Pythagoras’ Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most famous and useful results in mathematics. It connects the three sides of a right-angled triangle and allows us to find missing lengths. Named after the ancient Greek mathematician Pythagoras, the theorem has been used for thousands of years in building, navigation and design. In this revision guide, you will learn what the theorem says, how to use its formula, and how to apply it in both two-dimensional and three-dimensional problems. We will also explore Pythagorean triples, common mistakes to avoid, and plenty of worked examples to build your confidence.

勾股定理是数学中最著名、最实用的定理之一。它将直角三角形的三条边联系起来，使我们能够求出未知的边长。这一定理以古希腊数学家毕达哥拉斯的名字命名，几千年来一直应用于建筑、航海和设计领域。在这份复习指南中，你将学习定理的内容、如何运用公式，以及如何在二维和三维问题中应用它。我们还将探讨勾股数组、需要注意的常见错误，并通过大量例题帮助你建立信心。

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 longest side (the hypotenuse) is equal to the sum of the squares of the other two sides. This relationship only works for right-angled triangles. It is written as a² + b² = c², where c is the hypotenuse and a and b are the two shorter sides. The theorem allows us to calculate one side when the other two are known.

勾股定理指出，在任何直角三角形中，最长边（斜边）的平方等于另外两条边的平方和。这一关系只适用于直角三角形。定理写作 a² + b² = c²，其中 c 代表斜边，a 和 b 代表两条直角边。当已知两边长度时，通过该定理可以计算出第三边的长度。

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2. The Hypotenuse | 斜边

The hypotenuse is always the side opposite the right angle. It is the longest side in a right-angled triangle because it faces the largest angle (90°). When labelling a triangle, it is common to use lower-case letters a and b for the legs, and c for the hypotenuse. Remember, c must be the side directly across from the right angle, no matter how the triangle is rotated.

斜边始终是直角所对的边。它是直角三角形中最长的一条边，因为它正对着最大的角（90°）。标记三角形时，通常用小写字母 a 和 b 表示直角边，用 c 表示斜边。注意，无论三角形如何旋转，c 必须始终是直角正对的边。

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3. The Formula | 公式

The standard formula is a² + b² = c². To use it, square the two shorter sides, add them together, and the result equals the square of the hypotenuse. When solving for a missing leg, rearrange the equation: a² = c² − b² or b² = c² − a². Always remember to take the square root as the final step to find the actual side length.

标准公式为 a² + b² = c²。使用时，先将两条直角边的长度分别平方，然后把结果相加，所得之和就等于斜边的平方。当需要求某一直角边时，可调整公式：a² = c² − b² 或 b² = c² − a²。务必记住最后一步是开平方，才能得到实际边长。

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4. A Simple Example | 简单示例

Imagine a right-angled triangle with legs of 3 cm and 4 cm. Using Pythagoras’ theorem: 3² + 4² = 9 + 16 = 25, so c² = 25. Taking the square root, c = 5 cm. This is the classic 3-4-5 triangle, which is a Pythagorean triple. Always include units in your final answer and check that the hypotenuse is larger than either leg.

设想一个直角三角形，两条直角边分别为 3 cm 和 4 cm。应用勾股定理：3² + 4² = 9 + 16 = 25，因此 c² = 25。开平方后得出 c = 5 cm。这就是经典的 3-4-5 三角形，它是一组勾股数组。最终答案一定要带有单位，并检查斜边是否大于任意一条直角边。

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5. Visual Proof | 几何证明

One way to ‘see’ the theorem is to draw squares on each side of a right-angled triangle. The area of the square built on the hypotenuse equals the sum of the areas of the squares on the two legs. For the 3-4-5 triangle, the square on side 3 has area 9, on side 4 has area 16, and on side 5 has area 25. So 9 + 16 = 25 perfectly matches the theorem.

一种直观“看见”定理的方法是，在直角三角形的每条边上各画一个正方形。画在斜边上的正方形面积，恰好等于画在两条直角边上正方形面积之和。对于 3-4-5 三角形，边长为 3 的正方形面积为 9，边长为 4 的正方形面积为 16，边长为 5 的正方形面积为 25。9 + 16 = 25 完全符合定理。

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6. Finding the Hypotenuse | 求斜边长度

When given both legs, finding the hypotenuse is straightforward: square each leg, add them, and take the square root. For example, legs 5 m and 12 m give 5² + 12² = 25 + 144 = 169. The square root of 169 is 13, so the hypotenuse is 13 m. Always round your answer appropriately if the square root is not exact.

已知两条直角边求斜边非常简单：将每一直角边平方，相加后开平方。例如，直角边为 5 m 和 12 m，则 5² + 12² = 25 + 144 = 169。169 的平方根是 13，因此斜边为 13 m。若开方结果不是整数，应按要求适当四舍五入。

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7. Finding a Shorter Side | 求直角边长度

If you know the hypotenuse and one leg, rearrange the equation to find the missing leg. Suppose c = 10 cm and a = 6 cm. Then b² = c² − a² = 100 − 36 = 64, so b = 8 cm. Be careful not to add the squares by mistake. Subtraction is required because the hypotenuse square is always the largest number.

如果已知斜边和一条直角边，可重新排列方程求出未知的直角边。假设 c = 10 cm，a = 6 cm，那么 b² = c² − a² = 100 − 36 = 64，因此 b = 8 cm。注意不要误用加法——这里需要减法，因为斜边的平方总是最大的那个数。

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8. Pythagorean Triples | 勾股数组

A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². The most common examples are (3, 4, 5), (5, 12, 13), (8, 15, 17) and their multiples, such as (6, 8, 10). Recognising these triples can save time in exams and help check your work. Not all right triangles have whole-number side lengths, but many exam questions use triples to keep answers neat.

勾股数组是指能够满足 a² + b² = c² 的三个正整数。最常见的例子有 (3, 4, 5)、(5, 12, 13)、(8, 15, 17) 以及它们的倍数，如 (6, 8, 10)。识别这些数组可以节省考试时间，也有助于检查答案。并非所有直角三角形的边长都是整数，但许多考题会使用勾股数组以保持答案整洁。

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9. Applying Pythagoras in Real Life | 实际应用

Pythagoras’ theorem appears in countless real-world situations. Builders use it to check whether walls are perpendicular by measuring a 3-4-5 triangle. Navigators find the shortest distance between two points by treating latitude and longitude differences as legs of a right triangle. Even screen sizes of televisions and monitors are calculated using the diagonal, which acts as the hypotenuse of the screen’s width and height.

勾股定理在现实生活中随处可见。建筑工人利用 3-4-5 三角形来检验墙体是否垂直。航海人员通过将经纬度差视作直角三角形的直角边，求出两点之间的最短距离。就连电视和显示器的屏幕尺寸也是通过其对角线计算的，这条对角线相当于屏幕宽和高所构成的直角三角形的斜边。

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10. Pythagoras in 3D | 三维空间中的勾股定理

In three dimensions, we often need to find the length of a space diagonal inside a rectangular box (cuboid). The formula becomes √(x² + y² + z²), where x, y and z are the length, width and height of the box. This is a double application of Pythagoras: first find the diagonal of the base, then use it with the height to find the space diagonal.

在三维空间中，我们常常需要求长方体内部空间对角线的长度。此时公式变为 √(x² + y² + z²)，其中 x、y 和 z 分别代表长方体的长、宽、高。这本质上是勾股定理的两次运用：先求出底面对角线，再与高结合求出空间对角线。

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11. Common Mistakes | 常见错误

Many students forget to square the side lengths before adding, or they add the lengths first and then square the sum, which is incorrect. Another frequent error is mixing up the hypotenuse and a leg, especially when a triangle is rotated. It is also easy to forget to take the square root at the end. Always label the sides carefully, and ask yourself whether your final answer makes sense (the hypotenuse should be the longest side).

许多学生忘记先将边长的数值平方再相加，或者先将边长相加再求平方，这都是错误的。另一个常见错误是混淆斜边和直角边，尤其在三角形经过旋转之后。还容易忘记最后一步开平方。务必仔细标记各边，并问自己最终答案是否合理（斜边应是最长的一条边）。

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12. Practice Questions | 练习题

Try these problems: (1) A right triangle has legs 9 cm and 12 cm. Find the hypotenuse. (2) A ladder of 13 m leans against a wall. Its base is 5 m from the wall. How high up the wall does it reach? (3) A cuboid measures 3 cm by 4 cm by 12 cm. Calculate the length of the space diagonal. Check your answers: (1) 15 cm, (2) 12 m, (3) 13 cm. Practise regularly to master the steps.

试一试这些题目：(1) 一直角三角形的直角边为 9 cm 和 12 cm，求斜边。(2) 一架 13 m 长的梯子斜靠墙壁，梯脚离墙 5 m，问梯子顶端能达到多高？(3) 一个长方体的长、宽、高分别为 3 cm、4 cm、12 cm，求空间对角线的长度。核对答案：(1) 15 cm，(2) 12 m，(3) 13 cm。坚持练习才能熟练掌握解题步骤。

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Published by TutorHao | Maths Revision Series | aleveler.com

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