📚 Pythagoras' Theorem | 勾股定理
Pythagoras' theorem is a fundamental rule in geometry that describes the relationship between the sides of a right‑angled triangle. It is named after the ancient Greek mathematician Pythagoras and is one of the most important tools you will use throughout KS3, IGCSE and beyond. This article covers everything you need to master Pythagoras' theorem, from basic definitions to real‑life applications, all designed to match the Cambridge KS3 Mathematics curriculum.
勾股定理是几何学中描述直角三角形三条边关系的基本定理，以古希腊数学家毕达哥拉斯命名，是你在 KS3、IGCSE 乃至更高阶数学学习中最重要的工具之一。本文涵盖了掌握勾股定理所需的所有要点，从基础定义到实际应用，完全贴合 Cambridge KS3 数学课程要求。
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 angle exactly equal to 90°. The side opposite this right angle is called the hypotenuse, and it is always the longest side. The other two sides are called the legs, and they form the right angle.
直角三角形是其中一个角恰好等于 90° 的三角形。直角所对的边称为斜边，它永远是最长的一条边。另外两条边称为直角边，它们构成了直角。
In the diagram, if ∠ hypotenuse, while a and b are the legs. Recognising the hypotenuse correctly is the first and most crucial step in using Pythagoras' theorem.
在图中，如果 ∠C = 90°，那么边 c（∠C 的对边）就是斜边，而 a 和 b 是直角边。正确识别斜边是使用勾股定理的第一步，也是最关键的一步。
2. The Formula a² + b² = c² | 公式 a² + b² = c²
Pythagoras' theorem states that in any right‑ c², where c represents the length of the hypotenuse and a, b are the lengths of the legs.
勾股定理指出，在任何一个直角三角形中，斜边的平方等于两条直角边的平方之和。用符号表示就是 a² + b² = c²，其中 c 代表斜边长度，a 和 b 代表两条直角边的长度。
This formula only works for right‑angled triangles. Do not try to apply it to other types of triangles. You can rearrange the equation to find any missing side as long as you know the other two.
这个公式只适用于直角三角形，不要尝试在其他类型的三角形中使用。只要你知道其中两条边的长度，就可以通过重新整理方程来求出任意一条未知的边。
3. Finding the Hypotenuse | 求斜边长度
quare root. For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.
如果已知两条较短的边（直角边）的长度，你可以将数值代入公式然后开平方根来求斜边。例如，若 a = 3 cm，b = 4 cm，则 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5 cm。
Always remember to find the square root at the end, not just to leave the answer as c². Also, watch the units: if the legs are in metres, the hypotenuse will be in metres as well.
务必记住最后要求平方根，不要把答案只停留在 c² 这一步。同时注意单位：如果直角边以米为单位，斜边同样以米为单位。
4. Finding a Shorter Side | 求直角边长度
ate the unknown side. For example, to find b, use b² = c² − a², then take the square root.
有时候你知道斜边和一条直角边，需要求另一条直角边。从 a² + b² = c² 入手，重新整理方程以隔离未知边。例如，要求 b，可用 b² = c² − a²，然后再开平方根。
Suppose a right‑ 9 − 25 = 144, so b = √144 = 12 m. This method always involves subtraction, never addition.
假设一个直角三角形的斜边长为 13 m，一条直角边长为 5 m，那么 b² = 13² − 5² = 169 − 25 = 144，所以 b = √144 = 12 m。这种方法总是用减法，永远不会是加法。
5. Pythagorean Triples | 勾股数组
cause 3² + 4² = 5². Other common triples include (5, 12, 13), (7, 24, 25) and (8, 15, 17).
勾股数组是指满足 a² + b² = c² 的三组正整数。最著名的勾股数组是 (3, 4, 5)，因为 3² + 4² = 5²。其他常见的数组还有 (5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。
Recognising these triples can save you time in exams. If you spot a triangle with sides 6 and 8, you can immediately know the hypotenuse is 10 because (6, 8, 10) is just the (3, 4, 5) triple multiplied by 2.
识别这些数组可以让你在考试中节省时间。如果你看到一个三角形两边为 6 和 8，马上就能知道斜边是 10，因为 (6, 8, 10) 只是将 (3, 4, 5) 每个数乘以 2 得到的。
6. Applying Pythagoras’ Theorem to Solve Problems | 应用勾股定理解决问题
Pythagoras' theorem is not just for abstract triangles; it can be used to solve real‑life problems. For instance, if you know the height a ladder reaches up a wall and the distance the base of the ladder is from the wall, you can find the length of the ladder.
勾股定理不仅用于抽象三角形，还能用来解决现实问题。例如，如果你知道梯子靠在墙上达到的高度以及梯脚到墙的距离，你就可以计算出梯子的长度。
he wall. The ladder length L is the hypotenuse: L² = 6² + 2.5² = 36 + 6.25 = 42.25, so L = √42.25 = 6.5 m.
设想一个梯子竖直靠在墙上，高度为 6 m，梯脚距墙底 2.5 m。梯子长度 L 就是斜边：L² = 6² + 2.5² = 36 + 6.25 = 42.25，所以 L = √42.25 = 6.5 m。
7. Using Pythagoras in 2D Coordinate Geometry | 在二维坐标几何中使用勾股定理
You can use Pythagoras' the horizontal distance is (x₂ − x₁) and the vertical distance is (y₂ − y₁). The straight‑line distance d between them is the hypotenuse: d² = (x₂ − x₁)² + (y₂ − y₁)².
你可以利用勾股定理求坐标网格上两点之间的距离。如果两点坐标为 (x₁, y₁) 和 (x₂, y₂)，那么水平距离为 (x₂ − x₁)，竖直距离为 (y₂ − y₁)。两点间的直线距离 d 就是斜边：d² = (x₂ − x₁)² + (y₂ − y₁)²。
25, so AB = 5 units. This distance formula is a direct application of Pythagoras’ theorem.
例如，求 A(2, 3) 和 B(5, 7) 之间的距离：AB² = (5 − 2)² + (7 − 3)² = 3² + 4² = 9 + 16 = 25，所以 AB = 5 个单位。这个距离公式是勾股定理的直接应用。
8. Is a Triangle Right‑Angled? | 三角形是直角三角形吗？
The converse of Pythagoras’ theorem can be used to check if a triangle is right‑ ere c is the longest side), then the triangle has a right angle opposite the longest side.
勾股定理的逆定理可用于检验一个三角形是否为直角三角形。如果三边长度满足 a² + b² = c²（其中 c 为最长边），那么最长边所对的角就是直角。
est: 9² + 12² = 81 + 144 = 225, and 15² = 225. Since they are equal, the triangle is right‑d with the hypotenuse as 15 cm. This is very useful in geometric proofs and constructions.
对于一个边长分别为 9 cm、12 cm 和 15 cm 的三角形，检验：9² + 12² = 81 + 144 = 225，而 15² = 225，两者相等，因此该三角形是直角三角形，斜边为 15 cm。这在几何证明和作图中非常有用。
9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法
ays set up the equation carefully: if the unknown is a leg, use b² = c² − a², not c² + a².
一个常见错误是：在求直角边时本该用减法却用了加法。一定要仔细建立方程：如果未知边是直角边，应使用 b² = c² − a²，而不是 c² + a²。
Another mistake is forgetting to take the square root at the end. Leaving the answer as, say, 25 instead of 5 will lose marks. Also, misidentifying the hypotenuse can lead to completely wrong results. Always put a small arrow pointing to the right angle first.
另一个错误是最后忘记开平方根。把答案写成 25 而不是 5 会丢分。此外，误判斜边也会导致结果完全错误。建议先在直角处标一个小箭头，以此确认斜边。
10. Proof of Pythagoras' Theorem | 勾股定理的证明
There are many proofs of Pythagoras' theorem, one of the simplest being the rearrangement proof. Take four identical right‑angled triangles with legs a and b and hypotenuse c. Arrange them to form a large square of side (a + b). The area of the large square can be expressed in two ways.
勾股定理有许多种证明方法，其中最简单的一种是重组证明法。取四个完全相同的直角三角形，直角边为 a 和 b，斜边为 c。把它们拼成一个边长为 (a + b) 的大正方形。这个大正方形的面积可以用两种方式表达。
ed inner square of side c and four triangles each of area (½ab). So (a + b)² = c² + 4(½ab) ⇒ a² + 2ab + b² = c² + 2ab ⇒ a² + b² = c². This visual proof helps understand why the theorem works.
总面积 = (a + b)²。同时，大正方形由一个倾斜放置、边长为 c 的内部正方形和四个面积为 (½ab) 的三角形组成。因此 (a + b)² = c² + 4(½ab) ⇒ a² + 2ab + b² = c² + 2ab ⇒ a² + b² = c²。这个直观的证明有助于理解定理为什么成立。
11. Practice Questions | 练习题目
Try these questions to build your confidence:
1. Find the hypotenuse of a right‑angled triangle with legs 7 cm and 24 cm.
2. A right‑angled triangle has a hypotenuse of 17 m and one leg of 8 m. Find the other leg.
3. Is a triangle with sides 10 cm, 24 cm and 26 cm right‑angled?
尝试以下题目来增强信心：
1. 求直角边为 7 cm 和 24 cm 的直角三角形的斜边长。
2. 一个直角三角形的斜边为 17 m，一条直角边为 8 m，求另一条直角边。
3. 边长为 10 cm、24 cm 和 26 cm 的三角形是直角三角形吗？
64 = 225, so b = 15 m. 3. 10² + 24² = 100 + 576 = 676, and 26² = 676, so yes, it is right‑angled.
答案：1. c² = 7² + 24² = 49 + 576 = 625，所以 c = 25 cm。2. b² = 17² − 8² = 289 − 64 = 225，所以 b = 15 m。3. 10² + 24² = 100 + 576 = 676，而 26² = 676，因此是直角三角形。
12. Summary and Key Points | 总结与要点
Pythagoras' theorem relates the sides of a right‑ ing the hypotenuse. Use addition when finding the hypotenuse and subtraction when finding a leg. Always check that the triangle has a right angle before applying the theorem.
勾股定理揭示了直角三角形三边的关系：a² + b² = c²，其中 c 为斜边。求斜边时用加法，求直角边时用减法。在应用定理之前，务必确认三角形有一个直角。
- Identify the hypotenuse correctly – it is opposite the right angle.
- Substitute carefully and solve step by step.
- Take the square root to find the length, not just the square.
- The converse tests if a triangle is right‑angled.
- Pythagorean triples can speed up your calculations.
- 正确识别斜边——它位于直角的对面。
- 仔细代入数值，按步骤求解。
- 求边长时要开平方根，而不仅仅是平方值。
- 逆定理可检验三角形是否为直角三角形。
- 勾股数组能加快计算速度。
Master this theorem and you will have a powerful tool for geometry, trigonometry and problem solving!
掌握这一定理，你就拥有了解决几何、三角学以及实际问题的一大有力工具！
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