📚 Cambridge KS3 Maths: Pythagoras' Theorem | 剑桥初中数学：勾股定理
This KS3 Cambridge revision guide covers the key facts you need for Pythagoras' theorem, with worked examples and exam-style tips. It is designed for students working through page 181, exercise 2 and similar questions, and it builds the foundation you will need later for trigonometry and coordinate geometry.
这本剑桥 KS3 复习指南涵盖勾股定理所需的关键知识点，配有例题和考试技巧。它适合正在学习第 181 页练习 2 及类似题目的学生，并为你以后学习三角函数和坐标几何打下基础。
1. What is Pythagoras' Theorem? | 什么是勾股定理？
Pythagoras' theorem is a rule that works only for right-angled triangles. It states that the square of the longest side, called the hypotenuse, is equal to the sum of the squares of the other two shorter sides.
勾股定理是一条只适用于直角三角形的规则。它指出，最长边（称为斜边）的平方等于另外两条较短边的平方和。
In symbols, for a right-angled triangle with legs a and b, and hypotenuse c:
用符号表示，对于直角边为 a 和 b、斜边为 c 的直角三角形：
a² + b² = c²
The theorem is named after the Greek mathematician Pythagoras of Samos (about 570-495 BC), although Babylonian and Indian mathematicians had already used the same relationship more than a thousand years earlier. In the Cambridge KS3 course you meet it as a practical tool for finding an unknown side; later, at IGCSE and A Level, it becomes the foundation of trigonometry, the sine rule, the cosine rule and coordinate geometry.
该定理以古希腊数学家萨摩斯的毕达哥拉斯（约公元前 570-495 年）命名，不过巴比伦和印度的数学家在一千多年前就已经使用过同样的关系。在剑桥 KS3 课程中，它是求未知边长的实用工具；到了 IGCSE 和 A Level，它又会成为三角函数、正弦定理、余弦定理和坐标几何的基础。
Three conditions must be true before you may use it: the triangle must contain a right angle, you must already know two of the sides, and the side you want must belong to that same triangle. If the triangle is not right-angled, this theorem does not apply at all - you would need the sine rule or the cosine rule instead.
使用它之前必须满足三个条件：三角形必须含有直角；你必须已经知道其中两条边；你要求的边必须属于同一个三角形。如果三角形不是直角三角形，这个定理完全不适用——你需要改用正弦定理或余弦定理。
2. The Hypotenuse and the Legs | 斜边与直角边
The hypotenuse is always the side opposite the right angle. It is the longest side in the triangle. The other two sides are called the legs, and they form the right angle.
斜边始终是直角三角形中直角对面的那条边。它是三角形中最长的边。另外两条边称为直角边，它们形成直角。
Always label the hypotenuse as c before substituting into the formula. This avoids many common mistakes.
在代入公式之前，始终将斜边标为 c。这可以避免许多常见错误。
A reliable four-step routine for labelling any diagram:
以下是一个可靠的"四步标记法"，适用于任何图形：
- Mark the right angle with a small square. 用一个小正方形标出直角。
- Find the side opposite that square and label it c. 找到该正方形对面的那条边，标为 c。
- Label the remaining two sides a and b; the order does not matter. 把剩下两条边标为 a 和 b，顺序无所谓。
- Write the equation a² + b² = c² with the numbers substituted. 把数字代入，写出等式 a² + b² = c²。
Rearranging the formula gives the three versions you will use most often:
把公式变形，可以得到最常用的三个版本：
c = √a² + b² a = √c² − b² b = √c² − a²
The first version is for finding the hypotenuse, so you add. The second and third versions are for finding a leg, so you subtract. Remembering "hypotenuse: add; leg: subtract" removes a large number of errors.
第一个版本用于求斜边，所以做加法；第二、第三个版本用于求直角边，所以做减法。记住"求斜边用加，求直角边用减"，可以消除大量错误。
3. The Formula a² + b² = c² | 公式 a² + b² = c²
The formula says that if you square both shorter sides and add the results, you get the square of the hypotenuse. To find the hypotenuse itself you must take the square root at the end. Forgetting this final square root is the single most common mistake in KS3 exams.
公式的含义是：把两条较短边分别平方后相加，就得到斜边的平方。若要得到斜边本身，最后必须开平方。忘记最后开平方是 KS3 考试中最常见的错误。
Read the formula as three separate operations:
可以把公式理解为三个独立的运算步骤：
- Square each of the two legs separately. 先把两条直角边分别平方。
- Add the two squares together. 再把两个平方结果相加。
- Take the square root of the total to get the length. 最后对总和开平方，得到长度。
Because the formula involves squares, it is very useful to recognise square numbers instantly. The table below lists the squares you should know by heart for KS3.
由于公式涉及平方，能够一眼认出完全平方数会非常有用。下表列出 KS3 阶段应当熟记的平方数。
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
If the total after adding is one of the numbers in the bottom row, the square root is a whole number and your answer will be exact. If it is not, you will need a calculator or you should leave the answer in surd form.
如果相加后的总和出现在下排数字中，那么平方根是整数，答案就是精确值。如果没有出现，你就需要用计算器，或者把答案保留为根式形式。
4. Finding the Hypotenuse: Worked Examples | 求斜边：例题精讲
When the two legs are known and you want the longest side, you add the squares. Work through the five lines below in the same order every time, and write each line down - method marks are awarded for correct working, not only for the final answer.
当已知两条直角边、要求最长边时，把平方相加。每次都按下面这五行顺序书写，并且每一行都要写出来——过程分是给正确步骤的，而不只是给最终答案。
Example 1: A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
例 1：一个直角三角形的两条直角边分别为 6 cm 和 8 cm，求斜边。
- Formula: a² + b² = c² 公式：a² + b² = c²
- Substitute: 6² + 8² = c² 代入：6² + 8² = c²
- Square: 36 + 64 = c² 平方：36 + 64 = c²
- Add: 100 = c² 相加：100 = c²
- Square root: c = √100 = 10 开平方：c = √100 = 10
The hypotenuse is 10 cm. Notice that the answer is longer than either leg, which is a useful check.
斜边是 10 cm。注意答案比任何一条直角边都长，这是一个很有用的检验方法。
Example 2: A right-angled triangle has legs of 5 m and 7 m. Find the hypotenuse correct to 3 significant figures.
例 2：一个直角三角形的两条直角边分别为 5 m 和 7 m，求斜边，结果保留 3 位有效数字。
Here 5² + 7² = 25 + 49 = 74, so c = √74 ≈ 8.60 m. Because 74 is not a perfect square, the exact answer is the surd √74, and the decimal is only an approximation. Always keep the full calculator value until the final rounding step.
这里 5² + 7² = 25 + 49 = 74，所以 c = √74 ≈ 8.60 m。由于 74 不是完全平方数，精确答案是根式 √74，小数只是近似值。请始终保留计算器的完整数值，直到最后一步才四舍五入。
5. Finding a Missing Leg | 求缺失的直角边
When the hypotenuse and one leg are known, you must subtract instead of add. Write the formula with the known values first, then rearrange before you do any arithmetic.
当已知斜边和一条直角边时，必须用减法而不是加法。先把已知数值代入公式，再变形，最后才做运算。
Example 3: The hypotenuse of a right-angled triangle is 13 cm and one leg is 5 cm. Find the other leg.
例 3：一个直角三角形的斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。
- Formula: a² + b² = c² 公式：a² + b² = c²
- Substitute: a² + 5² = 13² 代入：a² + 5² = 13²
- Square: a² + 25 = 169 平方：a² + 25 = 169
- Subtract 25: a² = 144 减去 25：a² = 144
- Square root: a = √144 = 12 开平方：a = √144 = 12
take the square root, which gives 13.9 cm. That answer is longer than the hypotenuse, which is impossible, so the check catches the mistake immediately.
缺失的直角边是 12 cm。这里常见的错误是把 169 + 25 算成 194，然后开平方得到 13.9 cm。这个答案比斜边还长，不可能成立，因此检验能立刻发现错误。
Example 4: = 8 cm. Again the answer is sensible: shorter than the hypotenuse but not absurdly small.
例 4：斜边为 10 cm，一条直角边为 6 cm。则 a² = 100 − 36 = 64，所以 a = 8 cm。答案同样合理：比斜边短，也不会小得离谱。
6. Surds, Square Roots and Rounding | 根式、平方根与四舍五入
Many KS3 answers do not come out as whole numbers. When the exact value is required, leave the answer as a surd such as √74 rather than writing a rounded decimal. When the question says "give your answer correct to 2 decimal places" or "to 3 significant figures", use your calculator and round only at the very end.
许多 KS3 题目的答案并不是整数。如果题目要求精确值，就把答案保留为根式，例如 √74，而不要写成四舍五入的小数。如果题目说"保留 2 位小数"或"保留 3 位有效数字"，就使用计算器，并且只在最后一步才取近似。
Useful square roots to recognise:
值得记住的常用平方根：
- √2 ≈ 1.414 √3 ≈ 1.732 √5 ≈ 2.236
- √10 ≈ 3.162 √13 ≈ 3.606 √17 ≈ 4.123
- √50 ≈ 7.071 √74 ≈ 8.602 √200 ≈ 14.142
For example, √74 mal places and also 8.60 to 3 significant figures. Do not round in the middle of a calculation: if a question needs two steps, keEp the unrounded value in your calculator memory and round once at the end. Rounding early can change the final digit and lose accuracy marks.
例如，√74 ≈ 8.602325267，保留 2 位小数是 8.60，保留 3 位有效数字也是 8.60。不要在计算中途取近似：如果一道题需要两步计算，就把未取整的数值存在计算器里，最后只取一次近似。过早取整会改变最后一位数字，从而失分。
Also check the units carefully. Lengths are measured in cm, m or km, and areas in cm², m² or km². The theorem gives a length, so the final answer should never carry a squared unit.
还要仔细检查单位。长度用 cm、m 或 km 表示，面积用 cm²、m² 或 km² 表示。勾股定理求的是长度，所以最终答案不应该带平方单位。
7. Pythagorean Triples and Their Multiples | 勾股数及其倍数
ctly. Recognising triples saves time in exams because you can write the answer down without any calculation.
勾股数是指三个整数恰好满足 a² + b² = c² 的一组数。记住这些勾股数可以在考试中节省时间，因为你可以直接写出答案而无需计算。
| Triple 勾股数 | Check 检验 |
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 8, 15, 17 | 64 + 225 = 289 |
| 7, 24, 25 | 49 + 576 = 625 |
| 20, 21, 29 | 400 + 441 = 841 |
| 9, 40, 41 | 81 + 1600 = 1681 |
Any multiple of a triple is also a triple. Multiplying 3, 4, 5 by 2 gives 6, 8, 10; by 3 gives 9, 12, 15; by 5 gives 15, 20, 25; by 10 gives 30, 40, 50. Similarly, 5, 12, 13 can be doubled to 10, 24, 26 or tripled to 15, 36, 39.
任何勾股数的倍数仍然是勾股数。把 3, 4, 5 乘以 2 得到 6, 8, 10；乘以 3 得到 9, 12, 15；乘以
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