Coordinate Geometry: Straight Lines | 坐标几何：直线

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

Coordinate geometry, also called analytic geometry, is a key IGCSE Mathematics topic that connects algebra and geometry. It allows you to describe points, lines, distances and midpoints using numbers and equations in the xy-plane.

坐标几何又称解析几何，是 IGCSE 数学的重要主题，它把代数与几何联系起来。它使你能用数字和方程在 xy 平面中描述点、直线、距离和中点。

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1. The Coordinate Plane | 坐标平面

The coordinate plane is formed by a horizontal x-axis and a vertical y-axis intersecting at the origin O(0, 0). Points are written as ordered pairs (x, y), where x is the horizontal position and y is the vertical position.

坐标平面由水平的 x 轴和竖直的 y 轴相交于原点 O(0, 0) 组成。点写成有序数对 (x, y)，其中 x 是水平位置，y 是竖直位置。

The plane is divided into four quadrants. In the first quadrant both x and y are positive; in the second quadrant x is negative and y is positive; in the third quadrant both are negative; and in the fourth quadrant x is positive and y is negative.

平面分为四个象限。第一象限中 x 和 y 都为正；第二象限中 x 为负、y 为正；第三象限两者都为负；第四象限中 x 为正、y 为负。

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2. Plotting and Reading Points | 描点与读点

To plot a point such as (3, -2), start at the origin, move 3 units to the right along the x-axis, then move 2 units down because y is negative. To read a point from a graph, count the horizontal and vertical distances from the origin.

要描出像 (3, -2) 这样的点，从原点出发，沿 x 轴向右移动 3 个单位，再因为 y 为负而向下移动 2 个单位。要从图形中读点，就从原点开始数水平距离和竖直距离。

Careful sign use is essential: moving left gives a negative x-coordinate, and moving down gives a negative y-coordinate. Many IGCSE mistakes come from confusing the order of coordinates; x always comes first.

正确使用符号非常重要：向左移动得到的 x 坐标为负，向下移动得到的 y 坐标为负。许多 IGCSE 错误来自混淆坐标顺序；x 总是写在前面。

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3. Gradient Between Two Points | 两点间的梯度

Gradient, often called slope, measures the steepness of a line. For two points A(x₁, y₁) and B(x₂, y₂), the gradient m is given by:

梯度（常称斜率）衡量直线的陡峭程度。对于两点 A(x₁, y₁) 和 B(x₂, y₂)，梯度 m 由下式给出：

m = (y₂ – y₁)/(x₂ – x₁)

The numerator is the change in y (rise), and the denominator is the change in x (run). A positive gradient means the line rises from left to right; a negative gradient means it falls from left to right.

分子是 y 的变化量（纵向增量），分母是 x 的变化量（横向增量）。梯度为正表示直线从左到右上升；梯度为负表示直线从左到右下降。

If the two points have the same y-coordinate, the line is horizontal and its gradient is 0. If the two points have the same x-coordinate, the line is vertical and its gradient is undefined.

如果两点的 y 坐标相同，则直线是水平的，其梯度为 0。如果两点的 x 坐标相同，则直线是竖直的，其梯度无定义。

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4. Distance and Midpoint Formulas | 距离与中点公式

The distance between two points A(x₁, y₁) and B(x₂, y₂) is found using Pythagoras’ theorem:

两点 A(x₁, y₁) 与 B(x₂, y₂) 之间的距离可用毕达哥拉斯定理求得：

d = √((x₂ – x₁)² + (y₂ – y₁)²)

The midpoint M of the segment AB is simply the average of the x-coordinates and the average of the y-coordinates:

线段 AB 的中点 M 就是 x 坐标的平均值和 y 坐标的平均值：

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

These two formulas appear frequently in IGCSE questions, especially when finding the perpendicular bisector of a line segment. Always simplify surds and leave exact answers unless the question asks for a decimal approximation.

这两个公式在 IGCSE 题目中经常出现，特别是在求线段的垂直平分线时。除非题目要求小数近似，否则应化简根式并保留精确答案。

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5. Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c

The most common form of a straight-line equation is y = mx + c, where m is the gradient and c is the y-intercept. The y-intercept is the point where the line crosses the y-axis, so its coordinates are (0, c).

直线方程最常见的形式是 y = mx + c，其中 m 是梯度，c 是 y 轴截距。y 轴截距是直线与 y 轴的交点，因此其坐标为 (0, c)。

For example, the line y = 2x – 5 has gradient 2 and crosses the y-axis at (0, -5). To sketch the line, plot the y-intercept first, then use the gradient to find another point.

例如，直线 y = 2x

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