Coordinate Geometry Key Points | 坐标几何考点精讲

📚 Coordinate Geometry Key Points | 坐标几何考点精讲

Coordinate geometry is a central topic in the Edexcel IGCSE Mathematics syllabus. It links algebraic equations to geometric points and lines on the Cartesian plane. Mastering formulas for distance, midpoint, gradient, and equations of straight lines, together with the conditions for parallel and perpendicular lines, is essential for problem-solving in both pure mathematics and real-life contexts.

坐标几何是 Edexcel IGCSE 数学大纲中的核心主题。它将代数方程与笛卡尔平面上的几何点和线联系起来。掌握距离、中点、斜率以及直线方程的公式，以及平行和垂直线的条件，对于解决纯数学问题和实际情境问题都至关重要。

1. Distance Formula | 距离公式

The distance between two points A(x₁, y₁) and B(x₂, y₂) is found by applying Pythagoras’ theorem. The formula is d = √[(x₂ – x₁)² + (y₂ – y₁)²]. Always subtract the coordinates in the same order and square the differences before adding.

两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离可通过勾股定理求得。公式为 d = √[(x₂ – x₁)² + (y₂ – y₁)²]。务必将坐标差按相同顺序相减，先平方再相加。

For example, the distance between (1, 3) and (4, 7) is √[(4-1)² + (7-3)²] = √(9+16) = √25 = 5.

例如，(1, 3) 和 (4, 7) 之间的距离为 √[(4-1)² + (7-3)²] = √(9+16) = √25 = 5。

2. Midpoint Formula | 中点公式

The midpoint M of the line segment joining (x₁, y₁) and (x₂, y₂) is simply the average of the x-coordinates and the average of the y-coordinates: M = ( (x₁+x₂)/2 , (y₁+y₂)/2 ). This can be visualised as finding the centre of the segment.

连接 (x₁, y₁) 和 (x₂, y₂) 的线段中点 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 )。这可以直观理解为求线段的中心。

For instance, the midpoint of (2, -1) and (6, 5) is ( (2+6)/2 , (-1+5)/2 ) = (4, 2).

例如，(2, -1) 和 (6, 5) 的中点是 ( (2+6)/2 , (-1+5)/2 ) = (4, 2)。

3. Gradient (Slope) of a Straight Line | 直线的斜率

The gradient m measures the steepness of a line. Given two points (x₁, y₁) and (x₂, y₂), m = (y₂ – y₁)/(x₂ – x₁). The gradient is positive if the line rises from left to right, negative if it falls, zero for a horizontal line, and undefined for a vertical line (division by zero).

斜率 m 衡量直线的倾斜程度。给定两点 (x₁, y₁) 和 (x₂, y₂)，m = (y₂ – y₁)/(x₂ – x₁)。若直线从左到右上升，斜率为正；若下降则为负；水平线斜率为零；垂直线斜率无定义（除以零）。

Always use the change in y over the change in x. In the equation y = mx + c, m also represents the tangent of the angle the line makes with the positive x-axis.

始终用 y 的变化量除以 x 的变化量。在方程 y = mx + c 中，m 也表示直线与正 x 轴夹角的正切值。

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

The most common form for 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, i.e., when x = 0). Once you know m and c, you can write the equation directly.

直线最常见的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距（即直线与 y 轴的交点，x = 0 时的 y 值）。一旦知道 m 和 c，就能直接写出方程。

For example, a line with gradient 3 passing through (0, -2) has equation y = 3x – 2.

例如，斜率为 3 且通过 (0, -2) 的直线方程为 y = 3x – 2。

Another form is ax + by + c = 0, which can be rearranged to y = mx + c. Edexcel questions often expect you to give the final answer in the form ax + by + c = 0 with integer coefficients.

另一种形式是 ax + by + c = 0，它可以整理为 y = mx + c。Edexcel 考题通常要求最终答案写成 ax + by + c = 0 的形式，且系数为整数。

5. Point-Gradient Form | 点斜式

If you know the gradient m and one point (x₁, y₁) on the line, the equation can be written as y – y₁ = m(x – x₁). This is extremely useful when the y-intercept is not immediately known. You then simplify to the required form.

如果知道斜率 m 和直线上一点 (x₁, y₁)，方程可写为 y – y₁ = m(x – x₁)。当 y 截距未知时，这个公式非常有用。然后将其化简为所需形式。

Suppose a line has gradient -2 and passes through (3, 5). The equation is y – 5 = -2(x – 3), which simplifies to y = -2x + 11 or 2x + y – 11 = 0.

假设一条直线的斜率为 -2，且经过点 (3, 5)。其方程为 y – 5 = -2(x – 3)，化简可得 y = -2x + 11 或 2x + y – 11 = 0。

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

When two points (x₁, y₁) and (x₂, y₂) are given, first calculate the gradient m = (y₂ – y₁)/(x₂ – x₁). Then substitute m and one of the points into y – y₁ = m(x – x₁). Alternatively, use the two-point form: (y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁), but the first method is safer and reduces errors.

当给出两点 (x₁, y₁) 和 (x₂, y₂) 时，先计算斜率 m = (y₂ – y₁)/(x₂ – x₁)，然后将 m 和其中一点代入 y – y₁ = m(x – x₁)。也可使用两点式：(y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁)，但第一种方法更保险，能减少错误。

For instance, to find the line through (2, 3) and (4, 7): m = (7-3)/(4-2) = 2. Using (2,3): y – 3 = 2(x – 2) → y = 2x – 1.

例如，求过 (2, 3) 和 (4, 7) 的直线：m = (7-3)/(4-2) = 2。利用 (2,3)：y – 3 = 2(x – 2) → y = 2x – 1。

7. Parallel Lines | 平行线

Two distinct lines are parallel if and only if they have the same gradient. So, for lines y = m₁x + c₁ and y = m₂x + c₂, parallelism means m₁ = m₂. In the form ax + by + c = 0, the ratio a:b determines the gradient, so parallel lines have proportional a and b.

两条不同的直线平行，当且仅当它们的斜率相等。因此，对于直线 y = m₁x + c₁ 和 y = m₂x + c₂，平行意味着 m₁ = m₂。在 ax + by + c = 0 形式中，a:b 的比值决定了斜率，因此平行线的 a 和 b 成比例。

To find a line parallel to y = 5x + 2 passing through (1, 4), keep the gradient 5 and use point-gradient form: y – 4 = 5(x – 1) → y = 5x – 1.

要找到一条平行于 y = 5x + 2 且通过 (1, 4) 的直线，保持斜率 5，并使用点斜式：y – 4 = 5(x – 1) → y = 5x – 1。

8. Perpendicular Lines | 垂直线

Two lines are perpendicular if the product of their gradients is -1, i.e., m₁ × m₂ = -1. This is equivalent to m₂ = -1/m₁ (the negative reciprocal). For example, if one line has gradient 3, a line perpendicular to it has gradient -1/3.

两条直线垂直，如果它们斜率的乘积为 -1，即 m₁ × m₂ = -1。这等价于 m₂ = -1/m₁（负倒数）。例如，若一条直线斜率为 3，则垂直于它的直线斜率为 -1/3。

Be careful: horizontal lines (gradient 0) are perpendicular to vertical lines (undefined gradient). Questions often ask for the equation of a perpendicular bisector – combine midpoint and perpendicular gradient.

注意：水平线（斜率为 0）与垂直线（斜率无定义）互相垂直。考题常要求写出垂直平分线的方程——需结合中点和垂直斜率。

9. Intersection of Two Lines | 两直线的交点

To find where two lines intersect, solve their equations simultaneously. If the lines are given as y = m₁x + c₁ and y = m₂x + c₂, set them equal: m₁x + c₁ = m₂x + c₂ and solve for x. Then substitute back to find y. If the lines are parallel, there is no intersection; if they are the same line, there are infinitely many intersection points.

要求两条直线的交点，需联立方程求解。若直线为 y = m₁x + c₁ 和 y = m₂x + c₂，令它们相等：m₁x + c₁ = m₂x + c₂，解出 x，再回代求 y。若两线平行，则无交点；若为同一直线，则有无数交点。

Alternatively, use elimination or substitution with the ax + by + c = 0 form. This skill is essential for finding vertices of polygons defined by lines.

或者，使用 ax + by + c = 0 的形式进行消元或代入。这个技巧对于求由直线定义的多边形的顶点至关重要。

10. Ratio of Line Segments (Internal Division) | 线段比例（内分点）

If a point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then the coordinates of P are P = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) ). This formula is derived from similar triangles and is often tested in word problems.

若点 P 将连接 A(x₁, y₁) 和 B(x₂, y₂) 的线段按比例 m:n 内分，那么 P 的坐标为 P = ( (mx₂ + nx₁)/(m+n) , (my₂ + ny₁)/(m+n) )。该公式由相似三角形推导而来，常在应用题中考查。

When P is the midpoint, m = n = 1, and the formula simplifies to the midpoint formula. Ensure you assign m and n correctly according to the order AP:PB.

当 P 为中点时，m = n = 1，公式即简化为中点公式。务必根据 AP:PB 的顺序正确分配 m 和 n。

11. Area of Triangle using Coordinates | 用坐标求三角形面积

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) can be found using the formula: Area = ½ | x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) |. The absolute value ensures the area is positive. This is sometimes presented as a determinant in a 3×3 matrix, but the given form is sufficient for IGCSE.

顶点为 (x₁, y₁)、(x₂, y₂)、(x₃, y₃) 的三角形面积可用公式：面积 = ½ | x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) |。绝对值确保面积为正。有时会以 3×3 矩阵行列式的形式给出，但对于 IGCSE，上述形式已足够。

If the three points are collinear, the area will be zero — a useful check for collinearity. Practice using the formula with coordinates given in any order; the absolute value handles sign changes.

如果三点共线，面积将为零——这是检验共线性的有效方法。练习以任意顺序使用坐标代入公式；绝对值会处理符号变化。

12. Applications and Problem Solving | 应用与解题技巧

Edexcel IGCSE exam questions often combine multiple concepts: you might be asked to find the equation of a perpendicular bisector of a chord, determine the coordinates of a point given a ratio, or calculate the area formed by intersecting lines. Start by sketching the diagram, identify known points and gradients, and then apply the relevant formula step by step. Always check that your final answer makes sense in the context (e.g., a gradient of a line falling to the right should be negative).

Edexcel IGCSE 考题常将多个概念结合：你可能需要求一条弦的垂直平分线方程、根据比例确定点的坐标，或计算相交直线所围成的面积。首先画出草图，确定已知点和斜率，然后逐步应用相关公式。始终检查最终答案在情境中是否合理（例如，向右下方倾斜的直线斜率应为负）。

Remember to write line equations in the form requested, typically ax + by + c = 0 with integer coefficients. When using the distance or midpoint formulas, be careful with negative signs. Regular practice with past papers will help you become confident and accurate.

记住，要按照题目要求的形式给出直线方程，通常是 ax + by + c = 0 且系数为整数。使用距离或中点公式时，注意负号。通过练习历年真题，你会变得自信且准确。

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Coordinate Geometry Key Points | 坐标几何 考点精讲