📚 Coordinate Geometry: Gradients, Midpoints and Lengths | 坐标几何：斜率、中点与长度
Coordinate geometry links algebra with geometry by using coordinates to describe points, lines and shapes on the Cartesian plane. In IGCSE Mathematics, you need to calculate gradients, midpoints and lengths accurately, and you also need to use these ideas to write and interpret linear equations.
坐标几何通过坐标来描述笛卡尔平面上的点、直线和图形，将代数与几何联系起来。IGCSE 数学要求你准确计算斜率、中点和长度，并运用这些概念写出和解读直线方程。
1. The Coordinate Plane and Coordinates | 坐标平面与坐标
A point on a Cartesian plane is written as an ordered pair (x, y), where x gives the horizontal position from the origin and y gives the vertical position. The two perpendicular axes divide the plane into four quadrants, and the signs of x and y change from quadrant to quadrant.
笛卡尔平面上的点写作有序数对 (x, y)，其中 x 表示距离原点的水平位置，y 表示垂直位置。两条互相垂直的坐标轴将平面分为四个象限，x 和 y 的符号因象限而异。
| Quadrant | 象限 | Sign of x | x 的符号 | Sign of y | y 的符号 | Example | 示例 |
|---|---|---|---|
| I | 一 | Positive | 正 | Positive | 正 | (3, 4) |
| II | 二 | Negative | 负 | Positive | 正 | (−3, 4) |
| III | 三 | Negative | 负 | Negative | 负 | (−3, −4) |
| IV | 四 | Positive | 正 | Negative | 负 | (3, −4) |
Accurate plotting is the foundation of coordinate geometry. Negative coordinates require extra care when substituting into formulas because sign errors are common.
准确描点是坐标几何的基础。负坐标在代入公式时需要格外小心，因为符号错误非常常见。
2. Gradient of a Straight Line | 直线的斜率
The gradient of a straight line measures how steep it is. It is defined as the change in y divided by the change in x between two points on the line.
直线的斜率衡量它的倾斜程度。它定义为直线上两点之间 y 的变化量除以 x 的变化量。
m = (y₂ − y₁) ÷ (x₂ − x₁)
If the gradient is positive, the line rises from left to right; if it is negative, the line falls. A horizontal line has gradient 0, and a vertical line has an undefined gradient.
如果斜率为正，直线从左到右上升；如果为负，直线下降。水平线的斜率为 0，垂直线的斜率没有定义。
mple: For points A(2, 5) and B(6, 13), the gradient is m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2.
示例：对于点 A(2, 5) 和 B(6, 13)，斜率为 m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2。
3. Midpoint of Two Points | 两点之间的中点
The midpoint of two points A(x₁, y₁) and B(x₂, y₂) is found by averaging the x-coordinates and the y-coordinates separately. This gives the point exactly halfway between A and B.
两点 A(x₁, y₁) 和 B(x₂, y₂) 的中点，是将 x 坐标和 y 坐标分别取平均值。这样得到的是 A 和 B 正中间的点。
M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
(3 + 7) ÷ 2) = (4, 5). This is useful when you need to find the centre of a line segment.
示例：对于 A(2, 3) 和 B(6, 7)，中点是 ((2 + 6) ÷ 2, (3 + 7) ÷ 2) = (4, 5)。当你需要求线段中点时，这非常有用。
find a missing endpoint. If M is the midpoint of AB and you know A and M, then B = 2M − A.
中点公式也可以反过来求缺失的端点。如果 M 是 AB 的中点，且已知 A 和 M，那么 B = 2M − A。
4. Distance Between Two Points | 两点之间的距离
The length of a straight line segment is calculated using Pythagoras' theorem. The distance between A(x₁, y₁) and B(x₂, y₂) is the square root of the sum of the squared differences in x and y.
直线段的长度可以用勾股定理计算。A(x₁, y₁) 与 B(x₂, y₂) 之间的距离，等于 x 差值的平方与 y 差值的平方之和的平方根。
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) 与 (4, 6) 之间的距离为 √[(4 − 1)² + (6 − 2)²] = √9 + 16 = √25 = 5。
The order of the points does not matter because the differences are squared. However, always subtract the coordinates in the same order in both the x and y parts.
两点的先后顺序并不重要，因为差值会被平方。但是，在 x 部分和 y 部分中必须使用相同的顺序进行相减。
5. Equation of a Straight Line | 直线方程
adient and c is the y-intercept. If you know the gradient and one point, you can substitute to find c.
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。如果你知道斜率和一点坐标，可以代入来求 c。
y = mx + c
r useful form is y − y₁ = m(x − x₁), which is helpful when a point and gradient are known.
另一种常用形式是 y − y₁ = m(x − x₁)，当已知一点和斜率时，这种形式很方便。
nt 2 passing through (1, 3) has equation y − 3 = 2(x − 1), which simplifies to y = 2x + 1.
示例：斜率为 2 且经过 (1, 3) 的直线方程为 y − 3 = 2(x − 1)，化简后得到 y = 2x + 1。
+ by = c. You can rearrange this into y = mx + c to identify the gradient and y-intercept.
有时直线方程以 ax + by = c 的形式给出。你可以把它整理成 y = mx + c 的形式，从而识别斜率和 y 轴截距。
For example, 2x + 3y = 6 rearranges to 3y = −2x + 6 or y = −2/3 x + 2.
例如，2x + 3y = 6 可以整理为 3y = −2x + 6，即 y = −2/3 x + 2。
6. Parallel and Perpendicular Lines | 平行线与垂直线
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to give −1, meaning one gradient is the negative reciprocal of the other.
平行线的斜率相同。垂直线的斜率乘积为 −1，也就是说一个斜率是另一个斜率的负倒数。
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1
For example, if a line has gradient 3, any line parallel to it also has gradient 3. A line perpendicular to it has gradient −1/3.
例如，如果一条直线的斜率为 3，那么与它平行的任何直线斜率也为 3。与它垂直的直线斜率为 −1/3。
heir gradients are 0 and undefined respectively, and they are perpendicular to each other.
水平线的形式为 y = k，垂直线的形式为 x = h。它们的斜率分别为 0 和没有定义，并且它们互相垂直。
7. Intercepts and Sketching Straight Lines | 截距与直线草图
The x-intercept
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