两点间距离公式推导是怎么回事,空间两点间距离公式怎么推导的

The distance formula is an algebraic expression that gives the shortest distance between two points in a two-dimensional space.

距离公式是一个代数表达式，它给出了二维空间中两点之间的最短距离。

You're sitting in math class trying to survive your latest pop quiz. The questions on Page 1 weren't too hard, but on the second page, you see a graph with two little dots on it labeled "Point 1" and "Point 2." And they're connected together by a diagonal line.

你坐在数学课上，试图通过最新的突击测验。第一页上的问题并不难，但在第二页上，你会看到一个图表，上面有两个小点，标着“点1”和“点2”。它们由一条对角线连接在一起。

Sweat trickles down your forehead as you read the prompt: "Find the distance between these points."

当你读到提示:“找出这些点之间的距离”时，汗水顺着你的额头流下来。

Don't panic — you don't even need a distance calculator to tackle this. The distance formula you're looking for is fairly straightforward and has ties to one of the most useful and famous concepts in all of mathematics: the Pythagorean theorem.

不要惊慌——你甚至不需要一个距离计算器来解决这个问题。你要找的距离公式是相当直接的，它与数学中最有用、最著名的一个概念有关:勾股定理。

Contents 内容

1. The Pythagorean Theorem Is Related to Distance Formula
2. 勾股定理与距离公式有关
3. Distance Formula and the Point Coordinate Plane
4. 距离公式与点坐标平面
5. How to Derive Distance Formula
6. 如何推导距离公式
7. Calculating the Distance Between Two Points
8. 计算两点之间的距离

勾股定理与距离公式有关

The Pythagorean theorem was named for the Greek philosopher Pythagoras. But he can't take sole credit for discovering it. Old Pythagoras lived from about 570 to 490 B.C.E. Yet more than 1,000 years before he was born, the ancient Babylonians were already aware of the geometric principle that now bears his name.

毕达哥拉斯定理以希腊哲学家毕达哥拉斯的名字命名。但他不能独享发现它的功劳。老毕达哥拉斯生活在约公元前570年至公元前490年。然而，在他出生前的1000多年前，古巴比伦人就已经知道了现在以他的名字命名的几何原理。

For those in need of a quick refresher, the Pythagorean theorem says:

对于那些需要快速复习的人，勾股定理是这样说的:

The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.

直角三角形斜边上的正方形面积等于其余边上的正方形面积之和。

We've got a couple of things to unpack here. A right triangle, also known as a right-angled triangle, is one that contains one 90-degree angle, also known as a right angle. The longest line on a right triangle is called the hypotenuse. (This is the line situated on the opposite side of the right angle.)

我们还有一些东西要整理。直角三角形，也叫直角三角形，是包含一个90度角的三角形，也叫直角。直角三角形中最长的一条线叫做斜边。(这是直角的对边。)

Now as we all know, a triangle may have three sides, but a square's got four.

我们都知道，三角形可能有三条边，但正方形有四条。

So imagine taking the hypotenuse of a right triangle and turning it into one of the four lines of a brand-new square. Then do the same thing to the other two sides in the original triangle. You'll end up with three individual squares.

想象一下，取一个直角三角形的斜边把它变成一个新正方形的四条线中的一条。然后对原来三角形的另外两条边做同样的事情。你会得到三个独立的方块。

As the Pythagorean theorem points out, the square you just made with the hypotenuse will have the same area as the other two squares put together. If the hypotenuse was labeled "c" and those other line segments were labeled "a" and "b," then we could express that idea like so:

正如毕达哥拉斯定理所指出的，你刚刚用斜边做的正方形的面积与其他两个正方形加起来的面积相等。如果斜边标注为c，其他线段标注为a和b，那么我们可以这样表达:

The Pythagorean theorem says a2 b2 = c2. The distance formula is derived by using the Pythagorean theorem.

毕达哥拉斯定理（勾股定理）说a2 b2 = c2。利用毕达哥拉斯定理推导了距离公式。

距离公式与点坐标平面

When most people hear the word "graph," they're picturing a chart with two lines — one vertical, one horizontal — that intersect each other at a right angle.

当大多数人听到“图表”这个词时，他们想象的是一张有两条线的图表——一条垂直线，一条水平线——它们相交成直角。

The vertical line is called the y-axis and its horizontal counterpart is the x-axis. Both lines work together to tell a story with data. Just take a look at this humorous graph from cartoonist Jorge Cham about somebody's not-so-relaxing vacation where the y-axis is labeled "stress" and the x-axis is labeled "time."

这条垂直线被称为y轴，它的水平对应物是x轴。这两条线一起用数据讲述一个故事。看看漫画家豪尔赫·查姆(Jorge Cham)画的这张关于某人不太放松的假期的幽默图表，y轴被标记为“压力”，x轴被标记为“时间”。

In order to make sense of where one point rests on your graph, you need to measure where it falls along the two dimensions (the x-axis and the y-axis). These are known as the point's coordinates. You need to find the coordinates for the first point and the second point before you can calculate the distance between them. You'll use the distance formula to measure the straight line segment connecting the two points.

为了理解一个点在图形上的位置，需要测量它在两个维度(x轴和y轴)上的位置。这些被称为点的坐标。在计算它们之间的距离之前，你需要找到第一个点和第二个点的坐标。你将使用距离公式来测量连接这两点的直线段。

如何推导距离公式

Enough preamble. The question you want answered is how to find the distance between two points on a graph (i.e., two sets of two coordinates).

足够的开场白。你想要回答的问题是如何找到图上两点之间的距离(即，两个坐标的两个集合)。

The first point and second point on your graph will each have an x coordinate and a y coordinate. You can calculate the shortest distance between these two points by using the Euclidean distance formula, which is a Pythagorean theorem-related algebraic expression. Here it is, folks:

图中的第一个点和第二个点都有一个x坐标和一个y坐标。您可以通过使用欧几里得距离公式计算这两点之间的最短距离，这是一个与勾股定理相关的代数表达式。下面就是，各位:

D = √(x2-x1)2 (y2-y1)2

Note that "D" means "distance." As for x2 and x1, they refer to the x coordinates of Point 2 and Point 1, respectively. Same goes for y2 and y1, except those are the two y coordinates.

注意“D”的意思是“距离”。其中x2和x1分别指点2和点1的x坐标。y2和y1也一样，只是它们是两个y坐标。

So to calculate the distance, our first step is to subtract x1 from x2. Then we have to multiply the resulting number by itself (or, in other words, "square" that number). After that, we must subtract y1 from y2 and then square the answer we get from doing so.

为了计算距离，第一步是用x2减去x1。然后我们必须将得到的数字乘以自身(或者，换句话说，“平方”这个数字)。之后，我们必须用y2减去y1，然后平方得到的结果。

This will leave us with two numbers we must add together. Then finally, take that number and find its square root. And that square root, ladies and gentlemen, is our distance.

这就剩下两个数字，我们必须把它们相加。最后，求出这个数的平方根。这个平方根，女士们先生们，就是我们的距离。

计算两点之间的距离

OK, so let's say Point 1 has an x coordinate of 2 and a y coordinate of 5. Let us also assume that Point 2's got an x coordinate of 9 and a y coordinate of 13.

假设点1的x坐标是2 y坐标是5。假设点2的x坐标是9 y坐标是13。

Plug those values into the handy dandy formula and you get this:

把这些值代入这个简单的公式，你会得到这个:

D = √(9-2)2 (13-5)2

What's 9 minus 2? Easy, 7. And 13 minus 5 is 8, of course.

9减2等于多少?容易,7。13 - 5当然是8。

So now we're left with this:

现在我们就剩下这个了:

D = √72 82

If you "square" 7 — as in, multiply the number by itself — you end up with 49. As for 8 squared that works out to 64. Let's plug those values into the equation.

如果你把7“平方”——也就是把这个数字自己相乘——你得到的结果是49。8的平方等于64。让我们把这些值代入方程

D = √49 64

Now we're cooking. Add 49 and 64 and you get 113.

现在我们开始做饭。49加64得113。

D = √113

What's the square root of 113? The answer is 10.63, so therefore:

113的平方根是多少?答案是10.63，因此:

D = 10.63

Go forth and ace that pop quiz!

去做突击测验吧!