两条平行线之间的距离是多少？

测量来自拉丁词mensurate 这意味着“测量”。所以测量章节是关于测量各种值的。在数学中，需要计算和测量不同的值，因此研究了测量。让我们试着测量两条平行线之间的距离。

平行线

任何具有相同斜率的线都称为平行线，或者它们可以被认为是两条或多条无限远相交的线。这意味着即使直线向无穷大延伸，它们也永远不会相交。当横切线以 θ 角与两条平行线相交时平行线的性质：

考虑到该图，属性定义如下：

• ∠a = ∠b 称为对应角
• ∠c = ∠a，它们被称为垂直对角
• ∠c + ∠b = 180°
• ∠c = ∠d，它们被称为交替内角。

坡

斜率定义为纵坐标（y坐标）相对于横坐标（x坐标）的变化率。它由字母m表示。它基本上说明了如果 x 从 x 变为 x+a 时 y 坐标将通过什么因素发生变化，其中 a 是实数。计算任何给定线的斜率的公式是：

slope =  y₂ – y₁/x₂ – x₁

斜率 (m) 也可以表示为函数.，用 θ 表示，即直线与 x 轴所成的角度。斜率 m = tan θ = (y ₂ – y ₁ )/(x ₂ – x ₁ )

两条平行线之间的距离是多少？

解决方案：

To measure the distance between two parallel lines. Let’s take two arbitrary parallel lines. Two parallel lines will have the same slope. The equation of a line is y = mx + c. So the equations of the two lines will be somewhat like this:

y = mx + c₁

y = mx + c₂

As shown in the above figure, the two lines are drawn with a slope equal to m. The distance between them has been taken as d.  The line y = mx + c₁ intercepts the y-axis at the point A(0, c₁) and the other line y = mx + c₂ intercepts the y-axis at B(0, c₂). The length AB is given by c₂ – c₁,  d can be calculated using trigonometry considering the triangle ABC. Considering the triangle ABC,  

d = AB × cos θ

Or, d = (c₂ – c₁) × cos θ

Or, d = (c₂ – c₁)/ (sec θ); since, cos θ = 1/sec θ

Or, d = (c₂ – c₁) / √(1 + tan² θ) ;  since, sec² θ = 1 + tan² θ

Or d = (c₂ – c₁)/√(1 + m²)

Therefore, the distance d between two parallel lines is given by,

$d= \frac{|c_2-c_1|}{\sqrt{1+m^2}}$

示例问题

问题 1：检查 5y = 45x +15 和 y = 9x + 3 的线是否平行。

解决方案：

Given the equation of lines are,

Equation 1: 5y = 45x + 15

Equation 2: y = 9x + 3

In order to check if these two lines are parallel or not, compare their slope and check if its equal.

Equation 1: 5y = 45x + 9

Taking 5 common in R.H.S 

5y = 5(9x + 3)

Or, y = 9x + 3

As, equation 1 and equation 2 are the same equation after evaluating and their slopes are same. So the given lines are parallel lines.

问题 2：求直线 y = 2x + 5 和 y = 2x + 10 之间的距离

解决方案：

m = slope = 2

c₁ = 5

c₂ = 10

Let the distance between the two lines is d

Using the formula we get,

|10 – 5|

d = √(1+2²)

d = √5 units

问题 3：求直线 y = 2x + 5 和 5y = 10x + 20 之间的距离

解决方案：

Equation of second line can be written as 5 × y = 5 × (2x + 4)

Or, y = 2x + 4

Therefore m = slope = 2

c₁ = 5

c₂ = 4

Let the distance between the two lines is d

Using the formula,

|5 – 4|

d = √(1 + 2²)

d = 1/√5 = 0.022 units