如何推导出距离公式？

用来覆盖两点之间长度的导线的长度就是两点之间的距离。如果两点位于同一水平线或同一垂直线上，则可以通过减去不相同的坐标来计算两点之间的距离。

距离公式可用于计算两条线段之间测量的距离。距离公式可用于计算多边形所有边的长度之和、多边形在坐标平面上的周长、多边形的面积等。三角形的边长也可以使用距离公式。三角形、不等边、等腰或等边的类型也可以使用距离公式确定。

两点之间的距离

该点的 x 坐标称为横坐标，而 y 坐标称为纵坐标。距离公式可用于计算 xy 平面两点之间的距离。有序对 (x, y) 表示点的坐标，其中 x 坐标定义为点到 x 轴的距离，y 坐标是点到 y 轴的距离。

以下公式可用于计算 2d 平面中两点之间的距离：

考虑给定坐标轴上的任意两个点 A(x ₁ ,y ₁ ) 和 B(x ₂ ,y ₂ )。以下两点之间的距离如下：

d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

如何推导出距离公式？

证明：

In the right-angled triangle ABC, we have, 

By the Pythagoras theorem,

AB² = AC² + BC²

Since, we have derived the formula of distance calculation between two points, therefore, 

The distance between points A and C is calculated as (x₂ – x₁)² 

The distance between points A and C is calculated as (y₂ – y^₁)² 

The distance, d is calculated as, 

d² = (x₂ – x₁)² + (y₂ – y₁)²

Now, 

Taking the square root on both sides,

d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

This is the called the distance between two points formula.

点到线的距离公式

一个点 (x _1, y ₁ ) 和一条线 ax + by + c = 0

d = $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

两条平行线之间的距离公式

两条平行线 ax + by + c ₁ = 0 和 ax + by + c ₂ = 0

d = $\frac{|c_2-c_1|}{\sqrt{a^2+b^2}}$

示例问题

问题 1. 计算点 X(5, 15) 和 Y(4, 14) 之间的距离

解决方案：

Here to find the distance between point A and B

Applying distance formula

d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

d = $\sqrt{(4-5)^2+(14-15)^2}\\ =\sqrt{1^2+1^2}\\ =\sqrt{2}\\ =\sqrt{2}$

Distance between X and Y is √2 or 1.41

问题 2. 求平行线 -6x + 20y + 10 = 0 和 -6x + 20y + 20 = 0 之间的距离。

解决方案：

The general equation of parallel lines is

Ax + By + C₁ = 0 and Ax + By + C₂ = 0, 

Here, 

A = -6, 

B = 20, 

C₁ = 10 and 

C₂ = 20.

Applying formula

d = $\frac{|c_2-c_1|}{\sqrt{a^2+b^2}}$

d = $\frac{|20-10|}{\sqrt{-6^2+20^2}}\\ \frac{|10|}{\sqrt{36+400}}\\ \frac{10}{\sqrt{436}}$

d = 10/√436

问题 3. 计算线 4a + 6b – 26 = 0 到点 (2, –4) 之间的距离。

解决方案：

The general equation of parallel lines is

A point (x₁, y₁) and a line ax + by + c = 0

Here,

A = 4, B = 6 and C = –26

Applying formula

d = $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$

d = $\frac{|4\times2+6\times(-4)+(-26)|}{\sqrt{4^2+6^2}}\\ =\frac{8-24-26}{\sqrt{16+36}}\\ =\frac{-42}{\sqrt{52}}$

d = -42/√52

问题 4. 计算点 A(-25, -5) 和 B(-16, -4) 之间的距离？

解决方案：

Here to find the distance between point A and B

Applying distance formula

d = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

d =$\sqrt{(-16-(-25))^2+(-4-(-5))^2}\\ =\sqrt{(-16+25)^2+(-4+5)^2}\\ =\sqrt{(9)^2+(1)^2}\\ =\sqrt{81+1}\\ =\sqrt{82}$

Distance between A and B is √82 or 9.05.