📚 Pythagoras' Theorem | 勾股定理
Pythagoras' theorem is one of the cornerstones of geometry, linking the three sides of any right-angled triangle. It is named after the ancient Greek mathematician Pythagoras, although the relationship was known to Babylonian and Indian scholars long before. Understanding this theorem is essential for solving countless problems involving distances, constructions, navigation and design. In the IGCSE Edexcel Mathematics syllabus, you are expected to apply the theorem confidently, both in 2D and 3D contexts, and to recognise its converse.
勾股定理是几何学的基石之一，它将任何一个直角三角形的三条边联系起来。该定理以古希腊数学家毕达哥拉斯命名，不过早在毕氏之前巴比伦和印度的学者就已经知道了这种关系。理解这个定理对于解决涉及距离、建筑、导航和设计的大量问题至关重要。在 IGCSE 爱德思数学大纲中，要求你能够熟练地将该定理应用于平面和三维情境，并识别它的逆定理。
1. What is Pythagoras' theorem? | 勾股定理是什么？
In any right-angled triangle, the side opposite the right angle is called the hypotenuse — it 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 (often called the legs). If we label the two shorter sides as a and b, and the hypotenuse as c, the relationship is written as:
在任何直角三角形中，直角对面的边称为斜边——它总是最长的边。勾股定理指出，斜边的平方等于另外两条边（通常称为直角边）的平方和。如果我们把两条较短的边标记为 a 和 b，斜边标记为 c，那么这种关系可以写成：
c² = a² + b²
his equation only works for right-angled triangles and provides a direct method to calculate an unknown side when the other two sides are known.
等价地，我们可以写成 a² + b² = c²。该方程仅适用于直角三角形，并且提供了一种在已知另外两边时直接计算未知边长的方法。
2. Identifying the hypotenuse | 识别斜边
The hypotenuse is always opposite the right angle and is the longest of the three sides. In a diagram, look for the square symbol marking the 90° angle — the side directly facing that symbol is the hypotenuse. Never assume a side is the hypotenuse just because it looks longest; always check the right angle position first.
斜边始终位于直角的对面，并且是三条边中最长的一条。在图形中，寻找标记 90° 角的方形符号——正好对着该符号的边就是斜边。决不要因为某条边看起来最长就认为它是斜边；一定要先确认直角的位置。
3. Finding the hypotenuse | 求斜边
take the square root. For example, in a right-angled triangle where a = 6 cm and b = 8 cm:
如果已知两条较短的边，将它们的长度代入 a² + b²，然后取平方根。例如，在一个直角三角形中 a = 6 cm, b = 8 cm：
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
Always include the units in your final answer. Remember that the square root gives a positive length — we discard the negative root because length cannot be negative.
最终答案中一定要带上单位。记住，平方根给出正的长度——我们舍去负根，因为长度不可能是负数。
4. Finding a shorter side | 求一条直角边
To find one of the shorter sides, rearrange the formula. If the hypotenuse c and one leg b are known, then:
要想求一条直角边，将公式重新排列。如果已知斜边 c 和一条直角边 b，那么：
a² = c² − b²
Then take the square root: a = √c² − b². For instance, if c = 13 cm and b = 5 cm,
然后取平方根：a = √c² − b²。例如，若 c = 13 cm, b = 5 cm，
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
Be careful with the order: it is always (hypotenuse)² minus (shorter leg)². Subtracting the wrong way will give a negative number inside the square root, which is impossible for a length.
注意顺序：总是（斜边）² 减去（直角边）²。用错误的方式相减会使得平方根里面出现负数，对于长度而言这是不可能的。
5. Pythagorean triples | 勾股数组
alled Pythagorean triples. Recognising them can save time in exams. Common examples include:
某些三个整数的集合恰好满足 a² + b² = c²，这些数称为勾股数组。识别它们能让你在考试中节省时间。常见的例子包括：
| a | b | c |
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
Notice that multiples of these triples also work — for example, 6, 8, 10 is a multiple of the 3-4-5 triple. Knowing the basic triples helps you spot right angles quickly and check your answers.
请注意这些数组的倍数也同样成立——例如，6, 8, 10 就是 3-4-5 这个数组的倍数。熟悉基本的勾股数组有助于你快速识别直角并检查答案。
6. A visual proof of the theorem | 定理的一个直观证明
One classic proof uses four identical right-angled triangles arranged inside a large square. Place the triangles with legs a and b so that they form a large square of side (a + b). The central empty space is a smaller square of side c. The total area of the large square can be expressed in two ways:
一个经典的证明利用了排列在一个大正方形内部的四个相同的直角三角形。将直角边为 a 和 b 的三角形放置，使它们构成一个边长为 (a + b) 的大正方形。中间的空当是一个边长为 c 的小正方形。大正方形的总面积可以用两种方式表示：
(a + b)² = a² + 2ab + b²
rea is also the sum of the four triangles plus the inner square: 4 × (½ab) + c² = 2ab + c²
这个面积也等于四个三角形的总面积加上内部正方形的面积：4 × (½ab) + c² = 2ab + c²
Equating the two expressions:
将两个表达式等同：
a² + 2ab + b² = 2ab + c²
². This visual argument shows why the theorem must be true for all right-angled triangles.
两边消去 2ab 即得到 a² + b² = c²。这个直观的论证说明了为什么这个定理对所有直角三角形都成立。
7. Applying the theorem to word problems | 将定理应用于文字题
Pythagoras' theorem often appears in real-life contexts. A ladder leaning against a wall forms a right-angled triangle with the wall and the ground. To find how high the ladder reaches, identify the hypotenuse (the ladder) and the known distance, then solve for the missing side. Always sketch a diagram and label the sides.
勾股定理经常出现在实际生活情境中。一架靠墙的梯子与墙壁和地面构成直角三角形。要计算梯子能够到达的高度，先确定斜边（梯子）以及已知的距离，然后求出未知的边长。始终要画出示意图并标注各边。
g the distance between two points on a coordinate grid: the horizontal and vertical differences are the legs.
类似地，矩形的对角线将它分成两个直角三角形。如果一个矩形的长为 L，宽为 W，那么对角线 d 满足 d² = L² + W²。这一想法可以扩展到求坐标网格上两点间的距离：水平差值和垂直差值就是两条直角边。
8. Pythagoras in 3D | 立体中的勾股定理
w². Then treat d₁ as one leg with height h as the other leg to find the space diagonal D:
该定理可以用来求长方体（矩形盒子）的空间对角线。如果盒子的长为 l，宽为 w，高为 h，那么连接相对顶点的对角线会穿过两个直角三角形。首先求底面对角线：d₁² = l² + w²。然后以 d₁ 为一条直角边，高 h 为另一条直角边，求出空间对角线 D：
D² = d₁² + h² = l² + w² + h²
Therefore D = √l² + w² + h². This formula is a direct extension of the 2D Pythagoras' theorem and is particularly useful in problems involving longest rod that can fit inside a box.
因此 D = √l² + w² + h²。这个公式是平面勾股定理的直接推广，在涉及能放入盒子内的最长杆子等问题时尤其有用。
9. The converse of Pythagoras' theorem | 勾股定理的逆定理
5 cm, 12 cm and 13 cm must contain a right angle, because 5² + 12² = 25 + 144 = 169 = 13².
逆定理指出：如果一个三角形的三边 a, b, c（其中 c 是最长边）满足 a² + b² = c²，那么这个三角形就是直角三角形。这提供了一种检验直角简单方法，而无需测量角度本身。例如，一个边长为 5 cm、12 cm 和 13 cm 的三角形必定含有一个直角，因为 5² + 12² = 25 + 144 = 169 = 13²。
When verifying a triangle, always make sure c is the longest side. If the equation does not hold, the triangle is not right-angled — it may be acute or obtuse.
检验三角形时，务必确保 c 为最长的边。如果等式不成立，那么该三角形就不是直角三角形——它可能是锐角三角形或钝角三角形。
10. Common mistakes and how to avoid them | 常见错误及其避免方法
any students forget to take the square root at the end, writing c = a² + b² instead of c = √a² + b². Always check whether you have squared or square rooted. Another common error is mixing up the legs and the hypotenuse. Label your triangle clearly and write the formula before substituting numbers.
许多学生会忘记最后取平方根，写成 c = a² + b²，而不是 c = √a² + b²。始终检查自己是求了平方还是开了平方。另一个常见错误是把直角边和斜边搞混。清楚地标注三角形，并在代入数字之前先写出公式。
Rounding too early can also cause inaccuracies. Keep full calculator values until the final step, especially when multiple calculations are involved. Lastly, don't assume a triangle is right-angled just because it looks like one — look for the 90° symbol or use the converse to confirm.
过早四舍五入也会导致不精确。最后一步之前要保留计算器上的完整数值，特别是在涉及多步计算的时候。最后，不要因为一个三角形看起来像直角三角形就认为它是——要去寻找 90° 符号，或者用逆定理来确认。
11. Exam-style tips for Edexcel IGCSE | 爱德思 IGCSE 考试提示
Edexcel IGCSE questions often embed Pythagoras' theorem in multi-step problems. You might need to find an unknown length before calculating an area or a perimeter. Show all your working clearly: write down the formula, substitute the given values, and show the square root step.
爱德思 IGCSE 考试题目经常将勾股定理嵌入多步骤的问题中。你可能会需要先求出一个未知长度，然后再计算面积或周长。清楚地展示所有步骤：写出公式，代入已知数值，并写出开平方的步骤。
Remember to include units in your final answer and check whether the question asks for an exact value (e.g. in surd form) or a decimal approximation. Using Pythagorean triples can help you verify your result quickly. Finally, read the question carefully to decide whether you need the original theorem or the converse.
记住在最终答案中包含单位，并检查题目要求的是精确值（例如以根式表示）还是小数近似值。利用勾股数组可以帮助你快速验证结果。最后，仔细读题，确定是使用原定理还是逆定理。
12. Summary and practice | 总结与练习
², where c is the hypotenuse. The formula can be rearranged to find any side. The converse lets you test for a right angle. Applications range from simple triangle calculations to 3D geometry and coordinate distances.
要点很直接：在直角三角形中，c² = a² + b²，其中 c 为斜边。公式可以重新排列来求任何一边。逆定理让你能够检验直角。应用范围从简单的三角形计算直到立体几何和坐标距离。
To build confidence, practise with a mix of numeric, algebraic and word-problem exercises. Draw a clear right-angled triangle for every problem, mark the given sides, and identify which side you need to find. Consistent practice will make Pythagoras' theorem a reliable tool for your IGCSE success.
为了建立自信，要通过数字题、代数题和文字题混合练习。每个问题都画出清晰的直角三角形，标出已知边长，并明确要求的是哪条边。坚持不懈的练习会使勾股定理成为你 IGCSE 成功路上一个可靠的工具。
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