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。

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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。

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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。

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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。

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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。

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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。

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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。

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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。

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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。

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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。

ent, 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。

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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：

1. Find the perpendicular bisector of the segment joining C(0, 0) and D(6, 4).
2. 求连接 C(0, 0) 和 D(6, 4) 的线段的垂直平分线方程。
3. The points E(2, 5) and F(−4, 1) define a segment. What is the equation of its perpendicular bisector?
4. 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？
5. Given G(−3, 2) and H(5, −2), find the perpendicular bisector and verify that it passes through the midpoint.
6. 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。

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11. Conclusion | 总结

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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