Pythagoras’ Theorem: Right-Angled Triangles and Its Converse | 勾股定理：直角三角形及其逆定理

📚 Pythagoras’ Theorem: Right-Angled Triangles and Its Converse | 勾股定理：直角三角形及其逆定理

Pythagoras’ theorem is a central topic in the Cambridge KS3 mathematics curriculum. It applies only to right-angled triangles and gives a relationship between the lengths of the three sides: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

勾股定理是剑桥 KS3 数学课程的核心主题。它只适用于直角三角形，并给出三条边长之间的关系：斜边的平方等于另外两条边的平方和。

1. What is Pythagoras’ theorem? | 什么是勾股定理？

In any right-angled triangle, the side opposite the right angle is called the hypotenuse. If the two shorter sides have lengths a and b, and the hypotenuse has length c, then the theorem states the following.

在任何直角三角形中，与直角相对的边叫做斜边。如果两条较短边的长度为 a 和 b，斜边为 c，那么定理表述如下。

c² = a² + b²

This formula can be rearranged to find any one of the three sides, provided the other two lengths are known. The theorem is named after the Greek mathematician Pythagoras.

这个公式可以变形来求三条边中的任意一条，只要另外两条边已知。该定理以古希腊数学家毕达哥拉斯命名。

2. The hypotenuse and the shorter sides | 斜边与较短直角边

The hypotenuse is always the longest side of a right-angled triangle. It is opposite the largest angle, which is the right angle of 90°.

斜边总是直角三角形中最长的边。它对着最大的角，即 90° 的直角。

The two shorter sides are often called the legs. They form the right angle and can be labelled in any order. In the formula c² = a² + b², c must be the hypotenuse, while a and b are the two legs.

两条较短的边通常称为直角边。它们构成直角，标注顺序可以互换。在公式 c² = a² + b² 中，c 必须是斜边，a 和 b 是两条直角边。

3. Finding the hypotenuse | 求斜边

To find the hypotenuse, substitute the two shorter sides into the formula and then take the square root of the result.

要求斜边，将两条较短边代入公式，然后对结果取平方根。

c = √(a² + b²)

For example, if a = 6 cm and b = 8 cm, then c² = 6² + 8² = 36 +

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