Root 2 simply is the hypotenuse of a right triangle

而 √2 只是两边长度为 1 的 直角三角形之斜边。

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

取四全等的直角三角形，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.

直角三角形是具有三条边的二维图形，其中两条是垂直的。

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

这于斜面 而这垂直于斜面这于斜面 而这垂直于斜面。

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

几何中最著的关系是直角三角形的边和斜边之间的关系。

Picture a right triangle where side A is the tree.

想直角三角形， 其中 A 边是树。

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

不知道为什么 菱形和等腰直角三角形似乎没有那么重要了。

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

根据毕达哥拉斯的逆定理，这就可以形成直角三角形，因此，便可形成直角。

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

旦我们画出那条径向线，加上距离 d，我们就得到了另直角三角形。

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

类似的实验表明， 除了锐角三角形之外的所有形状，包括直角三角形， 也会被唤醒。

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

如果我们在这里向左画条垂直线，我们就会得到直角三角形。

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

你有巴比伦的测量员，他们对直角三角形和矩形有自己独特的理解，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.

这表明直角三角形各边的方加起来等于其斜边的方。

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形边的长度。测量系统方差是该数字的方。

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定理对面上的每直角三角形都成立，而不是些数学家和测量人员所推测的呢？

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.

如果你真的想说服自己，你可以建转台，上面有三相同深度的正方形盒子，它们靠直角三角形相连。

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.

这种证明将直角三角形分为两部分，利用了如下定理：如果两三角形对应的角相同，那么它们的边的比例也是相同的。

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.

他的音乐，他的三角，他的金大腿：毕达哥拉斯关于音乐和命理的理论，关于直角三角形的勾股定理，以及毕达哥拉斯有条金大腿的神话，都在这里瞥。

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的千年以前，巴比伦的测量员对直角三角形和矩形有他们自己独特的理解，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的角度和边——在物理学中会经常出现，因为我们将直使用直角三角形。