Pythagoras' Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most useful results in geometry. It links the three sides of a right-angled triangle and allows us to find missing lengths in both pure mathematics and real-life situations. This revision guide covers the key ideas, worked examples, common mistakes and exam-style questions for Cambridge KS3 Mathematics.

勾股定理是几何中最有用的结论之一。它把直角三角形三条边的长度联系起来，帮助我们在纯数学和现实情境中求未知边长。本复习指南涵盖 Cambridge KS3 数学中勾股定理的核心概念、例题、常见错误和考试风格练习题。

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

In any right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides.

在任意一个直角三角形中，最长边的平方等于另外两条边的平方和。

a² + b² = c²

Here a and b are the two shorter sides that meet at the right angle, and c is the longest side opposite the right angle, called the hypotenuse.

这里 a 和 b 是相交成直角的两条较短边（直角边），c 是直角对面的最长边，称为斜边。

The theorem only works for right-angled triangles, so always check that one angle is 90° before using it.

该定理只适用于直角三角形，因此使用前一定要先确认其中一个角是 90°。

2. The Hypotenuse and Legs | 斜边与直角边

The hypotenuse is always the longest side and is always opposite the right angle. It is the side you find directly when using a² + b² = c².

斜边永远是最长边，永远位于直角对面。使用 a² + b² = c² 时，斜边是直接求解的边。

The legs are the two sides that form the right angle. They can be labelled a and b in any order, and swapping them does not change the result.

直角边是组成直角的两条边。它们可以任意标记为 a 和 b，互换标记不会影响计算结果。

Hypotenuse: the longest side, opposite the right angle. | 斜边：最长的边，直角对面。
Legs: the two sides that make the right angle. | 直角边：组成直角的两条边。
Label the legs as a and b in any order. | 直角边可以任意标记为 a 和 b。

3. The Formula a² + b² = c² | 公式 a² + b² = c²

The standard formula is usually written as c² = a² + b², where c is the hypotenuse. This form is used when you need to find the longest side.

标准公式通常写作 c² = a² + b²，其中 c 是斜边。当需要求最长边时使用这种形式。

You can rearrange the formula to find a shorter side: a² = c² − b² or b² = c² − a².

你也可以变形公式来求直角边：a² = c² − b² 或 b² = c² − a²。

Remember that after finding a² or b², you must take the square root to get the actual side length. Squaring gives an area-like value, not the side length itself.

记住求出 a² 或 b² 后，还需要开平方才能得到实际边长。平方得到的是类似面积的值，而不是边长本身。

4. Worked Example: Finding the Hypotenuse | 例题：求斜边

A right-angled triangle has legs of 6 cm and 8 cm. Find the length of the hypotenuse.

一个直角三角形的两条直角边分别为 6 cm 和 8 cm，求斜边的长度。

Substitute into c² = a² + b²:

代入 c² = a² + b²：

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

The hypotenuse is 10 cm. Notice that the two legs produce a whole-number hypotenuse because 6, 8 and 10 form a Pythagorean triple.

斜边为 10 cm。注意两条直角边得到了整数斜边，因为 6、8 和 10 构成一组勾股数组。

5. Worked Example: Finding a Shorter Side | 例题：求直角边

A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the other leg.

一个直角三角形斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。

Use the rearranged formula b² = c² − a²:

使用变形后的公式 b² = c² − a²：

b² = 13² − 5² = 169 − 25 = 144

b = √144 = 12 cm

So the missing leg is 12 cm. This is another example of a Pythagorean triple, namely 5-12-13.

所以缺失的直角边是 12 cm。这是勾股数组的另一个例子，即 5-12-13。

6. Checking for Right-Angled Triangles | 判断直角三角形

If the three side lengths of a triangle satisfy a² + b² = c², then the triangle is right-angled. This is the converse of Pythagoras’ theorem and is very useful for checking angles in practical work.

如果三角形三边满足 a² + b² = c²，那么这个三角形就是直角三角形。这是勾股定理的逆定理，在实际工作中常用来检查角度。

For example, do side lengths 5 cm, 12 cm and 13 cm form a right-angled triangle?

例如，边长为 5 cm、12 cm 和 13 cm 的三角形是直角三角形吗？

5² + 12² = 25 + 144 = 169 = 13²

So the condition holds, and the triangle is right-angled. If the two values are not equal, the triangle is not right-angled. For instance, 3, 4 and 6 give 3² + 4² = 25, but 6² = 36.

所以条件成立，该三角形是直角三角形。如果两个值不相等，这个三角形就不是直角三角形。例如 3、4、6 得到 3² + 4² = 25，而 6² = 36。

7. Pythagorean Triples | 勾股数组

A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². The smallest and most famous is 3-4-5.

勾股数组是满足 a² + b² = c² 的一组三个正整数。最小且最著名的是 3-4-5。

Other common triples include 5-12-13, 7-24-25 and 8-15-17. Recognising these can save time in calculations and checking answers.

其他常见勾股数组包括 5-12-13、7-24-25 和 8-15-17。熟悉这些数组可以节省计算时间，也有助于检查答案。

Triple \| 勾股数组	Check \| 验证
3, 4, 5	3² + 4² = 9 + 16 = 25 = 5²
5, 12, 13	5² + 12² = 25 + 144 = 169 = 13²
7, 24, 25	7² + 24² = 49 + 576 = 625 = 25²
8, 15, 17	8² + 15² = 64 + 225 = 289 = 17²

You can multiply every number in a triple by the same factor to get a new triple. For example, doubling 3-4-5 gives 6-8-10, which also works.

你可以把勾股数组中的每个数都乘以同一个倍数得到新的数组。例如 3-4-5 乘以 2 得到 6-8

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Pythagoras’ Theorem | 勾股定理