📚 Parallel and Perpendicular Lines | 平行线与垂直线
In coordinate geometry, the gradient of a straight line determines whether two lines are parallel or perpendicular. Parallel lines have equal gradients, while perpendicular lines have gradients whose product is -1. This simple rule is applied throughout the Pure Mathematics 1 straight-line chapter and appears frequently in Edexcel exam questions.
在坐标几何中，直线的斜率决定了两条直线是平行还是垂直。平行线的斜率相等，垂直线的斜率乘积为 -1。这一简单规则贯穿 Pure Mathematics 1 的直线章节，并且在 Edexcel 考试中经常出现。
1. Understanding the Gradient of a Line | 理解直线的斜率
The gradient of a line measures how steep the line is. It is the change in y divided by the change in x between two points on the line.
直线的斜率衡量这条线的倾斜程度。它等于直线上两点之间 y 的变化量除以 x 的变化量。
For two points (x₁, y₁) and (x₂, y₂), the gradient m is given by:
m = (y₂ – y₁) / (x₂ – x₁)
A positive gradient means the line slopes upwards from left to right; a negative gradient means it slopes downwards. If x₂ = x₁, the line is vertical and its gradient is undefined.
斜率为正表示直线从左向右上升；斜率为负表示直线从左向右下降。若 x₂ = x₁，则直线竖直，斜率不存在。
2. The Equation of a Straight Line | 直线方程
The most common form of a straight-line equation is y = mx + c, where m is the gradient and c is the y-intercept.
直线方程最常见的形式是 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
Another useful form is the point-gradient form, used when one point and the gradient are known:
y – y₁ = m(x – x₁)
Sometimes a line is written in the form ax + by + c = 0. To find its gradient, rearrange it into y = mx + c or use m = -a/b when b ≠ 0.
有时直线方程写成 ax + by + c = 0 的形式。要求斜率，可将其改写为 y = mx + c，或直接使用 m = -a/b（当 b ≠ 0 时）。
3. Parallel Lines | 平行线
Two lines are parallel if and only if they have the same gradient. If line 1 has gradient m₁ and line 2 has gradient m₂, then the lines are parallel when m₁ = m₂.
两条直线平行当且仅当它们的斜率相同。若直线 1 的斜率为 m₁，直线 2 的斜率为 m₂，则两条直线平行时满足 m₁ = m₂。
For example, y = 2x + 3 and y = 2x – 5 are parallel because both have gradient 2. Their y-intercepts are different, so they never meet.
例如，y = 2x + 3 与 y = 2x – 5 平行，因为它们的斜率都为 2。它们的 y 轴截距不同，因此它们永远不会相交。
4. Perpendicular Lines | 垂直线
Two lines are perpendicular if and only if the product of their gradients is -1. This means each gradient is the negative reciprocal of the other:
两条直线垂直当且仅当它们斜率的乘积为 -1。也就是说，每条直线的斜率都是另一条直线斜率的负倒数：
m₁ × m₂ = -1
Equivalently, m₂ = -1/m₁. For example, if one line has gradient 2/3, a perpendicular line must have gradient -3/2.
等价地，m₂ = -1/m₁。例如，若一条直线的斜率为 2/3，则与之垂直的直线斜率必须为 -3/2。
This condition does not work for vertical lines, because a vertical line has an undefined gradient. Instead, a vertical line is always perpendicular to a horizontal line.
这一条件不适用于竖直线，因为竖直线的斜率不存在。实际上，竖直线总是垂直于水平线。
5. Finding the Equation of a Parallel Line | 求平行线方程
To find the equation of a line parallel to a given line and passing through another point, first identify the gradient of the given line. Then substitute the known point into the point-gradient form.
要求一条与已知直线平行且过另一点的直线方程，首先要确定已知直线的斜率，然后将已知点代入点斜式。
Example: find the equation of the line parallel to y = 3x – 1 passing through (2, 4).
例：求与 y = 3x – 1 平行且过点 (2, 4) 的直线方程。
The given line has gradient 3, so the parallel line also has gradient 3. Using y – y₁ = m(x – x₁):
已知直线斜率为 3，因此平行线斜率也为 3。使用 y – y₁ = m(x – x₁)：
y – 4 = 3(x – 2) ⇒ y = 3x – 2
You can check that the point (2, 4) satisfies y = 3x – 2, and its gradient is 3.
可以验证点 (2, 4) 满足 y = 3x – 2，并且该直线斜率为 3。
6. Finding the Equation of a Perpendicular Line | 求垂直线方程
To find a line perpendicular to a given line, first determine the gradient of the given line, then take its negative reciprocal. After that, use the known point in the point-gradient formula.
要求一条与已知直线垂直的直线方程，先求出已知直线的斜率，再取其负倒数。然后利用已知点代入点斜式。
Example: find the equation of the line perpendicular to 2x + 3y = 6 and passing through (-1, 4).
例：求与 2x + 3y = 6 垂直且过点 (-1, 4) 的直线方程。
First rewrite 2x + 3y = 6 as y = -2/3 x + 2. The original gradient is -2/3, so the perpendicular gradient is 3/2.
先将 2x + 3y = 6 改写为 y = -2/3 x + 2。原直线斜率为 -2/3，因此垂直斜率为 3/2。
y – 4 = 3/2 (x + 1) ⇒ y = 3/2 x + 11/2
In general form, this is 3x – 2y + 11 = 0.
化为一般式为 3x – 2y + 11 = 0。
7. Special Cases: Horizontal and Vertical Lines | 特殊情形：水平线与竖直线
Horizontal lines have the form y = c and gradient 0. Vertical lines have the form x = k and their gradient is undefined.
水平线的方程为 y = c，斜率为 0。竖直线的方程为 x = k，其斜率不存在。
Two horizontal lines are always parallel, and two vertical lines are always parallel as well. A horizontal line and a vertical line are always perpendicular.
两条水平线总是平行，两条竖直线也总是平行。水平线与竖直线总是互相垂直。
When dealing with vertical lines, do not force the product rule m₁ × m₂ = -1. Instead, use the fact that a vertical line is perpendicular to any horizontal line.
处理竖直线时，不要强行使用斜率乘积规则 m₁ × m₂ = -1。应利用“竖直线垂直于任意水平线”这一事实。
8. Perpendicular Bisectors | 垂直平分线
A perpendicular bisector of a line segment is a line that passes through the midpoint of the segment and is perpendicular to the segment.
一条线段的垂直平分线是经过该线段中点且与该线段垂直的直线。
To find it, calculate the midpoint of the two given points, then find the gradient of the original segment and take its negative reciprocal. Finally, use the point-gradient form with the midpoint.
要求垂直平分线，先计算两点的中点，再求原线段斜率的负倒数，最后用中点代入点斜式。
Example: for A(1, 2) and B(5, 6), the midpoint is (3, 4). The gradient of AB is 1, so the perpendicular gradient is -1. The perpendicular bisector is:
例如：对于 A(1, 2) 和 B(5, 6)，中点为 (3, 4)。AB 的斜率为 1，因此垂直斜率为 -1。垂直平分线为：
y – 4 = -1(x – 3) ⇒ y = -x + 7
9. Worked Exam Example | 考试例题精解
Question: The line L passes through A(0, 1) and B(2, 5). (a) Show that L is parallel to 2y = 4x + 1. (b) Find the equation of the line perpendicular to L through the midpoint of AB.
题目：直线 L 经过 A(0, 1) 和 B(2, 5)。(a) 证明 L 与 2y = 4x + 1 平行。(b) 求过 AB 中点且垂直于 L 的直线方程。
Solution: The gradient of L is (5 – 1)/(2 – 0) = 2. The line 2y = 4x + 1 can be rewritten as y = 2x + 1/2, so its gradient is also 2. Therefore L is parallel to 2y = 4x + 1.
解答：L 的斜率为 (5 – 1)/(2 – 0) = 2。直线 2y = 4x + 1 可改写为 y = 2x + 1/2，其斜率也为 2。因此 L 与 2y = 4x + 1 平行。
The midpoint of AB is (1, 3). Since the gradient of L is 2, the perpendicular gradient is -1/2. Using point-gradient form:
AB 的中点为 (1, 3)。因为 L 的斜率为 2
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