📚 IGCSE Coordinate Geometry: Gradient, Distance, Midpoint and Equations of Lines | IGCSE 坐标几何：斜率、距离、中点与直线方程
Coordinate geometry is a core topic in IGCSE Mathematics that connects algebra with geometry. It gives you a way to describe points, lines and distances using coordinates on the Cartesian plane. You will need to calculate gradients, midpoints, distances and equations of straight lines confidently for both Paper 2 and Paper 4.
坐标几何是 IGCSE 数学的核心主题之一，它将代数与几何联系起来。它提供了一种在笛卡尔平面上使用坐标描述点、线和距离的方法。你需要熟练计算斜率、中点、距离和直线方程，以应对 Paper 2 和 Paper 4 的考试。
1. The Coordinate Plane | 坐标平面
The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They meet at the origin, written as (0, 0). A point is identified by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
笛卡尔平面由两条互相垂直的数轴组成：x 轴（水平）和 y 轴（垂直）。它们相交于原点，写作 (0, 0)。一个点由有序对 (x, y) 确定，其中 x 是距原点的水平距离，y 是垂直距离。
For example, the point (−3, 4) means move 3 units to the left and 4 units up from the origin. The first value is always the x-coordinate; the second is always the y-coordinate.
例如，点 (−3, 4) 表示从原点向左移动 3 个单位、向上移动 4 个单位。第一个值始终是 x 坐标，第二个值始终是 y 坐标。
2. Plotting Points and Reading Coordinates | 描点与读取坐标
To plot a point such as (2, −5), start at the origin, move 2 units right along the x-axis, then move 5 units down parallel to the y-axis. Mark the position clearly with a cross or dot.
要描画点 (2, −5)，从原点开始，沿 x 轴向右移动 2 个单位，再平行于 y 轴向下移动 5 个单位。用叉号或点清楚地标出该位置。
When reading coordinates from a grid, always read the x-value first and the y-value second. A common exam error is reversing the order, so check the scale on each axis before answering.
从网格中读取坐标时，始终先读 x 值，再读 y 值。一个常见的考试错误是顺序颠倒，因此在作答前要检查每个轴上的刻度。
3. Distance Between Two Points | 两点之间的距离
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
The distance between two points A(x₁, y₁) and B(x₂, y₂) is found using Pythagoras' theorem. The difference in x-coordinates gives the horizontal leg, and the difference in y-coordinates gives the vertical leg.
两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离利用勾股定理求得。x 坐标之差给出水平直角边，y 坐标之差给出竖直直角边。
Example: for A(1, 2) and B(4, 6), the distance is √[(4 − 1)² + (6 − 2)²] = √3² + 4² = √25 = 5 units.
例如：对于 A(1, 2) 和 B(4, 6)，距离为 √[(4 − 1)² + (6 − 2)²] = √3² + 4² = √25 = 5 个单位。
Always leave exact answers in surd form unless the question asks for a decimal. If a diagram is given, identify the horizontal and vertical changes carefully, including negative signs.
除非题目要求小数，否则将精确答案保留为根号形式。如果给出图形，请仔细确定水平和竖直变化，包括负号。
4. Midpoint of a Line Segment | 线段的中点
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
The midpoint of a segment is simply the average of the x-coordinates and the average of the y-coordinates. This is useful for finding the centre of a line segment or checking symmetry.
线段的中点就是 x 坐标的平均值和 y 坐标的平均值。这对于求线段的中心或检查对称性非常有用。
ample: for A(3, −2) and B(−1, 8), the midpoint is ((3 + (−1)) / 2, (−2 + 8) / 2) = (1, 3).
例如：对于 A(3, −2) 和 B(−1, 8)，中点为 ((3 + (−1)) / 2, (−2 + 8) / 2) = (1, 3)。
Be careful with negative signs: adding a negative coordinate is the same as subtracting its absolute value. Many marks are lost through simple sign errors in midpoint calculations.
注意负号：将负坐标相加等同于减去它的绝对值。在中点计算中，许多失分源于简单的符号错误。
5. Gradient of a Straight Line | 直线的斜率
m = (y₂ − y₁) / (x₂ − x₁)
Gradient measures the steepness of a line and is often written as m. It is the change in y divided by the change in x, sometimes called 'rise over run'.
斜率衡量直线的倾斜程度，通常写作 m。它是 y 的变化量除以 x 的变化量，有时称为“铅垂变化与水平变化之比”。
A positive gradient means the line rises from left to right; a negative gradient means it falls. A horizontal line has gradient 0, and a vertical
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