: Find the distance between (1, 2) and (5, 5).

例题：求 (1, 2) 和 (5, 5) 之间的距离。

d = √[(5 − 1)² + (5 − 2)²] = √(4² + 3²) = √(16 + 9) = √25 = 5

Distance is always non-negative. Do not forget to

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