📚 Pythagoras' Theorem | 勾股定理
Pythagoras' theorem is one of the most famous results in mathematics. It connects the three sides of a right‑angled triangle and allows us to solve countless problems in geometry, construction, navigation and even 3D shapes. Whether you are calculating the length of a ladder needed to reach a window or checking whether a corner is perfectly square, this theorem is an essential tool. In this article we will explore the theorem step by step, from its basic statement to practical applications, common pitfalls and exam‑style questions.
勾股定理是数学中最著名的定理之一。它将直角三角形的三条边联系起来，使我们能够解决几何、建筑、导航甚至三维图形中的大量问题。无论是计算达到窗户所需的梯子长度，还是检查墙角是否完全垂直，勾股定理都是一个不可或缺的工具。在本文中，我们将逐步探索该定理，从基本表述到实际应用、常见错误以及考试类型题目。
1. Introduction to Right‑Angled Triangles | 直角三角形的介绍
A right‑angled triangle is a triangle that contains one angle of exactly 90°. The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are often referred to as the 'legs' of the triangle. In diagrams, the right angle is usually marked with a small square, and the hypotenuse is placed opposite it.
直角三角形是包含一个恰好为 90° 角的三角形。直角对面的边总是最长的边，被称为斜边。其他两条边通常称为三角形的“直角边”。在图形中，直角通常用小方块标记，斜边则位于其对侧。
For example, consider a triangle with vertices A, B and C, with the right angle at B. Then the side AC is the hypotenuse, while AB and BC are the shorter sides. Recognising the hypotenuse correctly is the first key step when using Pythagoras' theorem.
例如，考虑一个顶点为 A、B 和 C 的三角形，直角在 B 处。那么边 AC 是斜边，而 AB 和 BC 是较短的边。正确识别斜边是应用勾股定理的第一个关键步骤。
2. 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. This relationship only holds for right‑angled triangles and is used to find an unknown side length when the other two are known.
勾股定理表明，在任何直角三角形中，斜边长度的平方等于其他两条边长度的平方和。这个关系仅适用于直角三角形，用于在已知其他两条边时求未知边长。
Historically, the theorem is named after the Greek mathematician Pythagoras, although evidence shows that Babylonian and Indian mathematicians knew the relationship centuries earlier. Today it is taught worldwide because of its simplicity and power.
从历史上看，这个定理以希腊数学家毕达哥拉斯命名，尽管证据表明巴比伦和印度的数学家早在几个世纪前就已经知道了这个关系。如今它因其简洁和强大而全球教授。
3. The Formula: a² + b² = c² | 公式：a² + b² = c²
The standard formula is written as:
标准公式写为：
c² = a² + b²
Here, c represents the length of the hypotenuse, while a and b represent the lengths of the other two sides. It does not matter which shorter side is labelled a or b, as addition is commutative. The side labelled c must always be the hypotenuse.
这里，c 代表斜边长度，而 a 和 b 代表其他两条边的长度。哪个直角边标记为 a 或 b 并不重要，因为加法是可交换的。标记为 c 的边必须始终是斜边。
a², then take the square root. This rearranged form is just as important as the original.
公式可以变形以求出任意一边。例如，如果我们需要求一条直角边，我们使用 a² = c² − b² 或 b² = c² − a²，然后取平方根。这种变形形式与原公式同样重要。
4. Finding the Hypotenuse | 求斜边长度
e a = 3 cm and b = 4 cm. Then c² = 3² + 4² = 9 + 16 = 25. Taking the square root gives c = √25 = 5 cm. This is the classic 3‑4‑5 triangle.
当已知两个直角边时，求斜边很简单。假设 a = 3 厘米，b = 4 厘米。那么 c² = 3² + 4² = 9 + 16 = 25。取平方根得到 c = √25 = 5 厘米。这就是经典的 3‑4‑5 三角形。
Always remember to take the square root at the end. A common error is to forget this step and give the squared value as the final answer. In exams, marks are often awarded for the correct substitution and the final square root step.
始终记得最后取平方根。一个常见错误是忘记这一步，把平方后的值当作最终答案。在考试中，通常正确的代入和最后的平方根步骤都能够得分。
Example: a = 6 m, b = 8 m.
例子：a = 6 米，b = 8 米。
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 m.
5. Finding a Shorter Side | 求直角边长度
the hypotenuse c and one shorter side b are known, we compute a² = c² − b², then find a = √c² − b². It is vital to subtract correctly, not add.
为了求直角边，我们重新整理公式。如果已知斜边 c 和一条直角边 b，我们计算 a² = c² − b²，然后得出 a = √c² − b²。正确地做减法而非加法至关重要。
t a wall, with its foot 2 m away from the wall. How high up the wall does the ladder reach? Here the ladder is the hypotenuse (c = 5 m), the unknown height is a, and the distance from the wall is b = 2 m.
假设一个 5 米长的梯子靠在墙上，梯脚距离墙 2 米。梯子顶端达到多高？这里梯子是斜边（c = 5 米），未知高度是 a，到墙的距离是 b = 2 米。
a² = 5² − 2² = 25 − 4 = 21, so a = √21 ≈ 4.58 m (to 3 s.f.).
Note that the answer is left in surd form or rounded as required. Careful rounding is part of the mathematical process.
注意答案可以保留为根式形式或按要求四舍五入。仔细取整是数学过程的一部分。
6. Pythagorean Triples | 勾股数组
mous triple is (3, 4, 5). Others include (5, 12, 13), (7, 24, 25) and multiples like (6, 8, 10) which is just a scaled version of the 3‑4‑5 triple.
勾股数组由三个正整数 a、b 和 c 组成，满足 a² + b² = c²。最小且最著名的是 (3, 4, 5)。其他还有 (5, 12, 13)、(7, 24, 25) 以及如 (6, 8, 10) 这样的倍数，后者是 3‑4‑5 的放大版。
Knowing common triples can save time in exams. If you spot that two sides of a right triangle are part of a triple, you can instantly write down the missing side without calculating. However, always verify that the triangle is indeed right‑angled before applying a triple.
熟悉常见的勾股数可以在考试中节省时间。如果你发现直角三角形的两条边属于某个勾股数的一组，你可以直接写出缺失的边长而无需计算。不过，在应用勾股数之前，一定要确认三角形确实是直角三角形。
7. Applying Pythagoras in Word Problems | 勾股定理的应用题
Real‑world problems often require us to form a right‑angled triangle and then apply Pythagoras. Typical situations include finding the diagonal of a rectangle, the distance between two points on a coordinate grid, or the shortest path across a park. The key is to sketch a diagram and label the known and unknown sides clearly.
现实问题往往需要我们构造一个直角三角形，然后应用勾股定理。典型情况包括求矩形的对角线、坐标网格上两点之间的距离或穿过公园的最短路径。关键是画出示意图并清楚地标明已知边和未知边。
Example: A rectangular field measures 40 m by 30 m. A path runs diagonally across it. Find the length of the diagonal path. The diagonal splits the rectangle into two right‑angled triangles, each with legs 40 m and 30 m. The diagonal is the hypotenuse, so:
例子：一个矩形田地长 40 米、宽 30 米。一条小径沿对角线穿过。求对角路径的长度。对角线将矩形分成两个直角三角形，每个直角边为 40 米和 30 米。对角线是斜边，因此：
diagonal² = 40² + 30² = 1600 + 900 = 2500, diagonal = √2500 = 50 m.
Another classic problem involves two ships sailing at right angles to each other. If one travels 12 km east and the other 9 km north, the distance between them is the hypotenuse of a right triangle with legs 12 and 9, giving 15 km (a 3‑4‑5 triple scaled by 3).
另一个经典问题涉及两艘船分别向东和向北航行。如果一艘向东行驶 12 公里，另一艘向北 9 公里，它们之间的距离就是直角边为 12 和 9 的斜边，距离为 15 公里（3‑4‑5 勾股数组放大 3 倍）。
8. Check: Is a Triangle Right‑Angled? | 判断三角形是否为直角三角形
The converse of Pythagoras' theorem is also true: If the squares of the two shorter sides add up to the square of the longest side, then the triangle is right‑angled. This provides a simple test for right angles.
勾股定理的逆命题同样成立：如果两条较短边的平方和等于最长边的平方，那么这个三角形就是直角三角形。这为直角提供了一个简单的检验方法。
iangle contains a right angle opposite the 17 cm side. This is particularly useful in construction and carpentry to ensure corners are square.
对于一个边长分别为 8 厘米、15 厘米和 17 厘米的三角形，检验 8² + 15² = 64 + 225 = 289，而 17² = 289。由于它们相等，该三角形在 17 厘米边对面包含一个直角。这在建筑和木工中特别有用，以确保角落方正。
9. Pythagoras in 3D Shapes (Introduction) | 在三维形状中的勾股定理简介
Pythagoras' with sides p, q and r is d² = p² + q² + r². This is derived by applying Pythagoras twice.
勾股定理可以自然地扩展到三维问题，我们常常需要求长方体（长方体）的空间对角线长度。一个边长分别为 p、q 和 r 的长方体，其空间对角线长度 d 的公式为 d² = p² + q² + r²。这是通过两次应用勾股定理推导出来的。
= p² + q² + r². This technique appears frequently in KS3 extension work and later in GCSE.
第一步：求底面矩形的对角线：x² = p² + q²。第二步：将 x 和 r 视为一个新的直角三角形的直角边，空间对角线为斜边：d² = x² + r² = p² + q² + r²。此技巧经常出现在 KS3 拓展内容和以后的 GCSE 中。
10. Common Mistakes to Avoid | 常见错误避免
Even with a simple theorem, errors are frequent. The most common mistakes are:
即使是一个简单的定理，错误也常常出现。最常见的错误有：
1. Adding the squares when you should subtract (finding a shorter side). Always check which side is the unknown. If it is not the hypotenuse, use subtraction.
1. 在应该做减法时却做加法（求直角边）。总是检查未知边是哪个。如果不是斜边，就用减法。
2. Forgetting to take the square root at the end. Remember that the formula gives the square of the side, not the side itself.
2. 忘记最后取平方根。记住公式给出的是边长的平方，而不是边长本身。
3. Misidentifying the hypotenuse. The hypotenuse is always opposite the right angle. Look for the square marking the right angle.
3. 错误识别斜边。斜边总是在直角对面。寻找标记直角的小方块。
4. Using mixed units without converting. Ensure all lengths are in the same unit before calculating.
4. 未转换单位就混用。在计算前确保所有长度都使用相同单位。
5. Rounding too early. Keep exact values until the final answer, then round if needed.
5. 过早取整。保持准确值直到最后答案，然后根据需要取整。
11. Practice Questions Walkthrough | 练习讲解
Let us work through two exam‑style questions to solidify understanding.
让我们做两道考试式题目以巩固理解。
Question 1: A right triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the length of the other side.
题目 1：一个直角三角形的斜边为 13 厘米，一条直角边为 5 厘米。求另一条直角边的长度。
3² − 5² = 169 − 25 = 144, so a = √144 = 12 cm. (Check: 5, 12, 13 is a Pythagorean triple.)
解答：设未知边为 a。那么 a² = 13² − 5² = 169 − 25 = 144，因此 a = √144 = 12 厘米。（检验：5、12、13 是一组勾股数。）
Question 2: A kite string is 20 m long. The kite is directly above a point on the ground 12 m from the person holding the string. How high is the kite? Assume the string is taut.
题目 2：一个风筝的线长 20 米。风筝位于地面上与放风筝者水平距离 12 米的正上方。风筝多高？假设线是拉直的。
leg (b = 12 m), and the height is a. a² = 20² − 12² = 400 − 144 = 256, so a = √256 = 16 m.
解答：风筝线为斜边（c = 20 米），水平距离为一直角边（b = 12 米），高度为 a。a² = 20² − 12² = 400 − 144 = 256，因此 a = √256 = 16 米。
12. Summary | 总结
Pythagoras' that you will use again and again. Remember the formula c² = a² + b² applies only to right‑ ct. Be aware of common triples to speed up your work, and always check your answer is reasonable. With plenty of practice, applying this theorem becomes second nature, building confidence for more advanced topics like trigonometry and coordinate geometry.
勾股定理是几何学的一个基石，你会反复用到。记住公式 c² = a² + b² 只适用于直角三角形。首先识别斜边，然后仔细代入已知值。当未知是一条直角边时，变形为减法。熟悉常见的勾股数以加快解题速度，并且始终检查答案是否合理。经过大量练习，应用该定理会变得得心应手，为以后三角学和坐标几何等更高级的内容建立信心。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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