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.

希腊天文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。

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，这就可以成一个直角三角，因此，便可成直角。

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直角三角边和边之间关系。

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

这个平行于面 而这个垂直于面 这个平行于面 而这个垂直于面。

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，对于一个直角三角，一边平方加上另一边平方等于边平方。

So the magnitude of this over the hypotenuse of this right triangle.

直角三角边是多少 直角三角边是多少。

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 只是两边长度为 1 直角三角之边。

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，边长 c。

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成立，而不是一些数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.

这种证明将一个直角三角分为两个部分，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。

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盒子，它们靠一个直角三角相连。

Picture a right triangle where side A is the tree.

想象一个直角三角， 其中 A 边是树。

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.

他们都为着名勾股定理提出了简洁证明，该定理指出， 对于一个直角三角，一条直角边平方加上另一条直角边平方等于边平方。

And so what they thought, and this is what the ancient historians described, is that they effectively tried to draw a right angle triangle with the two entrances at each end of the hypotenuse.

他们认为， 古代历史家描述就是，他们实际上试图画一个直角三角， 两个入口分别位于边两端。

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——我们用来计算三角角度和边——物理中会经常出现，因为我们将一直使用直角三角。

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

不知道为什么 菱和等腰直角三角似乎没有那么重要了。

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，都这里一瞥。

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度。测量系统方差是该数字平方。

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。