IGCSE AQA Maths: Coordinate Geometry Key Points | IGCSE AQA 数学：坐标几何考点精讲

📚 IGCSE AQA Maths: Coordinate Geometry Key Points | IGCSE AQA 数学：坐标几何考点精讲

Coordinate geometry (also known as straight-line graphs) is a core topic in the IGCSE AQA Mathematics specification. It connects algebraic equations with graphical representations, allowing you to analyse points, lines, gradients, distances, and intersections. Mastering these techniques is essential for success on both the foundation and higher tier papers.

坐标几何（也称直线图像）是 IGCSE AQA 数学考试的核心内容。它将代数方程与图形表示联系起来，使你能够分析点、直线、梯度、距离和交点。掌握这些技巧对基础卷和进阶卷的成功至关重要。

1. Understanding Gradient | 理解梯度

The gradient of a line measures its steepness and direction. It tells you how much y changes for every unit increase in x. A larger absolute value means a steeper line.

梯度度量直线的陡峭程度和方向。它表示 x 每增加一个单位，y 的变化量。绝对值越大，直线越陡。

If the line slopes upwards from left to right, the gradient is positive; if it slopes downwards, the gradient is negative. A horizontal line has a gradient of 0, and a vertical line has an undefined gradient (division by zero).

如果直线从左到右向上倾斜，梯度为正；向下倾斜，梯度为负。水平线的梯度为 0，铅垂线的梯度无定义（除以零）。

2. Calculating Gradient Between Two Points | 计算两点间的梯度

Given two points (x₁, y₁) and (x₂, y₂) on a straight line, the gradient m is found using the formula:

m = (y₂ − y₁) / (x₂ − x₁)

给定直线上两点 (x₁, y₁) 和 (x₂, y₂)，梯度 m 的计算公式为：

m = (y₂ − y₁) / (x₂ − x₁)

Always label your points carefully: (x₁, y₁) and (x₂, y₂). Subtracting in the wrong order gives the negative of the correct gradient. The formula is sometimes remembered as ‘rise over run’.

务必仔细标注你的点： (x₁, y₁) 和 (x₂, y₂)。减法顺序错误会得到正确梯度的相反数。这个公式有时被记作 ‘纵差除以横差’。

Example: For points A(2, 5) and B(6, 13), m = (13 − 5) / (6 − 2) = 8 / 4 = 2.

示例：对于点 A(2, 5) 和 B(6, 13)， m = (13 − 5) / (6 − 2) = 8 / 4 = 2。

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

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by an application of Pythagoras’ theorem:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离 d 由勾股定理得出：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Note that the distance is always a positive value, and you should leave exact answers in surd form (e.g. √13) unless a decimal approximation is requested.

注意距离总是正值，除非题目要求取近似小数，否则应保留根号形式（如 √13）。

It does not matter which point is labelled (x₁, y₁) and which is (x₂, y₂), because the squares eliminate any sign differences.

哪个点记为 (x₁, y₁)、哪个点记为 (x₂, y₂) 并不重要，因为平方会消除符号差异。

4. Midpoint of a Line Segment | 线段的中点

The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) has coordinates found by averaging the x-coordinates and averaging the y-coordinates:

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

连接 (x₁, y₁) 与 (x₂, y₂) 的线段的中点 M 的坐标由 x 坐标的平均值和 y 坐标的平均值给出：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Midpoints are especially useful when working with symmetry, perpendicular bisectors, or finding the centre of a line segment.

在处理对称性、垂直平分线或求线段中心时，中点特别有用。

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

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

线性方程最常见的形式是 y = mx + c，其中 m 是梯度，c 是 y 轴截距（直线与 y 轴交于点 (0, c)）。

To find the equation of a line given its gradient and one point, substitute the point’s coordinates to solve for c. For example, a line with gradient 3 passing through (2, 7): y = 3x + c → 7 = 3(2) + c → c = 1, so y = 3x + 1.

要由梯度和一个点求直线方程，代入该点坐标解出 c。例如，一条梯度为 3 且过点 (2, 7) 的直线： y = 3x + c → 7 = 3(2) + c → c = 1，因此方程为 y = 3x + 1。

6. Point-Slope Form | 点斜式形式

When you are given a point (x₁, y₁) on a line and its gradient m, you can write the equation directly using the point-slope form:

y − y₁ = m(x − x₁)

当你已知直线上一点 (x₁, y₁) 及其梯度 m 时，可直接使用点斜式写出方程：

y − y₁ = m(x − x₁)

This form is particularly helpful when you later need to rearrange into y = mx + c or ax + by + c = 0. It avoids the extra step of solving for c.

当你之后需要整理成 y = mx + c 或 ax + by + c = 0 的形式时，此式尤其方便，它省去了求解 c 的额外步骤。

7. Finding Intercepts | 求截距

The y-intercept is found by setting x = 0 in the equation of the line. The x-intercept is found by setting y = 0.

y 轴截距通过令直线方程中 x = 0 求得；x 轴截距通过令 y = 0 求得。

For example, for the line y = 2x − 6, the y-intercept is (0, −6); setting y = 0 gives 0 = 2x − 6 → x = 3, so the x-intercept is (3, 0).

例如，对于直线 y = 2x − 6，y 轴截距为 (0, −6)；令 y = 0 得 0 = 2x − 6 → x = 3，故 x 轴截距为 (3, 0)。

Intercepts are often used to sketch straight-line graphs quickly by plotting the two points and joining them.

截距常用于快速绘制直线图像：标出两点并连接即可。

8. Parallel and Perpendicular Lines | 平行与垂直直线

Parallel lines have the same gradient: m₁ = m₂. If two lines are parallel, their gradients are equal.

平行直线具有相同的梯度： m₁ = m₂。如果两直线平行，它们的梯度相等。

Perpendicular lines have gradients that are negative reciprocals of each other: m₁ × m₂ = −1. Equivalently, m₂ = −1 / m₁.

垂直直线的梯度互为负倒数： m₁ × m₂ = −1。等价地， m₂ = −1 / m₁。

Example: If a line has gradient 2/5, a line perpendicular to it has gradient −5/2. Be careful with signs, particularly when the original gradient is negative or a fraction.

示例：若一条直线的梯度为 2/5，则垂直于它的直线的梯度为 −5/2。注意符号，特别是当原梯度为负或分数时。

9. Intersection of Two Lines | 两直线的交点

The intersection point of two lines is found by solving their equations simultaneously. This often involves setting the expressions for y equal to each other if both are in the form y = …, or using substitution/elimination.

两直线的交点通过联立求解它们的方程得到。如果两者都写成 y = … 的形式，则常令两式相等；否则使用代入法或消元法。

For instance, lines y = 2x + 1 and y = −x + 7 intersect where 2x + 1 = −x + 7 → 3x = 6 → x = 2. Substitute x = 2 into either equation to get y = 5, so the intersection is (2, 5).

例如，直线 y = 2x + 1 与 y = −x + 7 的交点满足 2x + 1 = −x + 7 → 3x = 6 → x = 2。将 x = 2 代入任一方程得 y = 5，故交点为 (2, 5)。

Always check your solution in the other equation to ensure it satisfies both.

务必用另一方程检验你的解，确保它同时满足两式。

10. Perpendicular Bisector of a Segment | 线段的垂直平分线

The perpendicular bisector of a segment AB is a line that passes through the midpoint of AB and is perpendicular to AB. Its equation can be found in three steps:

线段 AB 的垂直平分线是一条通过 AB 中点且垂直于 AB 的直线。求其方程可分三步：

Step 1: Find the midpoint M of AB. Step 2: Find the gradient of AB, then use m_perp × m_AB = −1 to get the perpendicular gradient. Step 3: Use point-slope form with M and the perpendicular gradient, then rearrange to the desired form.

第一步：求 AB 的中点 M。第二步：求 AB 的梯度，然后利用 m_perp × m_AB = −1 得到垂直梯度。第三步：利用点斜式，代入 M 和垂直梯度，再整理为所需形式。

Example: For A(1, 2) and B(5, 6), midpoint M is (3, 4). Gradient m_AB = (6−2)/(5−1) = 1, so m_perp = −1. Equation: y − 4 = −1(x − 3) → y = −x + 7.

示例： A(1, 2) 和 B(5, 6)，中点 M 为 (3, 4)。梯度 m_AB = (6−2)/(5−1) = 1，故 m_perp = −1。方程： y − 4 = −1(x − 3) → y = −x + 7。

11. Verifying Points and Exam Tips | 验证点与考试技巧

To check whether a point lies on a given line, substitute its x-coordinate into the line’s equation; the resulting y should equal the point’s y-coordinate. If they do, the point lies on the line.

要检查一个点是否在给定直线上，将其 x 坐标代入直线方程；计算得到的 y 应与点的 y 坐标相等。若相等，则该点在直线上。

Always be comfortable rearranging equations into the form y = mx + c, as it immediately reveals the gradient and y-intercept. When tackling problems involving coordinate geometry, sketching a rough diagram can help you avoid sign errors and visualise relationships.

一定要熟练掌握将方程整理为 y = mx + c 的形式，因为这会立即显示梯度和 y 轴截距。处理坐标几何问题时，画一个粗略的示意图有助于避免符号错误并直观呈现关系。

Memorise the key formulas: gradient, distance, midpoint, and perpendicular gradient.

熟记关键公式：梯度、距离、中点以及垂直梯度。
Watch for horizontal and vertical lines: y = constant has gradient 0; x = constant has undefined gradient.

注意水平和竖直线： y = 常数的梯度为 0； x = 常数的梯度无定义。
Practice with fractions and negative numbers to avoid common mistakes.

多做涉及分数和负数的练习，避免常见错误。

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IGCSE AQA Maths: Coordinate Geometry Key Points | IGCSE AQA 数学：坐标几何 考点精讲