Root 2 simply is the hypotenuse of a right triangle

而 √2 只是两边长度 1 角三角形之斜边。

A right triangle is a two dimensional figure with three side, two of which are perpendicular.

角三角形是具有三条边二维图形，其中两条是垂。

And you should also see that this is a right triangle that I have set up over here.

这个平行于斜面 而这个垂于斜面这个平行于斜面 而这个垂于斜面。

Take four identical right triangles with side lengths a and b and hypotenuse length c.

取四个全等角三角形，两边别长 a 和 b，斜边长 c。

The most famous relation in geometry is the relation between the sides and the hypotenuse of a right triangle.

几何中最著名关系是角三角形边和斜边之间关系。

According to the converse of the Pythagorean theorem, that has to make a right triangle, and, therefore, a square corner.

根据毕达哥拉斯逆定理，这就可以形成一个角三角形，因此，便可形成角。

Picture a right triangle where side A is the tree.

想象一个角三角形， 其中 A 边是树。

Somehow, rhomboids and isosceles right triangles didn't seem so important.

知道什么 菱形和等腰角三角形似乎没有那么重要了。

Similar experiments show that all shapes other than acute triangles, including right triangles, will also wake up.

类似实验表明， 除了锐角三角形之外所有形状，包括角三角形， 也会被唤醒。

You've got Babylonian surveyors who have their own unique understanding of right triangles and rectangles, and they're using it.

你有巴比伦测量员，他们对角三角形和矩形有自己独特理解，并且他们正在使用它。

That states that the squares of the sides of a right triangle add up to the square of its hypotenuse.

这表明角三角形各边平方加起来等于其斜边平方。

And if you'd really like to convince yourself, you could build a turntable with three square boxes of equal depth connected to each other around a right triangle.

如果你真想说服自己，你可以建个转台，上面有三个相同深度正方形盒子，它们靠一个角三角形相连。

But how do we know that the theorem is true for every right triangle on a flat surface, not just the ones these mathematicians and surveyors knew about?

但是我们怎么知道这个定理对平面上每个角三角形都成立，而是一些数学家和测量人员所推测呢？

This proof divides one right triangle into two others and uses the principle that if the corresponding angles of two triangles are the same, the ratio of their sides is the same, too.

这种证明将一个角三角形两个部，利用了如下定理：如果两个三角形对应角相同，那么它们边比例也是相同。

You see, about a thousand years before the Greek astronomers were looking at the night sky. You've got Babylonian surveyors who have their own unique understanding of right triangles and rectangles, and they're using it.

在希腊天文学家观察夜空一千年以前，巴比伦测量员对角三角形和矩形有他们自己独特理解，他们还会使用它。

Trigonometry -- which we use to calculate the angles and sides of triangles -- is going to come up a lot in physics, because we'll be using right angle triangles all the time.

三角学——我们用来计算三角形角度和边——在物理学中会经常出现，因我们将一使用角三角形。

They all came up with elegant proofs for the famous Pythagorean theorem; the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse.

他们都对毕达哥拉斯定理（勾股定理）做出了精彩证明；这个定理是说，对于一个角三角形，一边平方加上另一边平方等于斜边平方。

They all came up with elegant proofs for the famous Pythagorean theorem, the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse.

他们都著名毕达哥拉斯定理提出了优雅证明，这条规则说对于角三角形，一侧平方加上另一侧平方等于斜边平方。