📚 Pythagoras' Theorem | 勾股定理
Pythagoras' theorem is one of the most famous results in geometry, linking the three sides of a right‑angled triangle. It provides a simple formula to calculate an unknown side length when the other two are known, and it has countless applications in construction, navigation, and design.
勾股定理是几何学中最著名的结论之一，它揭示了直角三角形三条边之间的关系。该定理给出了一个简洁的公式，当已知两条边的长度时可以用来计算未知边的长度，并在建筑、导航和设计等领域有着无数应用。
1. What is Pythagoras’ Theorem? | 什么是勾股定理？
Pythagoras’ theorem states that in any right‑angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The hypotenuse is always the longest side, lying directly opposite the right angle.
勾股定理指出，在任何直角三角形中，斜边长度的平方等于另外两条直角边长度的平方和。斜边永远是最长的那条边，正对着直角。
2. The Formula: a² + b² = c² | 公式：a² + b² = c²
The standard way to write the theorem is a² + b² = c², where c represents the length of the hypotenuse, and a and b represent the lengths of the two shorter sides (legs) that form the right angle.
该定理的标准写法是 a² + b² = c²，其中 c 表示斜边的长度，a 和 b 表示构成直角的两条较短边（直角边）的长度。
a² + b² = c²
3. Understanding the Hypotenuse | 理解斜边
The hypotenuse is the side opposite the right angle. It does not touch the right angle. When labelling a triangle, it is common to call the legs a and b, and the hypotenuse c. Always check that you are using c for the longest side.
斜边是直角的对边，不与直角相邻。给三角形标字母时，通常将两条直角边标为 a 与 b，将斜边标为 c。务必确认 c 始终代表最长的那条边。
4. Finding the Hypotenuse (Example) | 求斜边（例题）
the hypotenuse c is found by: c² = 3² + 4² = 9 + 16 = 25. Taking the square root gives c = √25 = 5 cm. The 3‑4‑5 triangle is the most famous Pythagorean triple.
如果三角形的两条直角边分别为 3 cm 和 4 cm，斜边 c 的计算方法是：c² = 3² + 4² = 9 + 16 = 25。取平方根得到 c = √25 = 5 cm。3‑4‑5 三角形是最著名的勾股数组合。
c = √a² + b²
Always remember to square the numbers first, add them, and then take the square root. Do not add a and b and then square – that gives a completely different (and incorrect) result.
始终要记住先求平方、再相加、最后开平方。切勿先加 a 和 b 后再平方——那样会得到完全不同的（也是错误的）答案。
5. Finding a Shorter Side (Example) | 求一条直角边（例题）
a hypotenuse of 10 cm and one leg of 6 cm: a² = c² − b² = 10² − 6² = 100 − 36 = 64, so a = √64 = 8 cm.
当已知斜边和一条直角边时，可以变换公式求缺失的短边。例如斜边长 10 cm，一条直角边长 6 cm：a² = c² − b² = 10² − 6² = 100 − 36 = 64，因此 a = √64 = 8 cm。
a = √c² − b²
This rearrangement is extremely useful: subtract the square of the known leg from the square of the hypotenuse, then take the square root. The side you find will be shorter than the hypotenuse, as expected.
这种变形非常实用：用斜边的平方减去已知直角边的平方，再开平方。求出的边应比斜边短，这符合预期。
6. Pythagorean Triples | 勾股数组
. Examples include (3, 4, 5), (5, 12, 13), (6, 8, 10) and (9, 12, 15). Multiples of a basic triple are also triples.
勾股数组是满足 a² + b² = c² 的三个正整数构成的组合。常见的例子有 (3, 4, 5)、(5, 12, 13)、(6, 8, 10) 和 (9, 12, 15)。基本勾股数组合的倍数同样也是勾股数组。
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
Recognising triples can save time in problem solving and helps check whether your calculated answer looks reasonable.
识别勾股数组可以在解题时节省时间，并有助于检验计算出的答案是否合理。
7. A Visual Proof | 一个直观的证明
One simple proof uses squares drawn on each side of a right‑angled triangle. The area of the large square on the hypotenuse equals the combined area of the squares on the two legs. Cutting and rearranging the shapes shows the physical equality.
一种简单的证明方法是在直角三角形的每条边上分别画出正方形。斜边上的大正方形面积恰好等于两条直角边上两个正方形的面积之和。通过剪切并重新拼合图形，可以直观地看到这种相等关系。
You can verify this with squared paper. Draw a 3‑4‑ 25 unit squares, while the 3×3 and 4×4 squares together contain 9 + 16 = 25 unit squares.
你可以用方格纸验证。画一个 3‑4‑5 三角形，在每条边上构造正方形，数一数小方格的数目。5×5 的正方形含有 25 个方格，而 3×3 和 4×4 的正方形合在一起共有 9 + 16 = 25 个方格。
8. Applying Pythagoras in Word Problems | 用勾股定理解应用题
The foot of the ladder is 1.5 m from the wall. How high up the wall does the ladder reach? Let the height be h. Then h² + 1.5² = 5² → h² + 2.25 = 25 → h² = 22.75 → h = √22.75 ≈ 4.77 m (to 2 d.p.).
一把 5 m 长的梯子斜靠在竖直墙壁上。梯脚距墙壁 1.5 m。梯子顶端能达到多高？设高度为 h，则 h² + 1.5² = 5² → h² + 2.25 = 25 → h² = 22.75 → h = √22.75 ≈ 4.77 m（保留两位小数）。
In navigation, a ship sails 40 km east and then 30 km north. The straight‑line distance from the starting point is √40² + 30² = √1600 + 900 = √2500 = 50 km. Pythagoras provides the shortest distance.
在航行问题中，一艘船先向东航行 40 km，再向北航行 30 km。从起点出发的直线距离为 √40² + 30² = √1600 + 900 = √2500 = 50 km。勾股定理给出了最短距离。
9. Pythagoras and the Distance Between Two Points | 勾股定理与两点间距离
₁)² + (y₂ − y₁)²]. This formula comes directly from Pythagoras’ theorem by drawing a right‑angled triangle with the line segment as hypotenuse.
在坐标格中，两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离公式为 √[(x₂ − x₁)² + (y₂ − y₁)²]。这个公式直接来自勾股定理：以两点间的线段为斜边构造直角三角形即可得出。
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
For example, the distance between (1, 2) and (4, 6) is √[(4−1)² + (6−2)²] = √3² + 4² = √9+16 = √25 = 5 units.
例如，(1, 2) 与 (4, 6) 之间的距离是 √[(4−1)² + (6−2)²] = √3² + 4² = √9+16 = √25 = 5 个单位。
10. The Converse of Pythagoras | 勾股定理的逆定理
If the three sides of a triangle satisfy a² + b² = c², then the triangle is right‑ lows you to test whether a given triangle contains a right angle without measuring angles.
如果一个三角形的三条边满足 a² + b² = c²，那么这个三角形一定是直角三角形。逆定理使我们在不测量角度的情况下就能判断一个三角形是否含有直角。
cm: 9² + 12² = 81 + 144 = 225, and 15² = 225. Since they are equal, the triangle is right‑angled with the right angle opposite the 15 cm side.
例如，边长分别为 9 cm、12 cm 和 15 cm 的三角形：9² + 12² = 81 + 144 = 225，而 15² = 225。它们相等，所以该三角形是直角三角形，且 15 cm 边所对的角为直角。
11. Common Mistakes | 常见错误
One common mistake is forgetting to take the square root at the end. Finding c² is not the final answer – you still need to take the positive square root to find the length. Another mistake is mixing up which side is the hypotenuse; always check it is opposite the right angle.
一个常见错误是最后忘记开平方。求出 c² 并不是最终答案——还需要取正的平方根才能得到边长。另一个错误是混淆哪条边是斜边；一定要确认它是直角的对边。
Forgetting to square the numbers before adding.
相加前忘记平方数字。
Adding the lengths first and then squaring the sum.
先将长度相加，再对和进行平方。
Applying the theorem to non‑right‑angled triangles.
将定理错误地应用于非直角三角形。
Using the negative square root for lengths (length is always positive).
对长度使用了负的平方根（长度总是为正）。
nstead of a² = c² − b². Always subtract the known leg squared from the hypotenuse squared.
在变形求短边时，有些学生会错写成 a² = b² − c²，而不是 a² = c² − b²。务必用斜边的平方减去已知直角边的平方。
12. Practice and Revision Tips | 练习与复习建议
To master Pythagoras' theorem, practise with both integer and decimal side lengths. Draw a clear right‑angled triangle for each problem, label the sides, and write the formula before substituting numbers. Check your answer by seeing if the hypotenuse is the longest side.
要熟练掌握勾股定理，就要练习整数边长和小数边长的题目。为每一道题画出清晰的直角三角形，标出各条边，在代入数字前写出公式。通过检查斜边是否是最长边来验证你的答案。
Use flowcharts: (1) Identify the right angle. (2) Label the hypotenuse c. (3) Decide whether you are finding the hypotenuse or a shorter side. (4) Choose the correct formula. (5) Substitute and calculate accurately.
使用流程图：(1) 辨认直角。(2) 将斜边标为 c。(3) 判断是求斜边还是求较短边。(4) 选择合适的公式。(5) 代入数值并准确计算。
Revision of squares and square roots up to 15² will speed up your work considerably.
复习 15 以内的平方与平方根会让你做题快得多。
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