Midpoints and Perpendicular Bisectors | 中点与垂直平分线

📚 Midpoints and Perpendicular Bisectors | 中点与垂直平分线

In coordinate geometry, midpoints and perpendicular bisectors connect algebra with geometric symmetry. They appear in A-Level Edexcel questions on straight lines, triangles, circles, and loci. This article explains midpoint calculation, perpendicular gradients, strategies for finding perpendicular bisector equations, and common exam applications.

在坐标几何中，中点与垂直平分线把代数与几何对称性联系起来。它们出现在 A-Level Edexcel 的直线、三角形、圆和轨迹问题中。本文讲解中点计算、垂直斜率关系、求垂直平分线方程的策略以及常见考试应用。

1. The Midpoint Formula | 中点公式

For two points A(x₁, y₁) and B(x₂, y₂), the midpoint M of segment AB is found by averaging the x-coordinates and averaging the y-coordinates.

对于两点 A(x₁, y₁) 和 B(x₂, y₂)，线段 AB 的中点 M 通过对两个 x 坐标取平均值、两个 y 坐标取平均值来求得。

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

The midpoint divides AB in the ratio 1:1, so it is the arithmetic mean of the endpoints. It always lies on the line segment and is the same distance from A and B.

中点以 1:1 的比例分割 AB，因此它是端点的算术平均值。它始终位于线段上，并且到 A 和 B 的距离相等。

For example, if A(2, 5) and B(8, 1), then M = ((2 + 8) / 2, (5 + 1) / 2) = (5, 3). This is the point exactly halfway between the two endpoints.

例如，若 A(2, 5) 且 B(8, 1)，则 M = ((2 + 8) / 2, (5 + 1) / 2) = (5, 3)。这个点恰好位于两个端点之间的正中间。

The midpoint formula is often the first step when constructing a perpendicular bisector, so it must be applied quickly and accurately.

中点公式通常是构造垂直平分线的第一步，因此必须快速且准确地运用。

2. Worked Example: Finding a Midpoint | 求中点的例题

Find the midpoint of the segment joining A(5, -3) and B(1, 9).

求连接 A(5, -3) 和 B(1, 9) 的线段的中点。

M = ((5 + 1) / 2, (-3 + 9) / 2) = (6 / 2, 6 / 2) = (3, 3)

The midpoint is (3, 3). Notice that the x-coordinate 3 lies exactly halfway between 5 and 1, and the y-coordinate 3 lies exactly halfway between -3 and 9.

中点为 (3, 3)。注意 x 坐标 3 恰好位于 5 和 1 的中间，y 坐标 3 恰好位于 -3 和 9 的中间。

If the midpoint and one endpoint are given, you can solve backwards to find the missing endpoint. Suppose M(4, 7) is the midpoint of AB and A(2, 3). Let B be (x, y).

如果已知中点和一个端点，可以反向求解另一个端点。假设 M(4, 7) 是 AB 的中点，且 A(2, 3)。设 B 为 (x, y)。

(2 + x) / 2 = 4 ⇒ x = 6
(3 + y) / 2 = 7 ⇒ y = 11

Therefore B is (6, 11). This reverse process is useful when a perpendicular bisector is known to pass through the midpoint of a side of a triangle.

因此 B 为 (6, 11)。当已知垂直平分线经过三角形某条边的中点时，这种反向求解过程非常有用。

3. Gradient of a Line Segment | 线段的斜率

The gradient m of a line segment joining A(x₁, y₁) and B(x₂, y₂) measures its steepness and direction.

连接 A(x₁, y₁) 和 B(x₂, y₂) 的线段的斜率 m 度量其倾斜程度和方向。

m = (y₂ – y₁) / (x₂ – x₁)

For A(1, 4) and B(7, 12), the gradient is m = (12 – 4) / (7 – 1) = 8 / 6 = 4 / 3. The positive gradient means the segment rises as x increases.

对于 A(1, 4) 和 B(7, 12)，斜率为 m = (12 – 4) / (7 – 1) = 8 / 6 = 4 / 3。斜率为正表示线段随 x 增大而上升。

If the y-coordinate decreases as x increases, the gradient is negative. For example, A(2, 9) and B(6, 1) give m = (1 – 9) / (6 – 2) = -8 / 4 = -2.

如果 y 坐标随 x 增大而减小，则斜率为负。例如，A(2, 9) 和 B(6, 1) 给出 m = (1 – 9) / (6 – 2) = -8 / 4 = -2。

When x₂ = x₁, the denominator is zero, so the gradient is undefined and the segment is vertical. When y₂ = y₁, the gradient is 0 and the segment is horizontal. These special cases are important when finding perpendicular bisectors.

当 x₂ = x₁ 时，分母为零，因此斜率无定义，线段是竖直的。当 y₂ = y₁ 时，斜率为 0，线段是水平的。在求垂直平分线时，这些特殊情况很重要。

4. Perpendicular Gradients | 垂直直线的斜率关系

Two non-vertical lines are perpendicular if and only if the product of their gradients is -1.

两条非竖直直线垂直，当且仅当它们斜率的乘积为 -1。

m₁ × m₂ = -1

If the original line has gradient m, then any perpendicular line has gradient -1 / m. This means you take the reciprocal of m and change its sign.

如果原直线的斜率为 m，那么任何与它垂直的直线斜率为 -1 / m。这意味着取 m 的倒数并改变符号。

For example, if m = 2 / 3, the perpendicular gradient is -3 / 2. If m = -4, the perpendicular gradient is 1 / 4. If m = 1, the perpendicular gradient is -1.

例如，如果 m = 2 / 3，则垂直斜率为 -3 / 2。如果 m = -4，则垂直斜率为 1 / 4。如果 m = 1，则垂直斜率为 -1。

A common error is to forget the sign change or only take the reciprocal. Always check by multiplying the two gradients together: they must give exactly -1.

常见错误是忘记变号或只取倒数。始终通过将两个斜率相乘来检验：它们的乘积必须恰好为 -1。

If a line is vertical, its gradient is undefined, and any perpendicular line is horizontal with gradient 0.

如果一条直线是竖直的，其斜率无定义，那么任何与它垂直的直线都是水平的，斜率为 0。

5. Definition of a Perpendicular Bisector | 垂直平分线的定义

The perpendicular bisector of a segment AB is the line that passes through the midpoint of AB and is perpendicular to AB.

线段 AB 的垂直平分线是经过 AB 的中点并垂直于 AB 的直线。

Equivalently, every point on the perpendicular bisector is equidistant from A and B. This makes the bisector a locus: the set of all points P such that PA = PB.

等价地，垂直平分线上的每个点到 A 和 B 的距离相等。这使得平分线成为一条轨迹：满足 PA = PB 的所有点 P 的集合。

The perpendicular bisector is also the axis of symmetry of segment AB. If you reflect A across this line, it maps to B, and the segment folds onto itself.

垂直平分线也是线段 AB 的对称轴。如果将 A 沿这条线反射，它会映射到 B，线段自身重合。

In circle geometry, the perpendicular bisector of any chord passes through the centre of the circle. This fact is often used to find the centre given two chords.

在圆几何中，任何弦的垂直平分线都经过圆心。这个事实常用于已知两条弦时求圆心。

6. Finding the Equation of a Perpendicular Bisector | 求垂直平分线方程

To find the equation of the perpendicular bisector of segment AB, follow a clear four-step method.

要求线段 AB 的垂直平分线方程，遵循清晰的四步方法。

Step 1: Find the midpoint M of AB using the midpoint formula. The perpendicular bisector must pass through this point.

步骤 1：用中点公式求 AB 的中点 M。垂直平分线必须经过这个点。

Step 2: Find the gradient of AB using m = (y₂ – y₁) / (x₂ – x₁).

步骤 2：用 m = (y₂ – y₁) / (x₂ – x₁) 求 AB 的斜率。

Step 3: Find the perpendicular gradient by taking the negative reciprocal, -1 / m. This is the gradient of the perpendicular bisector.

步骤 3：通过取负倒数 -1 / m 求垂直斜率。这就是垂直平分线的斜率。

Step 4: Use the point-gradient form y – y₁ = m(x – x₁) with the midpoint and the perpendicular gradient, then simplify to the required form.

步骤 4：将中点和垂直斜率代入点斜式 y – y₁ = m(x – x₁)，然后化简为所需形式。

The final answer may be given as y = mx + c or as ax + by + c = 0 unless the question specifies a particular form.

除非题目指定特定形式，最终答案可以写成 y = mx + c 或 ax + by + c = 0。

After finding the equation, check that the midpoint satisfies it and that the product of the original gradient and perpendicular gradient is -1.

求出方程后，检查中点是否满足方程，以及原斜率与垂直斜率的乘积是否为 -1。

7. Worked Example: Perpendicular Bisector Equation | 垂直平分线方程例题

Find the equation of the perpendicular bisector of the segment joining A(3, 2) and B(7, 10).

求连接 A(3, 2) 和 B(7, 10) 的线段的垂直平分线方程。

Midpoint M = ((3 + 7) / 2, (2 + 10) / 2) = (5, 6)
Gradient of AB = (10 – 2) / (7 – 3) = 8 / 4 = 2
Perpendicular gradient = -1 / 2

Using y – y₁ = m(x – x₁) at M(5, 6):

在 M(5, 6) 处使用 y – y₁ = m(x – x₁)：

y – 6 = -1/2 (x – 5)

Multiply both sides by 2: 2y – 12 = -x + 5. Rearranging gives x + 2y – 17 = 0, or y = -1/2 x + 17/2.

两边乘以 2：2y – 12 = -x + 5。整理得 x + 2y – 17 = 0，或 y = -1/2 x + 17/2。

Check: the midpoint (5, 6) gives 5 + 2(6) – 17 = 0, so it lies on the line. The product of gradients is 2 × (-1/2) = -1, so the line is perpendicular to AB.

检验：中点 (5, 6) 代入得 5 + 2(6) – 17 = 0，所以它在此直线上。斜率乘积为 2 × (-1/2) = -1，所以该直线垂直于 AB。

This structured method works for any segment, even when coordinates are fractions or negative numbers.

即使坐标是分数或负数，这种结构化方法也适用于任何线段。

8. Equidistance Property and Locus | 等距性质与轨迹

Since every point on the perpendicular bisector is equidistant from A and B, we can derive the equation directly from the distance formula.

由于垂直平分线上的每个点到 A 和 B 的距离相等，我们可以直接从距离公式推导其方程。

√((x – x₁)² + (y – y₁)²) = √((x – x₂)

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Midpoints and Perpendicular Bisectors | 中点与垂直平分线