📚 Pythagoras' Theorem: Complete Guide with Worked Examples | 毕达哥拉斯定理：完整指南与解题示例
Pythagoras' theorem is one of the most important results in KS3 mathematics and one of the few facts that students carry with them all the way to A Level. It connects the three sides of a right-angled triangle and provides a reliable method for finding an unknown side when two others are known. This article explains the theorem step by step, from labelling the triangle correctly, through finding the hypotenuse and finding a shorter side, to Pythagorean triples, the converse test, coordinate and 3D applications, worked examples and the common mistakes to avoid.
毕达哥拉斯定理是 KS3 数学中最重要的结论之一，也是少数会一直陪伴学生到 A Level 的知识点。它把直角三角形三条边联系起来，并在已知两条边时提供了一种可靠的方法来求未知边。本文逐步解释该定理：从正确标注三角形，到求斜边、求较短边，再到勾股数、逆定理判定、坐标与立体图形的应用、解题示例以及需要避免的常见错误。
1. What is Pythagoras' Theorem? | 什么是毕达哥拉斯定理？
A right-angled triangle contains one angle of exactly 90°. The side opposite this right angle is called the hypotenuse, and because the right angle is the largest angle in the triangle, the hypotenuse is always the longest side. Pythagoras' theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The theorem is named after the Greek mathematician Pythagoras, although historians have found evidence that Babylonian and Indian mathematicians used the same relationship more than a thousand years earlier.
直角三角形包含一个恰好为 90° 的角。这个直角的对边叫做斜边；因为直角是三角形中最大的角，所以斜边总是最长的一条边。毕达哥拉斯定理指出，斜边的平方等于另外两条边的平方之和。该定理以古希腊数学家毕达哥拉斯命名，不过历史学家发现，巴比伦和印度的数学家在更早一千多年就已经在使用这一关系。
a² + b² = c²
ly the same equation - but c must always be the side opposite the right angle. If c is not the hypotenuse, the statement a² + b² = c² is simply false.
这里 c 表示斜边，a 和 b 表示两条较短的边。a 和 b 的顺序完全无关紧要，互换位置得到的方程完全相同；但 c 必须是直角的对边。如果 c 不是斜边，那么 a² + b² = c² 这个等式就是错误的。
s the first check you should make in any question: before touching your calculator, confirm that the triangle really has a right angle, or that one is given in the diagram by the small square symbol in the corner.
该定理只适用于直角三角形。如果三角形不包含 90° 角，就不能直接使用公式 a² + b² = c²。这是你在任何题目中应做的第一个检查：在拿起计算器之前，先确认三角形真的有直角，或者图中角落的直角小方块符号已经给出了直角。
2. Labelling the Triangle Correctly | 正确标注三角形
Before you calculate, label the three sides. Put the right angle in the bottom left of your sketch so that the hypotenuse runs diagonally from the bottom right corner up to the top left corner. The side directly across from the right angle is the hypotenuse; label it c. The other two sides, which meet at the right angle, are labelled a and b. These two sides are often called the legs of the triangle.
计算前，先标注三条边。将直角放在草图的左下角，这样斜边就会从右下角斜向左上角。与直角正对的边是斜边，把它标记为 c。另外两条在直角处相交的边标记为 a 和 b，它们通常被称为三角形的两条"直角边"。
It is helpful to write the letters on the diagram rather than keeping them in your head. If you mix up the hypotenuse with a shorter side, your answer will be incorrect: you will end up adding a leg to the hypotenuse when you should be subtracting, or taking the square root of a value that is far too large. Always ask: which side is opposite the 90° angle? That side is c, every single time.
把字母写在图上会很有帮助，而不要只在脑海里记着。如果把斜边和较短边混淆，你的答案就会出错：本该相减时你却把一条直角边加到了斜边上，或者对一个明显过大的数开平方。永远问自己：哪条边是 90° 角的对边？那条边就是 c，每一次都是如此。
Watch out for naming conventions. In more advanced geometry, a triangle is often named ABC with the vertices labelled, and in the sine rule and cosine rule the letter c means the side opposite angle C, which is not necessarily the hypotenuse. In Pythagoras work, however, c conventionally means the hypotenuse. If a diagram already uses different letters, do not blindly rename everything - identify the hypotenuse first, then decide which letter plays the role of c in your equation.
注意字母命名的习惯差异。在更高阶的几何中，三角形常按顶点命名为 ABC，而在正弦定理和余弦定理中，字母 c 表示角 C 的对边，它不一定是斜边。但在毕达哥拉斯定理的应用中，c 通常指斜边。如果图里已经使用了别的字母，不要盲目全部改名——先找出斜边，再决定在你的等式中哪个字母扮演 c 的角色。
3. Finding the Hypotenuse | 求斜边
When the two shorter sides are known and the hypotenuse is unknown, the theorem is used in its original form: square both legs, add the results, then take the square root. The square root step is essential - many students stop at the sum of the squares and write the wrong answer.
当已知两条较短边、要求斜边时，定理按原式使用：先把两条直角边平方，把结果相加，再开平方。开平方这一步必不可少——很多学生停在平方和就写下答案，结果当然是错的。
c = √a² + b²
m. Notice that the square root of 100 has two solutions, +10 and -10, but a length can never be negative, so only +10 is accepted.
例题：一个直角三角形的两条直角边分别为 6 cm 和 8 cm。则 c² = 6² + 8² = 36 + 64 = 100，由于 c 是长度，必须为正，所以 c = √100 = 10 cm。注意 100 的平方根有两个，+10 和 -10，但长度不可能为负，所以只取 +10。
ctly the same way. If the legs are 5.3 m and 7.1 m, then c² = 28.09 + 50.41 = 78.5, so c = √78 the full value on your calculator display until the final rounding step; rounding 28.09 to 28 before adding would already have introduced an error.
小数运算完全一样。如果两条直角边是 5.3 m 和 7.1 m，则 c² = 28.09 + 50.41 = 78.5，所以 c = √78.5 ≈ 8.86 m（保留两位小数）。在最后一步取近似值之前，请让计算器显示保留完整数值；如果先把 28.09 约成 28 再加，就已经引入了误差。
instead of pressing the square root button. For legs of 4 and 6, c² = 16 + 36 = 52, so c = √52 = √4 × 13 = 2√13. Leaving the answer as √52 is usually accepted, but 2√13 is cleaner and preferred when the question says "give your answer in exact form".
有些考题要求精确值而不是小数。这时应当化简根式，而不是按计算器的开方键。若两条直角边为 4 和 6，则 c² = 16 + 36 = 52，所以 c = √52 = √4 × 13 = 2√13。写成 √52 通常也能得分，但 2√13 更简洁；当题目要求"给出精确形式"时，更推荐后者。
4. Finding a Shorter Side | 求较短边
btract b² from both sides to obtain a² = c² - b². The hypotenuse square must always be on its own on the larger side of the subtraction.
当已知斜边和一条较短边时，必须先重新整理方程，再代入数字。由 a² + b² = c² 出发，两边同时减去 b²，得到 a² = c² - b²。斜边的平方必须单独放在减法的被减数位置。
a² = c² - b² b² = c² - a²
smaller than the hypotenuse, which is a quick sanity check that you have not made a slip.
例题：一个直角三角形的斜边为 13 cm，一条直角边为 5 cm。则 a² = 13² - 5² = 169 - 25 = 144，所以 a = √144 = 12 cm。答案小于斜边，这是一个快速的合理性检验，说明你没有算错。
d the square root of a negative number does not exist among real lengths. If your working produces a negative value inside the square root, you have almost certainly subtracted the hypotenuse square from a leg square.
再看一个例子：斜边 15 m，一条直角边 9 m。则 b² = 15² - 9² = 225 - 81 = 144，所以 b = 12 m。注意如果顺序颠倒，会得到 81 - 225 = -144，而负数在实数范围内没有平方根（不能表示长度）。如果你的运算在根号内出现了负数，几乎可以肯定你是用直角边的平方去减斜边的平方了。
5. Pythagorean Triples | 勾股数
exactly. Recognising the common triples saves time in exams and lets you check answers instantly, because you know in advance that the result should be a whole number.
勾股数是指恰好满足 a² + b² = c² 的三个整数。记住常见的勾股数可以节省考试时间，并让你立刻检查答案，因为你知道结果本应是一个整数。
| Triple | 勾股数组 | Check | 验证 | Multiples | 倍数形式 |
| 3, 4, 5 | 9 + 16 = 25 | 6, 8, 10; 9, 12, 15; 30, 40, 50 |
| 5, 12, 13 | 25 + 144 = 169 | 10, 24, 26; 15, 36, 39 |
| 8, 15, 17 | 64 + 225 = 289 | 16, 30, 34 |
| 7, 24, 25 | 49 + 576 = 625 | 14, 48, 50 |
| 9, 40, 41 | 81 + 1600 = 1681 | 18, 80, 82 |
a² + b²) = k²c² = (kc)². So 3, 4, 5 scales to 6, 8, 10, then 9, 12, 15, and so on. This is why a triangle with legs 30 and 40 has a hypotenuse of 50 rather than some awkward decimal.
任何勾股数的倍数仍然是勾股数，因为把每条边都乘以相同的比例系数 k，可得 (ka)² + (kb)² = k²(a² + b²) = k²c² = (kc)²。所以 3, 4, 5 可以扩展到 6, 8, 10，再到 9, 12, 15，依此类推。这就是为什么两条直角边为 30 和 40 的三角形，斜边正好是 50，而不是某个别扭的小数。
If a question produces an answer that is not a whole number, do not panic or assume you are wrong. Many questions deliberately use non-integer sides. Check whether the question asks for a decimal to a given accuracy, or for an exact surd, and answer accordingly.
如果题目算出的答案不是整数，不必慌张，也不要立刻认为自己做错了。很多题目是故意使用非整数边长的。看清题目是要求保留若干位小数的近似值，还是要求精确的根式，再按需作答。
6. The Converse: Testing for a Right Angle | 逆定理：判断是否为直角三角形
The converse of Pythagoras' st side, then the triangle must be right-angled. This is the standard method for testing a triangle when no angle is marked.
毕达哥拉斯定理的逆定理方向相反：如果三角形的三条边满足 a² + b² = c²（其中 c 是最长边），那么这个三角形一定是直角三角形。当图中没有标出任何角度时，这就是判断三角形的标准方法。
hen 9² + 12² = 81 + 144 = 225 and 15² = 225. The two values are equal, so the triangle is right-angled, and the right angle lies between the sides of length 9 and 12.
方法：先找出最长边并记为 c，把三条边分别平方，然后比较。以边长 9、12、15 为例：最长边是 15。则 9² + 12² = 81 + 144 = 225，而 15² = 225，两者相等，所以该三角形是直角三角形，直角位于长度为 9 和 12 的两条边之间。
angle is not right-angled. In fact 52 < 64, which means the triangle is obtuse-angled; if the sum had been greater than 64, the triangle would be acute-angled.
以边长 4、6、8 为例：最长边是 8。则 4² + 6² = 16 + 36 = 52，而 8² = 64。因为 52 ≠ 64，所以该三角形不是直角三角形。事实上 52 < 64，说明这个三角形是钝角三角形；如果平方和大于 64，则三角形为锐角三角形。
Examiners often phrase this as "show that the triangle is right-angled" or "is this triangle right-angled? You must justify your answer." A justification must include both computed values and an explicit comparison statement, not just the number 225 written twice.
考题常写成"证明该三角形是直角三角形"或"这个三角形是直角三角形吗？必须说明理由"。完整的理由必须同时写出两个计算结果以及明确的比较语句，而不能只是把 225 写两遍。
7. Coordinates: Distance Between Two Points | 坐标：两点之间的距离
On a coordinate grid, Pythagoras' theorem gives the distance between any two points. The horizontal and vertical gaps form the two legs of a right-angled triangle, and the straight-line distance between the points is the hypotenuse.
在坐标网格中，毕达哥拉斯定理可以求出任意两点之间的距离。水平间距和竖直间距构成直角三角形的两条直角边，两点之间的直线距离就是斜边。
d = √((x₂ - x₁)² + (y₂ - y₁)²)
= 9 + 16 = 25, so d = 5 units. This is the familiar 3, 4, 5 triple appearing again in a coordinate setting.
例题：求点 (1, 2) 与点 (4, 6) 之间的距离。水平差为 4 - 1 =
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