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

取四个全等直角三角形，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二维图形，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 直角三角形之斜边。

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.

根毕达哥逆定理，这就可以形成一个直角三角形，因此，便可形成直角。

Picture a right triangle where side A is the tree.

想象一个直角三角形， 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形来推断这一点。

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。

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形。

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，包括直角三角形， 也会被唤醒。

When you draw a vector, it's a lot like the HYPOTENUSE of a right angle triangle.

当你画一个箭头时，它很像 直角三角形斜边。

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，他们对直角三角形和矩形有自己独特理解，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形。

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平方加起来等于其斜边平方。

For example, the isosceles or the scalene, or really any much either of these two can also be a " right angle triangle" .

例如，等腰或斜角，或者这两者中任何一个都可以是“直角三角形”。

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长度。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形都成立，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，它们靠一个直角三角形相连。

And the reasoning here is they both have that triple  ticked side, a double ticked side, and they're both 90-degree triangles, so this follows by the  90-degree side-side-angle congruence relation.

这里理由是它们都有三条三重标记边，两条双标记边， 并且它们都是90度直角三角形，所以这符合90度边边角全等关系。

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形分为两个部分，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。