📚 Pythagoras’ Theorem (Cambridge KS3, p241) | 勾股定理 (剑桥 KS3, p241)
This article is designed to help you master Pythagoras’ theorem, a key concept in the Cambridge KS3 Mathematics syllabus. Based on the exercise from page 241 (p241_1), we will explore how to find missing sides in right-angled triangles, understand the theorem’s statement and proof, and apply it to real-world problems.
本文旨在帮助你精通勾股定理，这是剑桥 KS3 数学大纲中的一个关键概念。基于第 241 页的练习 (p241_1)，我们将探索如何求直角三角形的缺失边长，理解定理的陈述与证明，并将其应用于实际问题。
1. What is a Right-Angled Triangle? | 什么是直角三角形？
A right-angled triangle is a triangle that has one interior angle equal to 90°. This special angle is often marked with a small square in diagrams. The side opposite the right angle is always the longest side, known as the hypotenuse.
直角三角形是其中一个内角等于 90° 的三角形。这个特殊的角在图中通常用一个小的正方形标记。与直角相对的边总是最长的边，称为斜边。
The two sides that form the right angle are called the legs (or catheti). In Pythagoras’ theorem, these are usually labelled a and b, while the hypotenuse is labelled c.
形成直角的两条边称为直角边。在勾股定理中，这两条边通常标记为 a 和 b，而斜边标记为 c。
2. Labelling the Triangle Correctly | 正确标记三角形
Correct labelling is essential before applying the theorem. The hypotenuse (c) must be the side opposite the right angle. The legs (a and b) can be assigned in any order to the remaining two sides that meet at the right angle.
在应用定理之前，正确的标记至关重要。斜边 (c) 必须是与直角相对的边。直角边 (a 和 b) 可以按任意顺序分配给相交于直角的其余两条边。
Always start by identifying the right angle, then find the side directly facing it – that is your hypotenuse. The other two sides are automatically your legs a and b.
始终从识别直角开始，然后找出正对直角的边 – 那就是你的斜边。另外两条边自然就是你的直角边 a 和 b。
3. Statement of Pythagoras’ Theorem | 勾股定理的陈述
Pythagoras’ theorem states that in any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This relationship only holds for right-angled triangles.
勾股定理指出，在任何直角三角形中，斜边 (c) 的长度的平方等于其他两条边 (a 和 b) 长度的平方和。这种关系仅适用于直角三角形。
c² = a² + b²
This simple equation is the foundation for solving countless geometry problems, from finding missing side lengths to checking if a triangle is right-angled.
这个简单的方程是解决无数几何问题的基础，从求缺失的边长到检查三角形是否为直角三角形。
4. Understanding the Formula a² + b² = c² | 理解公式 a² + b² = c²
The formula tells us that if we construct squares on each of the three sides, the area of the square on the hypotenuse equals the combined areas of the squares on the two legs. For example, if a = 3 cm and b = 4 cm, then a² = 9 cm², b² = 16 cm², so c² = 25 cm², giving c = 5 cm.
该公式告诉我们，如果我们在三条边上各构造一个正方形，那么斜边上的正方形面积等于两条直角边上正方形面积之和。例如，如果 a = 3 cm，b = 4 cm，则 a² = 9 cm²，b² = 16 cm²，因此 c² = 25 cm²，得到 c = 5 cm。
This geometric interpretation helps many students visualise why the theorem works. Remember that you are dealing with squared lengths, so the units will be squared (e.g., cm²) until you take the square root to find a side length.
这种几何解释帮助许多学生直观地理解定理为什么成立。记住，你处理的是长度的平方，因此在你开平方根求边长之前，单位将是平方单位（例如，cm²）。
5. Finding the Hypotenuse (c) | 求斜边 (c)
When you know the lengths of both legs (a and b), you can find the hypotenuse by squaring them, adding the results, and then taking the square root. The steps are: substitute, square, add, square root.
当你知道两条直角边（a 和 b）的长度时，你可以通过对它们求平方、相加，然后开平方根来求出斜边。步骤是：代入、平方、相加、开平方根。
c = √(a² + b²)
Always check that your answer for c is longer than either a or b, because the hypotenuse is the longest side. If your calculated c is shorter, you may have mislabelled the triangle.
始终检查你求出的 c 是否比 a 或 b 都要长，因为斜边是最长的边。如果算出的 c 更短，你可能错误标记了三角形。
6. Finding a Shorter Side (a or b) | 求直角边 (a 或 b)
If you know the hypotenuse and one leg, you can find the missing leg by rearranging the formula. To find leg a: a² = c² – b², so a = √(c² – b²). Similarly, b = √(c² – a²).
如果你知道斜边和一条直角边，你可以通过重新排列公式来求出缺失的直角边。求直角边 a：a² = c² – b²，因此 a = √(c² – b²)。类似地，b = √(c² – a²)。
It is a common mistake to add the squares instead of subtracting when finding a leg. Remember that the hypotenuse is the largest side, so its square minus the square of one leg gives the square of the other leg.
在求直角边时，一个常见的错误是将平方相加而不是相减。记住，斜边是最大的边，所以它的平方减去一条直角边的平方得到另一条直角边的平方。
7. Worked Example 1: Finding the Hypotenuse | 示例 1：求斜边
A right-angled triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse.
一个直角三角形的两条直角边长分别为 6 cm 和 8 cm。求斜边的长度。
Solution: Use c² = a² + b². Let a = 6, b = 8. Then c² = 6² + 8² = 36 + 64 = 100. c = √100 = 10 cm.
解：使用 c² = a² + b²。设 a = 6，b = 8。则 c² = 6² + 8² = 36 + 64 = 100。c = √100 = 10 cm。
Notice that 6, 8, 10 form a Pythagorean triple (each side is a whole number), which makes this triangle a scaled version of the 3-4-5 triangle.
注意，6、8、10 构成一组勾股数（每条边都是整数），这使得该三角形是 3-4-5 三角形的放大版本。
8. Worked Example 2: Finding a Leg | 示例 2：求一直角边
A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the length of the other leg.
一个直角三角形的斜边为 13 cm，一条直角边为 5 cm。求另一条直角边的长度。
Solution: Let c = 13, b = 5, find a. a² = c² – b² = 13² – 5² = 169 – 25 = 144. a = √144 = 12 cm.
解：设 c = 13，b = 5，求 a。a² = c² – b² = 13² – 5² = 169 – 25 = 144。a = √144 = 12 cm。
Again, this is a Pythagorean triple (5-12-13). Recognising common triples can save you time during exams.
这同样是一组勾股数（5-12-13）。识别常见的勾股数可以帮你在考试中节省时间。
9. Real-life Applications | 实际应用
Pythagoras’ theorem is widely used in navigation, construction, and design. For example, if a ladder of length L leans against a wall, with its foot d metres from the wall, the height h reached on the wall can be found using h² = L² – d².
勾股定理广泛应用于导航、建筑和设计。例如，如果一架长为 L 的梯子斜靠在墙上，梯脚离墙 d 米，则墙上达到的高度 h 可通过 h² = L² – d² 求得。
Another common application is finding the shortest distance between two points on a coordinate grid, which is the length of the line segment connecting them: distance = √((x₂–x₁)² + (y₂–y₁)²). This formula comes directly from Pythagoras.
另一个常见应用是求坐标网格上两点之间的最短距离，即连接两点的线段长度：距离 = √((x₂–x₁)² + (y₂–y₁)²)。这个公式直接源于勾股定理。
10. Pythagorean Triples | 勾股数组
A Pythagorean triple consists of three positive integers a, b, c that satisfy a² + b² = c². The smallest and most famous triple is (3, 4, 5). Other examples include (5, 12, 13), (8, 15, 17), and (7, 24, 25).
勾股数组由三个正整数 a、b、c 组成，它们满足 a² + b² = c²。最小且最著名的数组是 (3, 4, 5)。其他例子包括 (5, 12, 13)、(8, 15, 17) 和 (7, 24, 25)。
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Multiples of a triple also work: doubling (3,4,5) gives (6,8,10). Learning these triples helps you quickly identify right triangles and verify answers without a calculator.
勾股数组的倍数同样有效：将 (3,4,5) 加倍得到 (6,8,10)。学习这些数组有助于你快速识别直角三角形，并在不使用计算器的情况下验证答案。
11. The Converse of Pythagoras’ Theorem | 勾股定理的逆定理
The converse states that if a triangle has side lengths a, b, c and a² + b² = c², then the triangle is right-angled, with the right angle opposite side c. This is extremely useful for checking whether a given triangle is right-angled.
逆定理指出，如果一个三角形的边长为 a、b、c，且 a² + b² = c²，那么这个三角形就是直角三角形，直角位于边 c 的对侧。这对于检查给定的三角形是否为直角三角形非常有用。
For example, to test if sides 9 cm, 12 cm, and 15 cm form a right triangle, compute 9² + 12² = 81 + 144 = 225, and 15² = 225. Since they are equal, the triangle is right-angled.
例如，要检验边长 9 cm、12 cm 和 15 cm 是否构成直角三角形，计算 9² + 12² = 81 + 144 = 225，而 15² = 225。两者相等，所以该三角形是直角三角形。
12. Common Mistakes and Tips | 常见错误与技巧
Mistake 1: Adding the squares when finding a leg instead of subtracting. Always remember: c² = a² + b², so a² = c² – b². Use the correct rearrangement.
错误 1：在求直角边时，将平方相加而不是相减。始终牢记：c² = a² + b²，因此 a² = c² – b²。使用正确的变形。
Mistake 2: Forgetting to take the square root at the end. After you find c² = 25, you must calculate c = √25 = 5, not leave 25 as the side length.
错误 2：最后忘记开平方根。在你求得 c² = 25 之后，必须计算 c = √25 = 5，而不能将 25 当作边长。
Mistake 3: Mislabelling the hypotenuse. The hypotenuse is always opposite the right angle and is the longest side. Double-check before substituting numbers.
错误 3：错误标记斜边。斜边总是与直角相对，并且是最长的边。代入数字之前请仔细检查。
Tip: Draw a sketch and label sides clearly. Write down the formula. Substitute numbers with units. Solve step by step, and ask yourself whether your final answer is reasonable (e.g., hypotenuse longer than legs).
技巧：画出草图并清楚标注各边。写下公式。将带单位的数字代入。逐步求解，并问自己最终答案是否合理（例如，斜边应长于直角边）。
Published by TutorHao | Mathematics Revision Series | aleveler.com
Find Cambridge KS3 Maths Textbooks on eBay UK
New, used and second-hand copies of textbooks and revision guides are often much cheaper than retail — check current listings and prices before you buy.
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导