📚 A-Level Maths: Finding Midpoints and Perpendicular Bisectors | A-Level数学：中点与垂直平分线的求解
In coordinate geometry, a straight line segment is defined by its two endpoints. Two of the most important skills you need in Edexcel A-Level Pure Mathematics are finding the midpoint of a segment and writing the equation of its perpendicular bisector. These skills appear in many exam problems and are essential for later topics such as circles, vectors, and loci.
在解析几何中，一条线段由其两个端点确定。在 Edexcel A-Level 纯数学中，你需要掌握的最重要技能之一就是求线段的中点，以及写出其垂直平分线的方程。这些技能在许多考试题目中都会出现，并且对于后续的圆、向量和轨迹等主题至关重要。
1. The Midpoint Formula | 中点公式
For a line segment with endpoints A(x₁, y₁) and B(x₂, y₂), the midpoint M is the point exactly halfway between the two endpoints.
对于端点分别为 A(x₁, y₁) 和 B(x₂, y₂) 的线段，中点 M 就是位于两个端点正中间的点。
To find it, average the x-coordinates and average the y-coordinates. This gives the formula below.
要求中点，只需将两个 x 坐标取平均值，再将两个 y 坐标取平均值，得到如下公式。
M = ((x₁ + x₂)/2 , (y₁ + y₂)/2)
The same formula applies whether the coordinates are positive, negative, or fractional.
无论坐标是正数、负数还是分数，这个公式都适用。
2. Worked Example: Finding the Midpoint | 例题：求中点
Example. Find the midpoint of the segment joining A(2, 3) and B(8, −1).
例题：求连接 A(2, 3) 和 B(8, −1) 的线段的中点。
First, add the x-coordinates: 2 + 8 = 10, then divide by 2 to get 5.
首先将 x 坐标相加：2 + 8 = 10，再除以 2 得到 5。
Then add the y-coordinates: 3 + (−1) = 2, then divide by 2 to get 1.
然后将 y 坐标相加：3 + (−1) = 2，再除以 2 得到 1。
Therefore the midpoint is M(5, 1).
因此中点坐标为 M(5, 1)。
3. The Gradient of a Line | 直线的斜率
Before we can find a perpendicular bisector, we need to be confident with gradients. The gradient of a line measures its steepness.
在求垂直平分线之前，我们需要熟练掌握斜率的概念。直线的斜率衡量其倾斜程度。
For two points A(x₁, y₁) and B(x₂, y₂), the gradient is calculated as follows.
对于两点 A(x₁, y₁) 和 B(x₂, y₂)，斜率按如下方式计算。
m = (y₂ − y₁)/(x₂ − x₁)
Notice that the order must be consistent: subtract the y-coordinates in the same order as you subtract the x-coordinates.
注意运算顺序必须一致：用第二个点的坐标减去第一个
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