📚 Pythagoras' Theorem | 勾股定理
The Cambridge KS3 mathematics curriculum introduces one of the most powerful tools in geometry – Pythagoras' theorem. This theorem describes the special relationship between the three sides of a right‑angled triangle. It is named after the ancient Greek mathematician Pythagoras, although the principle was known to Babylonian and Indian scholars centuries earlier. Understanding this theorem will help you solve problems involving lengths, areas, and real‑life measurements, and it also lays the foundation for trigonometry in later stages.
剑桥 KS3 数学课程介绍几何学中最强大的工具之一——勾股定理。这个定理描述了直角三角形三条边之间的特殊关系。它以古希腊数学家毕达哥拉斯命名，尽管这一原理在几个世纪前就已被巴比伦和印度的学者所知。理解这个定理将帮助你解决有关长度、面积和现实生活测量的问题，也为后续的三角学打下基础。
1. Understanding Right‑Angled Triangles | 认识直角三角形
A right‑angled triangle is a triangle that contains one angle equal to 90°. The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides, which form the right angle, are referred to as the legs or shorter sides. In diagrams, the right angle is usually marked with a small square.
直角三角形是包含一个等于 90° 角的三角形。直角相对的边总是最长的一边，称为 斜边。另外两条形成直角的边称为 直角边 或 较短的边。在图中，直角通常用小正方形标记。
Before applying the theorem, you must be confident in identifying which side is the hypotenuse. A common mistake is to assume the sloping side is automatically the hypotenuse; this is only true if the slope faces the right angle. Always check for the right‑angle symbol first.
在应用定理之前，你必须能够自信地辨认哪条边是斜边。一个常见的错误是认为倾斜的边自动就是斜边；只有这条边面向直角时才成立。务必先检查直角符号。
2. Statement of Pythagoras' Theorem | 勾股定理的表述
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. If we label the sides as a and b for the legs, and c for the hypotenuse, the relationship can be written as:
勾股定理表述为：在任何一个直角三角形中，斜边长度的平方等于另外两条边长度平方的和。如果我们将直角边标记为 a 和 b、斜边标记为 c，这种关系可以写成：
a² + b² = c²
This equation works only for right‑angled triangles. The sides a, b and c must be in the same units before substituting into the formula. Remember that c always represents the hypotenuse – the longest side.
这个等式仅适用于直角三角形。代入公式之前，边 a、b 和 c 必须使用相同单位。请记住 c 始终代表斜边——也就是最长边。
3. Labelling Triangles Correctly | 正确标记三角形
Correct labelling is the first step to a successful calculation. Choose one of the acute angles (not the right angle) and label the opposite side and adjacent side. However, for Pythagoras' theorem, the most important label is the hypotenuse (c). The legs can be assigned as a and b in either order, because a² + b² gives the same sum as b² + a².
正确标记是成功计算的第一步。选择一个锐角（不是直角），然后标记对边和邻边。然而对于勾股定理，最重要的标记是斜边 (c)。直角边可以任意分配为 a 和 b，因为 a² + b² 与 b² + a² 的和相同。
When a question gives lengths without labels, sketch the triangle and mark the right angle. Write 'c' opposite the right angle, and then put a and b on the remaining two sides. This habit will reduce sign‑switching errors later.
当题目给出没有标记的长度时，画出三角形的草图并标出直角。在直角对面写上 'c'，然后将 a 和 b 写在另外两条边上。这个习惯会减少后续的正负号错误。
4. Calculating the Hypotenuse | 计算斜边长度
eg, add the results, and then take the square root. For example, if a = 3 cm and b = 4 cm:
当已知两条直角边时求斜边，先将每条直角边平方，相加后开平方根。例如，如果 a = 3 cm 且 b = 4 cm：
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5 cm
Always check that your calculated hypotenuse is longer than either leg. If it is shorter, you have made a mistake. Use estimation to verify the size – here 5 cm is indeed longer than 3 cm and 4 cm.
始终检查你计算出的斜边是否比任何一条直角边长。如果它更短，说明出错了。用估算来验证大小——这里 5 cm 确实比 3 cm 和 4 cm 长。
With decimals, the process is the same: square the numbers carefully and use a calculator for the square root. Round your answer to a sensible degree of accuracy, typically 1 decimal place or as instructed.
涉及小数时，步骤相同：仔细平方各个数字，并用计算器求平方根。将答案四舍五入到合理的精确度，通常保留 1 位小数或按照题目要求。
5. Calculating a Shorter Side | 计算直角边长度
When the hypotenuse and one leg are known, rearrange the formula to find the missing leg. Subtract the square of the known leg from the square of the hypotenuse, then square root the result:
当已知斜边和一条直角边时，重新排列公式以求出缺失的直角边。用斜边的平方减去已知直角边的平方，然后将结果开平方根：
a² = c² − b² → a = √c² − b²
144 = 12 m. Notice that the missing leg is shorter than the hypotenuse, which makes sense.
例如，如果斜边为 13 m，一条直角边为 5 m，则 a² = 13² − 5² = 169 − 25 = 144，因此 a = √144 = 12 m。请注意缺失的直角边比斜边短，这是合理的。
Avoid the common pitfall of adding instead of subtracting. Write down the rearranged formula explicitly before substituting values, and double‑check your subtraction.
避免相加而不是相减的常见陷阱。在代入数值前明确写下重新排列的公式，并再次检查你的减法。
6. Pythagorean Triples | 勾股数（毕达哥拉斯三元组）
s example is (3, 4, 5). Other common triples include (5, 12, 13), (7, 24, 25) and (8, 15, 17). Multiples of these triples, such as (6, 8, 10) from doubling (3, 4, 5), also work.
勾股数是指能够恰好满足 a² + b² = c² 的三个正整数。最著名的例子是 (3, 4, 5)。其他常见的三元组包括 (5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。这些三元组的倍数，例如将 (3, 4, 5) 翻倍得到的 (6, 8, 10)，也同样成立。
Recognising these triples can save you calculation time in exams. If you spot that two sides belong to a known triple, you can immediately state the third side without squaring. However, always verify that the triangle is right‑angled before assuming a triple applies.
识别这些三元组可以节省你在考试中的计算时间。如果你发现两条边属于已知的勾股数，就可以立即说出第三条边而无需平方运算。但是，在假定适用勾股数之前，一定要确认三角形确实是直角三角形。
| a (leg) | b (leg) | c (hypotenuse) |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
7. Applying Pythagoras to Real‑Life Problems | 勾股定理在现实问题中的应用
Many practical questions require you to draw a right‑angled triangle from a description. A ladder leaning against a wall, a ship sailing on a bearing, or a diagonal of a rectangular garden all create right‑angled triangles where Pythagoras' theorem can be used. The key is to identify which side is unknown – often the direct distance, height, or width.
许多实际问题要求你根据描述画出直角三角形。靠在墙上的梯子、按方位航行的船只或矩形花园的对角线都能构成可以使用勾股定理的直角三角形。关键是确定哪条边是未知的——通常是直线距离、高度或宽度。
For example, a 6‑ r is placed 2 metres away from a vertical wall. How high up the wall does the ladder reach? potenuse, the distance from the wall is one leg, and the height is the other leg: height = √6² − 2² = √32 ≈ 5.66 m. Always sketch the situation.
例如，一架 6 米长的梯子放置在离垂直墙壁 2 米远的地方。梯子能到达墙壁多高的位置？这里梯子是斜边，离墙的距离是一条直角边，高度是另一条直角边：高度 = √6² − 2² = √32 ≈ 5.66 m。一定要画出草图。
Contextual questions may involve finding the length of a diagonal of a television screen, a football pitch, or a rectangular door. Convert all units to the same system before calculating.
情境题可能涉及求电视屏幕、足球场或矩形门的对角线长度。在计算之前，必须将所有单位转换成相同的体制。
8. Pythagoras in Three Dimensions | 三维空间中的勾股定理
In KS3 you may be introduced to finding the space diagonal of a cuboid – the longest line from one corner to the opposite corner through the inside. The 3D version of Pythagoras' theorem relates the length, width, and height of the box to the space diagonal d:
在 KS3 阶段你可能会接触到求长方体空间对角线的问题——即从一个角穿过内部到相对角的最长线段。勾股定理的三维形式将长方体的长、宽、高与空间对角线 d 联系在一起：
d = √l² + w² + h²
You can reach this formula by applying Pythagoras twice: first find the diagonal of the base (l² + w²), then treat that diagonal and the height as two legs of a new right‑angled triangle whose hypotenuse is the space diagonal.
你可以通过两次应用勾股定理得到这个公式：首先求出底面对角线 (l² + w²)，然后将这个对角线视为高度所在的直角边，斜边就是空间对角线。
For a box of length 4 cm, width 3 cm, and height 12 cm, the space diagonal is √4² + 3² + 12² = √16+9+144 = √169 = 13 cm. Notice how this builds on the 3‑4‑5 triple on the base and the 5‑12‑13 triple vertically.
对于一个长 4 cm、宽 3 cm、高 12 cm 的长方体，空间对角线长度为 √4² + 3² + 12² = √16+9+144 = √169 = 13 cm。注意底面的 3‑4‑5 三元组与垂直方向的 5‑12‑13 三元组是如何组合的。
9. Simple Proofs of the Theorem | 定理的简单证明
One of the visual proofs involves drawing a square on each side of a right‑ equal c² in one configuration and a² + b² in another. This demonstrates that a² + b² = c².
一个直观的证明是在直角三角形的每条边上各画一个正方形，然后重新排列面积。构造一个边长为 (a + b) 的大正方形，并在内部放入四个该三角形的副本，剩余面积在一种摆放方式中等于 c²，在另一种中等于 a² + b²。这就证明了 a² + b² = c²。
Another classic proof, credited to President James Garfield, uses a trapezium formed by two identical right‑angled triangles and one isosceles right triangle. The area calculated in two different ways yields the same relationship. These proofs help you understand why the theorem works, not just how to use it.
另一个经典证明来自于詹姆斯·加菲尔德总统，它利用两个相同的直角三角形和一个等腰直角三角形组成的梯形。通过两种不同方法计算面积得出相同的关系。这些证明帮助你理解定理为何成立，而不只是如何使用它。
10. Common Errors and How to Avoid Them | 常见错误与如何避免
Students often apply Pythagoras' theorem to non‑right‑ triangles. Always check for the 90° marker before you start. If the triangle is not right‑ ctly. You may need trigonometry or the sine/cosine rule later, but for KS3, stick to right‑angled triangles only.
学生经常将勾股定理应用于非直角三角形。开始解题前一定要检查 90° 标记。如果三角形不是直角，就不能直接使用 a² + b² = c²。以后你可能需要三角学或正弦/余弦定理，但在 KS3 阶段，仅限直角三角形。
ot symbol deliberately and underline it until the habit is automatic. Similarly, when finding a shorter side, always subtract before square‑rooting.
另一个常见错误是忘记最后开平方根。在计算出 c² = 25 后，有人会写 c = 25 而不是 c = 5。刻意写出根号并为它加下划线，直到习惯成自然。同样地，在求直角边时，一定要先相减再开平方根。
Misidentification of the hypotenuse is another trap. In a triangle where the right angle is not at the bottom, the hypotenuse may be drawn horizontally or at a strange angle. Rotate your page mentally and ask: 'Which side is opposite the right angle?' That side is c.
错误辨认斜边是另一个陷阱。在直角不在底部的三角形中，斜边可能水平绘制或以奇怪角度出现。在心里旋转页面并问自己：“哪条边正对直角？”那条边就是 c。
11. Practice Strategy and Exam Tips | 练习策略与考试技巧
Start with simple integer lengths and build confidence. Then move to decimal and word problems. Always write down the formula first, substitute with brackets if negative numbers could appear, and show all working steps. In Cambridge KS3 exams, method marks are often awarded for correct rearrangement and substitution even if the final answer is wrong.
从简单的整数长度开始建立信心，然后进阶到小数和应用题。始终先写下公式，如果可能出现负数就用括号代入数值，并展示所有运算步骤。在剑桥 KS3 考试中，即使最终答案错误，正确的公式重排和代入过程往往也能获得方法分。
Use estimation to check answers. For instance, if the legs are 5.1 and 6.8, the hypotenuse should be slightly above √25+36 = √61 easonable. Familiarity with perfect squares up to 15² speeds up mental arithmetic significantly.
用估算检查答案。例如，如果直角边是 5.1 和 6.8，斜边应略高于 √25+36 = √61 ≈ 7.8，所以 11.2 这样答案就不合理。熟悉 15² 以内的完全平方数能显著加快心算速度。
12. Linking Pythagoras to Other Topics | 将勾股定理与其他主题联系起来
Pythagoras' ts (x₁, y₁) and (x₂, y₂), the horizontal and vertical differences form the legs of a right‑ so the distance is √((x₂ − x₁)² + (y₂ − y₁)²). This formula is just Pythagoras on a grid.
勾股定理出现在坐标几何中的两点间距离。给定两点 (x₁, y₁) 和 (x₂, y₂)，水平差和垂直差构成直角三角形的两条直角边，因此距离为 √((x₂ − x₁)² + (y₂ − y₁)²)。这个公式不过是网格上的勾股定理。
e area directly using (1/2)×base×height and then find a missing altitude using Pythagoras.
在作图和方位角中，从点到直线的最短距离形成直角，因此勾股定理可以和比例绘图结合使用。即使在面积问题中，如果你知道直角三角形的边长，可以用 (1/2)×底×高 直接求面积，然后再用勾股定理求缺失的高。
Seeing these connections will help you answer multi‑step questions and prepare for the transition to IGCSE, where Pythagoras is frequently used in trigonometry, vectors and 3D geometry.
看清这些联系将帮助你解答多步综合题，并为过渡到 IGCSE 做好准备，在 IGCSE 中勾股定理常被用于三角学、向量和三维几何。
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