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

取四个全等直角三角，两边分别长 a 和 b，斜边长 c。

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

直角三角是具有三条边二维图，其中两条是垂直。

Root 2 simply is the hypotenuse of a right triangle

而 √2 只是两边长 1 直角三角之斜边。

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 而这个垂直于斜面这个平行于斜面 而这个垂直于斜面。

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

几何中最著关系是直角三角边和斜边之间关系。

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，这就以一个直角三角，因此，便直角。

Picture a right triangle where side A is the tree.

想象一个直角三角， 其中 A 边是树。

And then you are trying to infer that by constructing a right triangle.

然后你试图通过构建一个直角三角来推断这一点。

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

不知道什么 菱和等腰直角三角似乎没有那么重要了。

Once we draw that radial line, together with the distance d we have another right triangle.

一旦我们画出那条半径线，连同距离d我们就有了另一个直角三角。

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

类似实验表明， 除了锐角三角之外所有状，包括直角三角， 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，他们对直角三角和矩有自己独特理解，并且他们正在使用它。

And if we draw a vertical line up to the left here, we have  ourselves a right triangle.

如果我们在这里向上画一条竖直线，我们就得到了一个直角三角。

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

这表明直角三角各边平方加起来等于其斜边平方。

This is the length of measurement system standard deviation leg of the right triangle. The measurement system variance is the square of this number.

这是测量系统标准偏差直角三角边长。测量系统方差是该数字平方。

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直角三角都立，而不是一些数学家和测量人员所推测呢？

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，你以建个转台，上面有三个相同深正方盒子，它们靠一个直角三角相连。

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.

这种证明将一个直角三角分两个部分，利用了如下定理：如果两个三角对应角相同，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，巴比伦测量员对直角三角和矩有他们自己独特理解，他们还会使用它。

His musics, his trigon, his golden thigh: Pythagorean theories about music and numerology, the Pythagorean theorem about right triangles, and the myth that Pythagoras had a golden thigh are glanced here.

他音乐，他三角，他金大腿：毕达哥拉斯关于音乐和命理理论，关于直角三角勾股定理，以及毕达哥拉斯有一条金大腿神话，都在这里一瞥。