📚 Pythagoras' Theorem in Cambridge KS3 Maths | 剑桥 KS3 数学：勾股定理
Pythagoras' theorem is one of the most important results in Cambridge KS3 mathematics. It connects the three sides of a right-angled triangle and gives you a reliable way to find a missing length. This article explains the theorem step by step, with worked examples, common errors and exam-style practice.
勾股定理是剑桥 KS3 数学中最重要的结论之一。它将直角三角形的三条边联系起来，为你提供了一种可靠的方法来求缺失的边长。本文逐步解释该定理，并配有例题、常见错误和考试风格练习。
1. What Is Pythagoras' Theorem? | 什么是勾股定理？
Pythagoras' theorem only works in right-angled triangles. It says that the square of the longest side is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它表明最长边的平方等于另外两条边的平方之和。
a² + b² = c²
Here, c is the hypotenuse, the side opposite the right angle. The letters a and b are the two shorter sides.
这里 c 是斜边，也就是直角所对的边。字母 a 和 b 是两条较短的直角边。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is always the longest side of a right-angled triangle. It is also always opposite the right angle, never adjacent to it.
斜边始终是直角三角形中最长的边。它也总是与直角相对，从不与直角相邻。
For example, in a triangle with sides 3 cm, 4 cm and 5 cm, the side of 5 cm must be the hypotenuse because it is the largest value.
例如，在一个边长分别为 3 cm、4 cm 和 5 cm 的三角形中，5 cm 的边一定是斜边，因为它是最大的值。
3. The Formula a² + b² = c² | 公式 a² + b² = c²
You can rearrange the formula depending on which side is missing. To find the hypotenuse, add the squares of the two shorter sides and then take the square root.
你可以根据缺失的是哪条边来调整公式。要求斜边，先求两条较短边的平方和，然后再开平方。
c = √a² + b²
To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
要求一条直角边，用斜边的平方减去已知直角边的平方，然后再开平方。
a = √c² − b²
4. Worked Example: Finding the Hypotenuse | 例题：求斜边
A right-angled triangle has shorter sides 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 a = 6 and b = 8 into the formula:
将 a = 6 和 b = 8 代入公式：
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
Always remember to give units in your final answer.
请始终记得在最终答案中写出单位。
5. Worked Example: Finding a Shorter Side | 例题：求直角边
In a right-angled triangle, the hypotenuse is 13 cm and one shorter side is 5 cm. Find the missing side.
在一个直角三角形中，斜边为 13 cm，一条直角边为 5 cm。求缺失的边。
Use the rearranged formula with c = 13 and b = 5:
使用调整后的公式，令 c = 13 和 b = 5：
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
This checks because 5, 12, 13 is a well-known Pythagorean triple.
这是成立的，因为 5、12、13 是一组著名的勾股数。
6. Checking Right-Angled Triangles | 判断直角三角形
reverse. If the three sides satisfy a² + b² = c², then the triangle must be right-angled.
你可以反过来使用该定理。如果三条边满足 a² + b² = c²，那么这个三角形一定是直角三角形。
For example, check whether 4 cm, 5 cm and 6 cm form a right-angled triangle:
例如，判断 4 cm、5 cm 和 6 cm 是否构成直角三角形：
4² + 5² = 16 + 25 = 41, but 6² = 36
Since 41 ≠ 36, the triangle is not right-angled.
因为 41 ≠ 36，所以这个三角形不是直角三角形。
7. Real-Life Applications | 实际应用
Pythagoras' theorem is used in many real situations. For example, a ladder leaning against a wall forms a right-angled triangle with the ground.
勾股定理在许多实际情况中都有应用。例如，斜靠在墙上的梯子与地面构成一个直角三角形。
If a 5 m ladder is placed 2 m away from a wall, the height reached on the wall is:
如果一把 5 m 长的梯子底部离墙 2 m，那么它在墙上达到的高度为：
height = √5² − 2² = √25 − 4 = √21 ≈ 4.58 m
This kind of calculation is common in construction, navigation and design.
这类计算在建筑、导航和设计中都很常见。
8. Pythagorean Triples | 勾股数
A Pythagorean triple is a set of three whole numbers that satisfy the theorem exactly. These are useful for mental maths and quick checking.
勾股数是一组三个整数，能精确满足定理。它们对心算和快速检验很有用。
| Shorter sides | 直角边 | Hypotenuse | 斜边 |
|---|---|
| 3, 4 | 5 |
| 5, 12 | 13 |
| 7, 24 | 25 |
| 8, 15 | 17 |
Recognising these triples can save time in exams.
识别这些勾股数可以在考试中节省时间。
9. Common Mistakes | 常见错误
One common mistake is adding or subtracting the sides before squaring them. Always square first, then add or subtract.
一个常见错误是先对边进行加减，然后再平方。一定要先平方，然后再加减。
Wrong: (6 + 8)² = 196
错误：(6 + 8)² = 196
Right: 6² + 8² = 36 + 64 = 100
正确：6² + 8² = 36 + 64 = 100
Another mistake is using the hypotenuse in the wrong position. The hypotenuse must be on its own on one side of the equation.
另一个错误是把斜边放在错误的位置。斜边必须单独位于方程的一边。
10. Exam-Style Practice | 考试风格练习
Try these questions to test your understanding. Answers are given in the next section.
尝试以下问题来检验你的理解。答案在下一节中给出。
A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.
一个直角三角形两条直角边分别为 9 cm 和 12 cm。求斜边。
The hypotenuse is 17 cm and one shorter side is 8 cm. Find the other side.
斜边为 17 cm，一条直角边为 8 cm。求另一条直角边。
Do 6 cm, 7 cm and 9 cm form a right-angled triangle?
6 cm、7 cm 和 9 cm 能否构成直角三角形？
11. Answers and Explanations | 答案与解析
For the first question, compute c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15 cm.
第一题，计算 c² = 9² + 12² = 81 + 144 = 225，所以 c = √225 = 15 cm。
For the second question, compute b² = 17² − 8² = 289 − 64 = 225, so b = √225 = 15 cm.
第二题，计算 b² = 17² − 8² = 289 − 64 = 225，所以 b = √225 = 15 cm。
uestion, check 6² + 7² = 36 + 49 = 85, but 9² = 81. Since 85 ≠ 81, it is not right-angled.
第三题，检验 6² + 7² = 36 + 49 = 85，而 9² = 81。因为 85 ≠ 81，所以不是直角三角形。
12. Summary | 总结
Pythagoras' a² + b² = c² to find a missing hypotenuse, or a² = c² − b² to find a missing shorter side.
勾股定理只适用于直角三角形。记住 c 始终是斜边，使用 a² + b² = c² 求缺失的斜边，或使用 a² = c² − b² 求缺失的直角边。
Practise recognising Pythagorean triples and always show your working clearly with units.
练习识别勾股数，并始终清晰展示计算过程和单位。
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