The Mid-point Formula | 中点公式

📚 The Mid-point Formula | 中点公式

The mid-point formula is one of the most straightforward yet powerful tools in coordinate geometry. For two given points, it gives the exact coordinates of the point halfway between them.

中点公式是坐标几何中最直接而有力的工具之一。对于两个已知点，它可以给出位于两者正中间那一点的精确坐标。

1. Definition and Formula | 定义与公式

Given two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in a Cartesian plane, the midpoint \(M\) of the line segment \(AB\) is:

给定笛卡尔平面上的两点 \(A(x_1, y_1)\) 和 \(B(x_2, y_2)\)，线段 \(AB\) 的中点 \(M\) 为：

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

This formula is also called the average of the coordinates. Notice that we simply take the arithmetic mean of the x-coordinates and the arithmetic mean of the y-coordinates.

该公式也称为坐标的平均值。注意，我们只需取 x 坐标的算术平均值和 y 坐标的算术平均值。

The x-coordinate of the midpoint is the average of x₁ and x₂.

中点的 x 坐标是 x₁ 和 x₂ 的平均值。
The y-coordinate of the midpoint is the average of y₁ and y₂.

中点的 y 坐标是 y₁ 和 y₂ 的平均值。

2. Derivation from Averages | 从平均值推导

Suppose we move from point A to point B. Halfway along this journey, both the x-change and the y-change have been halved.

假设我们从点 A 移动到点 B。在这段旅程的中途，x 的变化量和 y 的变化量都被减半。

The x-coordinate of the midpoint is x₁ plus half of (x₂ – x₁):

中点的 x 坐标为 x₁ 加上 (x₂ – x₁) 的一半：

x = x₁ + ½(x₂ – x₁) = ½x₁ + ½x₂ = (x₁ + x₂)/2

Similarly, the y-coordinate is y₁ plus half of (y₂ – y₁). This confirms that the midpoint is simply the average of the two endpoints.

同理，y 坐标为 y₁ 加上 (y₂ – y₁) 的一半。这确认了中点就是两个端点的平均值。

3. Finding the Midpoint Between Two Points | 求两点之间的中点

To apply the formula, substitute the coordinates into the average expressions. No rearrangement is needed.

应用公式时，只需将坐标代入平均值表达式，无需重新整理。

Example: Find the midpoint of the segment joining (3, –2) and (7, 6).

示例：求连接 (3, –2) 和 (7, 6) 的线段的中点。

x = (3 + 7)/2 = 5, y = (–2 + 6)/2 = 2

M = (5, 2)

The midpoint is exactly halfway in both directions. The distance from (3, –2) to (5, 2) equals the distance from (5, 2) to (7, 6).

中点在两个方向上都是正中间。从 (3, –2) 到 (5, 2) 的距离等于从 (5, 2) 到 (7, 6) 的距离。

4. Finding an Endpoint Given the Midpoint | 已知中点求另一个端点

If you know the midpoint and one endpoint, you can reverse the formula to find the other endpoint. Let the known endpoint be \(A(x₁, y₁)\) and the midpoint be \(M(xₘ, yₘ)\). Then the unknown endpoint \(B(x₂, y₂)\) satisfies:

若已知中点和其中一个端点，可反向使用公式求出另一端点。设已知端点为 \(A(x₁, y₁)\)，中点为 \(M(xₘ, yₘ)\)。则未知端点 \(B(x₂, y₂)\) 满足：

x₂ = 2xₘ – x₁, y₂ = 2yₘ – y₁

Example: If M(4, –1) is the midpoint of AB and A(1, 3), find B.

示例：若 M(4, –1) 是 AB 的中点，且 A(1, 3)，求 B。

x₂ = 2×4 – 1 = 7, y₂ = 2×(–1) – 3 = –5

B(7, –5)

This technique is especially useful in vector and geometry problems involving symmetry.

该方法在涉及对称性的向量与几何问题中尤其有用。

5. Midpoint and Line Segments | 中点与线段

The midpoint divides a line segment into two equal parts. This property connects to concepts like perpendicular bisectors and medians of triangles.

中点将一条线段分成两个相等的部分。这一性质与垂直平分线、三角形的中线等概念紧密相连。

Every line segment has exactly one midpoint.

每条线段都有且只有一个中点。
The midpoint lies on the segment itself, not outside it.

中点位于线段上，而非线段之外。
If the coordinates are integers, the midpoint may be fractional; this is normal.

如果端点坐标为整数，中点可能为分数；这是正常的。

For a horizontal segment, the midpoint has the same y-coordinate as the endpoints. For a vertical segment, the midpoint has the same x-coordinate.

对水平线段，中点与端点有相同的 y 坐标；对竖直线段，中点与端点有相同的 x 坐标。

6. Midpoint in Coordinate Geometry | 坐标几何中的中点

In AQA A-level Mathematics, the midpoint formula often appears as a preliminary step when solving problems about circles and straight lines.

在 AQA A-level 数学中，中点公式常作为解决圆与直线问题的预备步骤出现。

For example, to find the centre of a circle when given the endpoints of a diameter, use the midpoint formula. The midpoint of the diameter is the centre of the circle.

例如，当给定直径的端点时，可用中点公式求圆心。直径的中点就是圆心。

Typical question: The endpoints of a diameter of a circle are (2, 5) and (8, –1). Find the centre.

典型问题：圆的直径端点为 (2, 5) 和 (8, –1)，求圆心。

Centre = ((2+8)/2, (5+(–1))/2) = (5, 2)

Once the centre is known, the radius can be found using the distance formula.

求得圆心后，可用距离公式求半径。

7. Midpoint in Three Dimensions | 三维空间中的中点

For points in three-dimensional space \((x₁, y₁, z₁)\) and \((x₂, y₂, z₂)\), the midpoint is:

对于三维空间中的点 \((x₁, y₁, z₁)\) 和 \((x₂, y₂, z₂)\)，中点为：

M = ( (x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2 )

The pattern extends naturally: each coordinate of the midpoint is the average of the corresponding coordinates of the endpoints.

该规律自然延伸：中点的每个坐标都是两端点对应坐标的平均值。

This version is often tested in AQA’s A-level Further Mathematics or in the coordinate geometry section of pure mathematics.

此扩展形式常在 AQA A-level 进阶数学或纯数坐标几何部分考查。

8. Worked Example – AQA Style | 典型例题（AQA 风格）

Question: The point M(3, 4) is the midpoint of the line segment joining P(–1, 2) and Q(a, b). Find the values of a and b.

题目：点 M(3, 4) 是连接 P(–1, 2) 和 Q(a, b) 的线段的中点。求 a 和 b 的值。

Solution: Using the midpoint formula:

解答：使用中点公式：

3 = (–1 + a)/2 ⇒ –1 + a = 6 ⇒ a = 7

4 = (2 + b)/2 ⇒ 2 + b = 8 ⇒ b = 6

Therefore Q = (7, 6).

因此 Q = (7, 6)。

This question combines the forward and reverse use of the formula, testing algebraic manipulation rather than just substitution.

该题结合了公式的正用与逆用，考查代数变形而非简单代入。

9. Common Mistakes | 常见错误

Students often make small arithmetic slips when using the midpoint formula. Here are the most frequent ones:

学生在使用中点公式时经常出现小计算错误。以下是最常见的几种：

Mistake \| 错误	Correction \| 正确做法
Adding x₁ and x₂ but forgetting to divide by 2.	Always divide the sum by 2.
Subtracting the coordinates instead of adding.	Use addition: (x₁ + x₂)/2, not (x₂ – x₁)/2.
Mixing the x and y coordinates when substituting.	Pair the x-coordinates together and the y-coordinates together.
When finding an endpoint, using x = (x₁ + xₘ)/2 instead of x = 2xₘ – x₁.	Multiply the midpoint by 2, then subtract the known endpoint.

Always check that your midpoint lies between the two endpoints. If it does not, revisit the calculation.

始终检查中点是否位于两端点之间。若不在，请重新检查计算。

10. Practice Questions | 练习问题

Try these questions yourself before checking the answers.

请先自行尝试以下问题，再核对答案。

1. Find the midpoint of (0, 0) and (10, –4).

1. 求 (0, 0) 和 (10, –4) 的中点。

2. The midpoint of AB is (5, 2). If A = (8, –1), find B.

2. AB 的中点为 (5, 2)。若 A = (8, –1)，求 B。

3. A circle has a diameter with endpoints ( –2, 7) and (4, –3). Find the centre.

3. 圆的直径端点为 ( –2, 7) 和 (4, –3)，求圆心。

Answers: 1. (5, –2) 2. (2, 5) 3. (1, 2)

答案：1. (5, –2) 2. (2, 5) 3. (1, 2)

The midpoint formula is simple to remember: average the x’s, average the y’s. Master this skill and you will solve coordinate geometry problems with confidence.

中点公式简记不难：x 取平均，y 取平均。掌握这一技巧，你就能自信地解决坐标几何问题。

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The Mid-point Formula | 中点公式