Pythagoras’ Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Imagine you are building a ramp for a skateboard, or perhaps you want to find the shortest path across a rectangular park. In both cases, you are dealing with right-angled triangles. One of the most powerful tools in mathematics to handle such situations is Pythagoras’ theorem. Named after the ancient Greek mathematician Pythagoras, this theorem reveals a fundamental relationship between the three sides of any right-angled triangle. It is a cornerstone of geometry and has been used for thousands of years in construction, navigation, and design. In this article, we will explore what Pythagoras’ theorem states, how to apply it correctly, and where it shows up in real life and further mathematics. By the end, you will be able to calculate unknown side lengths with confidence and even test whether a triangle is right-angled.

想象你正在为滑板搭建一个坡道，或者你想找到穿过一个矩形公园的最短路径。在这两种情况下，你都在处理直角三角形。数学中处理这类情况最强有力的工具之一就是勾股定理。这一定理以古希腊数学家毕达哥拉斯命名，揭示了任意直角三角形三条边之间的基本关系。它是几何学的基石，几千年来被用于建筑、导航和设计。在本文中，我们将探索勾股定理的内容、如何正确应用它，以及它在现实生活和更深入的数学中的应用。学完本文后，你将能够自信地计算未知边长，甚至检验一个三角形是否为直角三角形。

1. Right-Angled Triangles | 直角三角形

Before we can use Pythagoras’ theorem, we must recognise a right-angled triangle. A right-angled triangle is a triangle that has one interior angle exactly equal to 90°, often marked with a small square at that corner. The side opposite this right angle is always the longest side of the triangle, called the hypotenuse. The other two sides are sometimes called the legs or, more formally, the adjacent and opposite sides relative to a given angle, but for Pythagoras we group them as the two shorter sides. It is crucial to identify which side is the hypotenuse because the theorem is built around it.

在应用勾股定理之前，我们必须认识什么是直角三角形。直角三角形是恰好有一个内角为90°的三角形，通常在那个角处用一个小方块标记。直角所对的边总是三角形中最长的边，称为斜边。另外两条边有时称为直角边，或者更正式地相对于某个角称为邻边和对边，但在勾股定理中我们将它们统称为两条较短的直角边。确定哪条是斜边至关重要，因为定理正是围绕它建立的。

A common mistake is to assume the hypotenuse is simply the side that slopes or looks longest in the diagram. Always check for the right-angle symbol and identify the side directly across from it. In labelled triangles, the right angle is often at vertex C and the hypotenuse is side c, opposite C. In coordinate geometry, if you have two points forming a horizontal and vertical displacement, the straight-line distance between them is the hypotenuse.

一个常见的错误是以为斜边只是倾斜的那条边或者图上看起来最长的边。一定要检查直角符号，并找出正对着直角的边。在标注的三角形中，直角通常在顶点C处，斜边是边c，对着C。在坐标几何中，如果你有两个点，它们之间的水平距离和垂直距离构成两条直角边，而点与点之间的直线距离就是斜边。

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2. Identifying the Hypotenuse | 确定斜边

To avoid errors, always circle or highlight the right angle first. Then look across the triangle to find the side that does not touch the right angle; that is your hypotenuse. For example, in triangle ABC with the right angle at B, sides AB and BC meet at the right angle, so the side AC is the hypotenuse. It is irrelevant how the triangle is rotated; the hypotenuse is always the side opposite the right angle. In a diagram where the triangle is drawn with the hypotenuse as a sloping line, it is still the longest side.

为了避免错误，始终先圈出或高亮直角。然后看向三角形的对面，找到不与直角接触的边；那就是斜边。例如，在三角形ABC中，直角位于B点，AB和BC在直角处相交，因此AC边是斜边。三角形如何旋转无关紧要；斜边始终是直角所对的边。在斜边被画成倾斜线的图中，它仍然是最长的边。

Practice by drawing several right-angled triangles in different orientations. Label the right angle and then put a small square on the hypotenuse. This habit will help when the triangles are nested inside other shapes, such as in a cuboid or a pyramid, where you might need to apply the theorem twice.

通过在不同方向上画几个直角三角形来练习。标出直角，然后在斜边上画一个小方块。当三角形嵌套在其他形状中时，例如在长方体或棱锥中，这个习惯会很有用，因为在那些情形中你可能需要两次使用定理。

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3. The Statement of the Theorem | 定理表述

Pythagoras’ theorem states: 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 the hypotenuse is labelled c and the other two sides are a and b, then the theorem is written as:

c² = a² + b²

This formula is both simple and profound. The notation c² means c multiplied by itself, and similarly for a² and b². It is essential to remember that this relationship only holds for right-angled triangles. For any other triangle, a² + b² does not equal c² in this way, which leads to the converse of the theorem (Section 9).

勾股定理表明：在任何直角三角形中，斜边长度的平方等于另外两条边长度的平方之和。如果斜边用c表示，另外两边用a和b表示，那么定理可写为：

c² = a² + b²

这个公式既简单又深刻。c²表示c乘以自身，a²和b²同理。必须记住，这一关系仅在直角三角形中成立。对于任何其他三角形，a² + b²并不以这种方式等于c²，这就引出了勾股定理的逆定理（第9节）。

You can verify the theorem with simple numbers: if a = 3, b = 4, then a² + b² = 9 + 16 = 25, so c² = 25 and c = √25 = 5. This famous 3‑4‑5 triangle is often used by builders to create right angles without a protractor.

你可以用简单的数字验证定理：如果a = 3，b = 4，那么a² + b² = 9 + 16 = 25，因此c² = 25，c = √25 = 5。这个著名的3-4-5三角形常被建筑工人用来在没有量角器的情况下构造直角。

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4. A Visual Proof | 一种几何证明

One captivating way to understand Pythagoras’ theorem is through a geometric proof that does not rely on algebra. Imagine drawing a square on each side of a right-angled triangle. The area of the square drawn on the hypotenuse turns out to be exactly equal to the combined areas of the squares drawn on the other two sides. You can prove this by rearranging shapes. Cut out the squares on the two legs and rearrange the pieces to perfectly cover the square on the hypotenuse. This visual demonstration works for any right-angled triangle and shows why the relationship holds.

理解勾股定理的一个迷人方法是通过一个不依赖于代数的几何证明。想象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的面积恰好等于画在两条直角边上的正方形面积之和。你可以通过重新排列图形来证明这一点。剪下两条直角边上的正方形，重新排列碎片，使之完全覆盖斜边上的正方形。这种直观演示对任何直角三角形都成立，并揭示了这一关系为何成立。

Another famous proof, often attributed to the Indian mathematician Bhaskara, consists of drawing a square with side length c and placing four identical right-angled triangles inside it, leaving a smaller square in the middle. Calculating the total area in two different ways leads directly to c² = a² + b². Such proofs reinforce that the theorem is not just a formula to memorise but a deep geometric truth.

另一个著名的证明通常归功于印度数学家婆什迦罗，它画出一个边长为c的正方形，并在其中放置四个相同的直角三角形，中间留下一个小正方形。用两种不同方式计算总面积，直接得到c² = a² + b²。这些证明强化了定理不仅仅是一个需要记住的公式，而是一个深刻的几何真理。

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5. Finding the Hypotenuse | 求斜边

When you are given the lengths of the two shorter sides, you can find the hypotenuse using c = √(a² + b²). Always remember to take the square root at the end. For example, a ladder leaning against a wall forms a right-angled triangle with the ground. If the foot of the ladder is 2 m from the wall and the top reaches 6 m up the wall, the length of the ladder is the hypotenuse. Substituting a = 2 and b = 6 gives c² = 2² + 6² = 4 + 36 = 40, so c = √40. That simplifies to 2√10 metres, or approximately 6.32 m. Leaving your answer in surd form (√40 or simplified) is often more exact than a rounded decimal.

当已知两条直角边的长度时，你可以用 c = √(a² + b²) 求斜边。始终记得最后取平方根。例如，一架靠在墙上的梯子与地面构成一个直角三角形。如果梯脚离墙2米，梯顶在墙上6米高处，那么梯子的长度就是斜边。代入 a = 2，b = 6 得 c² = 2² + 6² = 4 + 36 = 40，所以 c = √40。这可以简化为2√10米，或者约为6.32米。将答案保留为根式形式（√40或化简后）通常比四舍五入的小数更精确。

At KS3, you are expected to use a calculator to find square roots when necessary, but also to simplify surds where possible. Always check that the hypotenuse you calculate is indeed longer than each of the other two sides; if it is not, you may have mixed up the sides.

在KS3阶段，你应能在必要时使用计算器求平方根，同时也要尽可能化简根式。始终检查你计算出的斜边是否确实比另外两边中的任一边都长；如果不是，你可能搞混了边的对应关系。

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6. Finding a Shorter Side | 求直角边

Sometimes the unknown length is one of the two shorter sides, not the hypotenuse. In that case, you need to rearrange the formula. If the hypotenuse c and one leg a are known, the other leg b is given by:

b = √(c² − a²)

It is vital to subtract the square of the known leg from the square of the hypotenuse, not the other way around. For instance, a television screen has a diagonal of 80 cm (hypotenuse) and a height of 40 cm. The width b can be found using b² = 80² − 40² = 6400 − 1600 = 4800, so b = √4800 ≈ 69.3 cm. Notice we subtracted, then took the square root.

有时未知长度是两条直角边之一，而非斜边。此时你需要重新整理公式。如果已知斜边c和一条直角边a，那么另一条直角边b由下式给出：

b = √(c² − a²)

必须用斜边的平方减去已知直角边的平方，而非反过来。例如，一台电视机的屏幕对角线长80厘米（斜边），高40厘米。宽度b可以用 b² = 80² − 40² = 6400 − 1600 = 4800 求得，因此 b = √4800 ≈ 69.3厘米。注意，我们先减，然后取平方根。

A common error is to put the numbers the wrong way round, for example writing b² = a² − c², which would lead to a negative number under the square root. Always ensure the hypotenuse is the largest number. In word problems, reading carefully to identify which side is being asked for can prevent such mistakes.

一个常见错误是把数字的顺序搞反，比如写成 b² = a² − c²，这会导致平方根下的数为负。始终确保斜边是最大的数。在应用题中，仔细阅读以确定要求的是哪条边，可以避免这类错误。

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7. Pythagorean Triples | 勾股数

Certain sets of three whole numbers satisfy Pythagoras’ theorem exactly. These are called Pythagorean triples. The simplest (3, 4, 5) is a triple because 3² + 4² = 5². Triples can be scaled up: multiplying all numbers by 2 gives (6, 8, 10), which also works. Other common triples are (5, 12, 13), (7, 24, 25) and (8, 15, 17). Knowing these can save time in calculations and help you spot right-angled triangles quickly. If two sides are given from a known triple, you can instantly write the third without using the formula.

某些三整数组恰好满足勾股定理，它们被称为勾股数。最简单的(3, 4, 5)就是一组勾股数，因为3² + 4² = 5²。勾股数可以按比例放大：将所有数乘以2得到(6, 8, 10)，同样成立。其他常见的勾股数有(5, 12, 13)、(7, 24, 25)和(8, 15, 17)。掌握这些可以节省计算时间，并帮助你快速识别直角三角形。如果已知的两边属于某一组勾股数，你可以直接写出第三边，而无需使用公式。

The table below lists a few Pythagorean triples that often appear in KS3 problems. The multiples row illustrates that if (a, b, c) is a triple, then (ka, kb, kc) is also a triple for any positive integer k.

Common Pythagorean Triples | 常见勾股数

a	b	c	Verification | 验证
3	4	5	9 + 16 = 25
5	12	13	25 + 144 = 169
7	24	25	49 + 576 = 625
8	15	17	64 + 225 = 289
6 (3×2)	8 (4×2)	10 (5×2)	36 + 64 = 100

Using triples can be a shortcut, but always check that the longest number is in the correct position as the hypotenuse unless the problem states otherwise.

使用勾股数可以是一条捷径，但始终要检查最长的数是否位于正确的斜边位置，除非题目另有说明。

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8. Applying Pythagoras in Real Life | 实际应用

Pythagoras’ theorem is not just an abstract classroom exercise; it appears in numerous practical contexts. Builders use it to ensure walls are perpendicular by measuring a 3‑4‑5 triangle. Ships and aircraft calculate the shortest distance between two points that are separated by both longitude and latitude, effectively forming a right-angled triangle on the Earth’s surface. In technology, the distance between two pixels on a screen is found using the theorem, as the screen is a grid of tiny squares. Even GPS devices rely on triangulation, which often involves several right-angled triangles.

勾股定理不只是一个抽象的课堂练习；它出现在许多实际场景中。建筑工人通过测量3-4-5三角形来确保墙壁垂直。船舶和飞机计算两个既经度又纬度分开的点之间的最短距离时，实际上是在地球表面形成了一个直角三角形。在技术领域，屏幕上两个像素之间的距离就用这一定理求出，因为屏幕是由微小的正方形网格构成的。就连GPS设备也依赖于三角测量，这往往会涉及多个直角三角形。

Another everyday example is determining whether a piece of furniture will fit through a doorway. You can measure the height and width of the doorway, then use Pythagoras to find its diagonal, which gives the maximum length of a straight object that can be carried through while remaining horizontal. This is exactly why movers tilt long sofas diagonally when going through narrow doors.

另一个日常例子是判断一件家具是否能够通过门框。你可以测量门框的高和宽，然后使用勾股定理求出对角线，这给出了一个保持水平的细长物体能够通过的最大长度。这正是搬家工人将长沙发斜着搬过窄门的原因。

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9. The Converse of Pythagoras | 勾股定理的逆定理

The converse of Pythagoras’ theorem states that if a triangle has sides of lengths a, b and c, and a² + b² = c², then the triangle is right-angled with the right angle opposite the side of length c. This provides a neat test for right angles without measuring any angle directly. For instance, given a triangle with sides 8 cm, 15 cm and 17 cm, calculate 8² + 15² = 64 + 225 = 289, which equals 17², so the triangle must be right-angled. You can then confidently use that fact to find heights, areas or other properties.

勾股定理的逆定理表明：如果一个三角形的三边长a、b和c满足a² + b² = c²，那么这个三角形是直角三角形，直角位于边长c的对边。这提供了一种无需直接测量角度即可检验直角的方法。例如，给定一个三边长分别为8厘米、15厘米和17厘米的三角形，计算8² + 15² = 64 + 225 = 289，等于17²，因此该三角形一定是直角三角形。然后你可以自信地利用这一事实去求高、面积或其他性质。

Watch out: some triangles may almost satisfy the equation but not exactly, which would indicate they are not perfectly right-angled. At KS3, you will mainly see integer or simple decimal side lengths where the equality holds precisely. The converse is particularly useful in coordinate geometry when you need to check if two line segments are perpendicular by forming triangles from their endpoints.

注意：有些三角形可能几乎满足等式，但并不完全相等，这表明它们并非精确的直角三角形。在KS3阶段，你主要会遇到整数或简单的边长为小数的三角形，等式精确成立。当你在坐标几何中需要检查两条线段是否垂直时，可以从它们的端点构造三角形来应用逆定理，这尤为有用。

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10. Pythagoras in 3D | 三维空间中的勾股定理

Pythagoras’ theorem extends naturally into three dimensions. Consider a rectangular box (cuboid) with length l, width w and height h. The longest diagonal that stretches from one bottom corner to the opposite top corner is called the space diagonal. To find its length, you apply Pythagoras twice. First, find the base diagonal d using the floor triangle: d² = l² + w². Then treat this base diagonal and the height as the two legs of a new right-angled triangle; the space diagonal s satisfies s² = d² + h². Combining gives s = √(l² + w² + h²).

勾股定理可以自然地延伸到三维。考虑一个长l、宽w、高h的长方体。从一个底角延伸到对顶角的最长对角线称为空间对角线。要求其长度，你需要两次使用勾股定理。首先，用地面的三角形求底面对角线d：d² = l² + w²。然后将这条底面对角线和高看作一个新的直角三角形的两条直角边；空间对角线s满足s² = d² + h²。合并得到s = √(l² + w² + h²)。

This is sometimes called the 3D Pythagorean theorem. It is very useful for finding the shortest distance an insect would crawl from one corner of a room to the opposite corner, or for determining the length of a rod that can be placed inside a box. At KS3, you might be asked to find the diagonal of a cube of side length a: s = √(a² + a² + a²) = √(3a²) = a√3.

这有时被称为三维勾股定理。它在求昆虫从房间一角爬到对顶角的最短距离，或者求能放入盒子内的直杆长度时非常有用。在KS3阶段，你可能会遇到求边长为a的立方体的对角线长：s = √(a² + a² + a²) = √(3a²) = a√3。

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11. Common Mistakes | 常见错误

Even though Pythagoras’ theorem is straightforward, students often make certain mistakes. The most frequent error is misidentifying the hypotenuse, especially when the diagram is not labelled clearly. Always look for the right-angle symbol first. Another common slip is adding when you should subtract (or vice versa) when finding a shorter side. Remember: for the hypotenuse, c = √(a² + b²); for a leg, leg = √(hypotenuse² − known leg²).

尽管勾股定理很直接，学生们还是会犯一些典型错误。最常见的错误是误判斜边，特别是当图形标注不清时。始终首先寻找直角符号。另一个常见的疏漏是在求直角边时把该加的当成减（反之亦然）。记住：求斜边时，c = √(a² + b²)；求直角边时，直角边 = √(斜边² − 已知直角边²)。

Students also sometimes forget to take the square root after squaring and adding or subtracting, leaving the answer as the square of the side. For instance, they might write “c = 25” instead of c = 5 after solving c² = 25. Finally, when simplifying surds, a common mistake is to incorrectly break down the number under the root, so practice simplifying √48 to 4√3 rather than leaving it as a messy decimal.

学生们有时也会在平方、相加或相减之后忘记取平方根，而把边长的平方作为答案。例如，在解出c² = 25之后，他们可能写“c = 25”而不是c = 5。最后，在化简根式时，一个常见错误是错误地分解根号内的数，因此要多练习把√48化简为4√3，而不是把它留成一个混乱的小数。

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12. Summary and Key Takeaways | 总结与要点

Pythagoras’ theorem is a vital relationship for any right-angled triangle: c² = a² + b², where c is the hypotenuse. To apply it correctly, first confirm the triangle is indeed right-angled and mark the hypotenuse. Then substitute the known values, solve for the unknown by either adding to find the hypotenuse or subtracting to find a leg, and finally take the square root. Keep an eye out for Pythagorean triples as shortcuts, and remember that the converse can test for a right angle. The theorem extends beautifully into three dimensions and is a practical tool in endless real-world situations.

勾股定理是任何直角三角形都必须掌握的重要关系：c² = a² + b²，其中c是斜边。要正确应用它，首先要确认三角形确实是直角三角形，并标出斜边。然后代入已知值，通过加法求斜边或减法求直角边来解出未知数，最后取平方根。留意勾股数作为捷径，并记住逆定理可用于检验直角。这一定理还能优雅地延伸至三维空间，并在无数现实情境中充当实用工具。

Regular practice with both numerical and word problems will build your confidence. Always draw a clear diagram, label the sides, and check that your answer makes sense (the hypotenuse must be the longest side). Master this theorem now, and you will have a strong foundation for trigonometry, vectors, and calculus later on.

通过数值题和应用题的经常练习，你将建立自信。始终画一个清晰的图示，标出各边，并检查答案是否合理（斜边必须是最长边）。现在掌握这一定理，你将会为以后的三角学、向量和微积分打下坚实的基础。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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