Example 3.2.2: Equation of a Perpendicular Bisector | 例题3.2.2：垂直平分线的方程

📚 Example 3.2.2: Equation of a Perpendicular Bisector | 例题3.2.2：垂直平分线的方程

In this example, we are given two points A(−2, 3) and B(4, 1). We need to find the equation of the perpendicular bisector of the line segment AB. This is a classic coordinate geometry question from the AQA A-Level Mathematics specification, and it tests your ability to find midpoints, gradients, and the equation of a straight line.

在本例中，我们已知两点 A(−2, 3) 和 B(4, 1)，需要求出线段 AB 的垂直平分线的方程。这是 AQA A-Level 数学考纲中的一个经典坐标几何题，考查大家求中点、斜率和直线方程的能力。

1. Understanding the Problem | 理解题目

A perpendicular bisector is a line that passes through the midpoint of a segment and is perpendicular to that segment. It means every point on this line is equidistant from A and B.

垂直平分线是指通过一条线段的中点并与该线段垂直的直线。也就是说，这条线上的任意一点到 A 和 B 的距离相等。

We must first recall the two important conditions: the line passes through the midpoint of AB, and its gradient is the negative reciprocal of the gradient of AB.

我们首先要记住两个重要条件：该直线通过 AB 的中点，并且它的斜率是 AB 斜率的负倒数。

2. Key Concepts Review | 回顾关键概念

To solve this problem, we need three formulas:

解决这个问题需要三个公式：

Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点公式： M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Gradient: m = (y₂ − y₁)/(x₂ − x₁)
斜率公式： m = (y₂ − y₁)/(x₂ − x₁)
Perpendicular gradient: m_perp = −1/m (when m ≠ 0)
垂直斜率： m_perp = −1/m（当 m ≠ 0 时）

We will use these formulas step by step.

我们将逐步使用这些公式。

3. Step 1: Find the Midpoint | 第 1 步：求中点

Given A(−2, 3) and B(4, 1), we substitute into the midpoint formula:

已知 A(−2, 3) 和 B(4, 1)，代入中点公式：

M = ((−2 + 4)/2, (3 + 1)/2) = (2/2, 4/2) = (1, 2)

So the midpoint of AB is (1, 2). This point will lie on the perpendicular bisector.

因此 AB 的中点坐标为 (1, 2)。该点必在垂直平分线上。

4. Step 2: Find the Gradient of AB | 第 2 步：求 AB 的斜率

Using the gradient formula with A as (x₁, y₁) and B as (x₂, y₂):

以 A 为 (x₁, y₁)，B 为 (x₂, y₂)，使用斜率公式：

m_AB = (1 − 3)/(4 − (−2)) = (−2)/(6) = −1/3

The gradient of AB is −1/3. This means AB slopes downwards from left to right.

AB 的斜率为 −1/3，表示 AB 从左到右向下倾斜。

5. Step 3: Find the Perpendicular Gradient | 第 3 步：求垂直斜率

If the original gradient is m, the perpendicular gradient is −1/m. With m = −1/3:

若原斜率为 m，则垂直斜率为 −1/m。已知 m = −1/3：

m_perp = −1 / (−1/3) = 3

Therefore the perpendicular gradient is 3. The perpendicular bisector has a positive slope, so it rises as x increases.

因此垂直斜率为 3。垂直平分线具有正斜率，随着 x 增大而上升。

6. Step 4: Write the Equation | 第 4 步：写出方程

We know the perpendicular bisector passes through (1, 2) and has gradient 3. Use the point-slope form:

我们知道垂直平分线通过 (1, 2)，且斜率为 3。使用点斜式：

y − y₁ = m(x − x₁)

y − 2 = 3(x − 1)

Simplify the equation:

化简方程：

y − 2 = 3x − 3

y = 3x − 1

Thus the equation of the perpendicular bisector is y = 3x − 1.

因此，垂直平分线的方程为 y = 3x − 1。

7. Verification | 验证

We should check that the line y = 3x − 1 passes through the midpoint (1, 2):

我们应验证直线 y = 3x − 1 是否经过中点 (1, 2)：

3(1) − 1 = 2

Since the left-hand side equals the y-coordinate of the midpoint, the point lies on the line.

因为左侧值等于中点的 y 坐标，所以该点在这条直线上。

We can also check that the product of the gradients is −1:

我们还可以检查两条直线的斜率乘积是否为 −1：

m_AB × m_perp = (−1/3) × 3 = −1

This confirms that the two lines are perpendicular.

这确认了两条直线互相垂直。

8. Common Pitfalls | 常见陷阱

Students often make mistakes in three areas:

学生常在以下三个方面犯错：

Incorrect midpoint: Forgetting to divide by 2, or mixing up x and y coordinates.
中点错误：忘记除以 2，或者混淆 x 和 y 坐标。
Wrong perpendicular gradient: Forgetting the negative sign, or accidentally using the reciprocal instead of the negative reciprocal.
垂直斜率错误：忘记负号，或者误用倒数而不是负倒数。
Sign errors: When substituting negative coordinates, especially in the gradient formula, be careful with signs.
符号错误：代入负坐标时，尤其是在斜率公式中，要小心符号。

9. Exam Tips | 考试技巧

In the AQA exam, always show your working clearly. Even if you make a small arithmetic slip, the method marks can still be awarded.

在 AQA 考试中，务必清晰展示计算过程。即使出现小的运算失误，方法分仍然可能得到。

When you find a perpendicular gradient, write m₁ × m₂ = −1 explicitly to show the examiner that you understand the condition.

在求垂直斜率时，写出 m₁ × m₂ = −1 的步骤，向考官展示你理解了垂直条件。

Always check that your final equation satisfies the midpoint coordinate. This quick check takes only a few seconds and can prevent errors.

始终检查你的最终方程是否满足中点坐标。这个快速检查只需几秒钟，可以避免错误。

10. Practice Questions | 类似练习题

Try these similar questions on your own:

请自己尝试以下类似题目：

Find the perpendicular bisector of the segment joining C(0, 0) and D(6, 4).
求连接 C(0, 0) 和 D(6, 4) 的线段的垂直平分线方程。
The points E(2, 5) and F(−4, 1) define a segment. What is the equation of its perpendicular bisector?
点 E(2, 5) 和 F(−4, 1) 构成一条线段，其垂直平分线的方程是什么？
Given G(−3, 2) and H(5, −2), find the perpendicular bisector and verify that it passes through the midpoint.
已知 G(−3, 2) 和 H(5, −2)，求垂直平分线并验证它经过中点。

Answers: (1) y = (−3/2)x + 6; (2) y = (3/2)x + 3; (3) y = 2x + 5.

答案：(1) y = (−3/2)x + 6；(2) y = (3/2)x + 3；(3) y = 2x + 5。

11. Conclusion | 总结

In this example, we found the perpendicular bisector of AB by first calculating the midpoint, then the gradient of AB, and finally using the negative reciprocal to write the equation of the line. The final answer y = 3x − 1 is both correct and fully verified.

在本例中，我们通过先求中点、再求 AB 的斜率、最后利用负倒数写出直线方程，得到了 AB 的垂直平分线。最终答案 y = 3x − 1 正确且经过了完整验证。

Remember the method:

记住这个方法的步骤：

Midpoint → gradient of segment → negative reciprocal → equation of line

求中点 → 求线段斜率 → 取负倒数 → 写出直线方程

With practice, this type of coordinate geometry question becomes straightforward.

通过练习，这类坐标几何题目会变得非常简单。

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Example 3.2.2: Equation of a Perpendicular Bisector | 例题3.2.2：垂直平分线的方程