GCSE Maths: Coordinate Geometry Revision | GCSE 数学：坐标几何考点精讲

📚 GCSE Maths: Coordinate Geometry Revision | GCSE 数学：坐标几何考点精讲

Coordinate geometry, also known as analytic geometry, is a key topic in GCSE Maths. It combines algebra with geometry, allowing you to describe lines, distances, and midpoints using coordinates. This revision guide covers all the essential concepts and formulas you need to know, from finding the gradient of a line to determining equations and intercepts. Master these skills to confidently tackle exam questions involving straight-line graphs.

坐标几何，也称解析几何，是 GCSE 数学中的一个重要主题。它将代数与几何相结合，使你能够用坐标描述直线、距离和中点。本复习指南涵盖所有必考概念和公式，从求直线斜率到确定方程和截距。掌握这些技能，自信应对涉及直线图像的考试题目。

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

The Cartesian plane is a two-dimensional surface formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. Any point on the plane can be uniquely represented by an ordered pair (x, y), where x is the horizontal distance from the origin and y is the vertical distance.

笛卡尔平面是由两条垂直数轴构成的二维平面：水平 x 轴和垂直 y 轴。平面上任意一点都可以用唯一的有序数对 (x, y) 表示，其中 x 是到原点的水平距离，y 是垂直距离。

The origin, labelled (0, 0), is the point where the axes intersect. The axes divide the plane into four quadrants. Quadrant I has positive x and y, Quadrant II has negative x and positive y, Quadrant III has both negative, and Quadrant IV has positive x and negative y. Understanding quadrants is essential for plotting points correctly.

原点，标记为 (0, 0)，是两轴交点。坐标轴将平面分为四个象限。第一象限 x 和 y 均为正，第二象限 x 负 y 正，第三象限两者均为负，第四象限 x 正 y 负。理解象限对于正确描点至关重要。

2. Distance Between Two Points | 两点间距离

The distance between two points A(x₁, y₁) and B(x₂, y₂) can be found using Pythagoras’ theorem. The horizontal difference is (x₂ − x₁) and the vertical difference is (y₂ − y₁). The straight-line distance is the hypotenuse of a right-angled triangle formed by these differences:

两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离可用勾股定理求得。水平差为 (x₂ − x₁)，垂直差为 (y₂ − y₁)。直线距离就是以这些差值为直角边的直角三角形的斜边：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula always gives a positive distance. It does not matter which point is taken as first; the squared differences eliminate negatives. For example, the distance between (1, 2) and (4, 6) is √[(4−1)² + (6−2)²] = √[3² + 4²] = √[9+16] = √25 = 5 units.

该公式总是给出正距离。哪个点作为第一个无关紧要；平方差消除了负号。例如，(1, 2) 和 (4, 6) 之间的距离为 √[(4−1)² + (6−2)²] = √[3² + 4²] = √[9+16] = √25 = 5 单位。

3. Midpoint of a Line Segment | 线段的中点

The midpoint of a line segment connecting two points (x₁, y₁) and (x₂, y₂) is simply the average of the x-coordinates and the average of the y-coordinates. The formula is:

连接两点 (x₁, y₁) 和 (x₂, y₂) 的线段的中点，就是 x 坐标的平均值和 y 坐标的平均值。公式如下：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

For instance, the midpoint between (−2, 5) and (4, −1) is ((−2+4)/2, (5+(−1))/2) = (2/2, 4/2) = (1, 2). This concept is often used in questions requiring you to find the centre of a line segment or to verify that a point lies exactly halfway.

例如，(−2, 5) 和 (4, −1) 的中点坐标为 ((−2+4)/2, (5+(−1))/2) = (2/2, 4/2) = (1, 2)。这个概念经常用于要求找线段中心或验证某点恰好位于一半位置的题目。

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

The gradient (or slope) of a line measures its steepness. It is defined as the change in y divided by the change in x between any two distinct points on the line. For points (x₁, y₁) and (x₂, y₂), the gradient m is:

直线斜率（或称坡度）衡量其倾斜程度。它定义为直线上任意两个不同点之间 y 的变化量除以 x 的变化量。对于点 (x₁, y₁) 和 (x₂, y₂)，斜率 m 为：

m = (y₂ − y₁) / (x₂ − x₁)

A positive gradient means the line slopes upwards from left to right, a negative gradient slopes downwards, a zero gradient is a horizontal line, and an undefined gradient (where denominator is zero) corresponds to a vertical line.

正斜率表示直线从左到右向上倾斜，负斜率向下倾斜，斜率为零是水平线，斜率未定义（分母为零）对应垂直线。

You can also determine the gradient from the equation y = mx + c, where m is the gradient. If the equation is in the form ax + by + c = 0, rearrange it to solve for y.

你也可以从方程 y = mx + c 中确定斜率，其中 m 就是斜率。如果方程形式为 ax + by + c = 0，可将其变形解出 y。

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

The most common form for the equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept (the point where the line crosses the y-axis). This is called the slope-intercept form.

最常见的直线方程形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴交点的 y 坐标）。这称为斜截式。

For example, y = 2x + 3 has gradient 2 and crosses the y-axis at (0, 3). The line y = −x + 5 has gradient −1 and y-intercept 5. To sketch the line, plot the y-intercept, then use the gradient to find another point. A gradient of 2 means for every 1 unit right, go 2 units up.

例如，y = 2x + 3 的斜率为 2，与 y 轴交于 (0, 3)。直线 y = −x + 5 的斜率为 −1，y 轴截距为 5。要画出直线，先标出 y 轴截距，然后利用斜率找到另一个点。斜率为 2 表示每向右 1 单位，向上 2 单位。

You may also see the general form ax + by + c = 0. To convert to y = mx + c, isolate y: by = −ax − c, then y = (−a/b)x − c/b, so m = −a/b and c = −c/b.

你还可能见到一般式 ax + by + c = 0。要转换为 y = mx + c，把 y 单独解出：by = −ax − c，从而 y = (−a/b)x − c/b，因此 m = −a/b，c = −c/b。

6. Finding the Equation from Two Points | 由两点求直线方程

To find the equation of a line passing through two given points, follow these steps: First, calculate the gradient m using m = (y₂ − y₁)/(x₂ − x₁). Then, substitute the coordinates of one point and m into the equation y − y₁ = m(x − x₁) or directly into y = mx + c to solve for c.

要求经过两给定点的直线方程，按以下步骤：首先，用 m = (y₂ − y₁)/(x₂ − x₁) 计算斜率 m。然后，将其中一个点的坐标和 m 代入 y − y₁ = m(x − x₁) 或直接代入 y = mx + c 来解出 c。

For example, if the line passes through (2, 5) and (4, 9): m = (9−5)/(4−2) = 4/2 = 2. Using (2, 5): y − 5 = 2(x − 2) → y − 5 = 2x − 4 → y = 2x + 1. Alternatively, substitute (2,5) into y = 2x + c: 5 = 2(2) + c → c = 1, giving y = 2x + 1.

例如，直线经过 (2, 5) 和 (4, 9)：m = (9−5)/(4−2) = 4/2 = 2。使用点 (2, 5)：y − 5 = 2(x − 2) → y − 5 = 2x − 4 → y = 2x + 1。或者代入 (2,5) 到 y = 2x + c：5 = 2(2) + c → c = 1，得到 y = 2x + 1。

Always check that both points satisfy the final equation.

务必检查两个点是否都满足最终方程。

7. Parallel and Perpendicular Lines | 平行线与垂直线

Parallel lines have the same gradient. Two lines are parallel if and only if m₁ = m₂. For example, y = 3x + 4 and y = 3x − 2 are parallel because both have gradient 3.

平行线具有相同斜率。两直线平行当且仅当 m₁ = m₂。例如，y = 3x + 4 和 y = 3x − 2 平行，因为二者的斜率都是 3。

Perpendicular lines meet at a right angle (90°). Their gradients satisfy the relationship m₁ × m₂ = −1, or equivalently m₂ = −1/m₁. For example, if a line has gradient 2, a line perpendicular to it will have gradient −1/2. The line y = 2x + 1 is perpendicular to y = −½ x + 5.

垂直线交于直角（90°）。它们的斜率满足关系 m₁ × m₂ = −1，或等价地，m₂ = −1

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GCSE Maths: Coordinate Geometry Revision | GCSE 数学：坐标几何 考点精讲