Pythagoras’ Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

In KS3 mathematics, Pythagoras’ theorem is one of the most powerful tools you will learn about right-angled triangles. It connects the three sides of a right-angled triangle in a precise algebraic relationship, allowing us to find missing lengths and solve a wide range of geometric problems. This article will guide you through the theorem, its proof, applications, common pitfalls and plenty of worked examples, all presented in clear, exam-focused language.

在 KS3 数学中，勾股定理是学习直角三角形时最有力的工具之一。它用一个精确的代数关系将直角三角形的三条边联系起来，使我们能够求出未知边长并解决各种几何问题。本文将以清晰的、紧扣考点的语言，带你逐步理解勾股定理、证明、应用、常见错误和大量例题。

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1. Right-Angled Triangles and the Hypotenuse | 直角三角形与斜边

A right-angled triangle is a triangle in which one angle is exactly 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 often labelled as a and b or as the ‘legs’ of the triangle. Being able to identify the hypotenuse correctly is the first essential step when using Pythagoras’ theorem.

直角三角形是指有一个角恰好是 90° 的三角形。直角所对的边总是最长的边，称为斜边。另外两条构成直角的边通常标记为 a 和 b，或称为直角边。正确识别斜边是运用勾股定理的第一步，也是至关重要的一步。

When you look at a diagram, the hypotenuse is the side that does not touch the right angle. In many diagrams, a small square is drawn in the right angle to help you see it quickly. Remember: the hypotenuse is always opposite the right angle.

当你观察图形时，斜边是不与直角接触的那条边。很多图形中会在直角处画一个小方格，帮助你快速识别。记住：斜边总是直角的对边。

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2. Statement of Pythagoras’ Theorem | 勾股定理的陈述

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. Using symbols:

c² = a² + b²

where c represents the hypotenuse and a, b represent the other two sides.

勾股定理指出：在任意直角三角形中，斜边长度的平方等于另外两条边长度的平方之和。用符号表示为：

c² = a² + b²

其中 c 表示斜边，a 和 b 表示另外两条边。

This relationship only works for right-angled triangles. It is a reversible statement – if the three sides of a triangle satisfy a² + b² = c², then the triangle must be right-angled (with c as the hypotenuse). This is called the converse of the theorem.

这种关系只适用于直角三角形。它也是一个可逆的命题——如果一个三角形的三条边满足 a² + b² = c²，那么这个三角形一定是直角三角形（c 为斜边）。这就是勾股定理的逆定理。

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3. Finding the Hypotenuse | 求斜边

To find the length of the hypotenuse when you know the lengths of the two shorter sides, simply substitute the known values into c² = a² + b² and then take the square root. For example, if a = 6 cm and b = 8 cm, then:

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

当已知两条直角边的长度时，求斜边只需将已知值代入 c² = a² + b²，再开平方即可。例如，若 a = 6 cm，b = 8 cm，则：

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

Always check that your answer is sensible – the hypotenuse must be longer than either of the other two sides. Putting the answer back into the equation to verify is a good habit: 10² = 100, and 6² + 8² = 100 indeed matches.

一定要检查答案是否合理——斜边必须比另外两条边都要长。养成将答案代回原式验证的好习惯：10² = 100，而 6² + 8² = 100，确实匹配。

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4. Finding a Shorter Side | 求一条直角边

When you know the hypotenuse and one leg, you can find the missing shorter side by rearranging the formula. If c is the hypotenuse and a is one leg, the missing side b satisfies:

b² = c² − a²

To find b, take the square root: b = √(c² − a²).

当已知斜边和一条直角边时，可以通过移项求出缺失的直角边。若 c 为斜边，a 为一条直角边，则缺失边 b 满足：

b² = c² − a²

再开平方得：b = √(c² − a²)。

For instance, if a right-angled triangle has hypotenuse 13 cm and one leg 5 cm, the other leg is:

b² = 13² − 5² = 169 − 25 = 144, so b = √144 = 12 cm

例如，一个直角三角形的斜边为 13 cm，一条直角边为 5 cm，则另一条直角边为：

b² = 13² − 5² = 169 − 25 = 144，因此 b = √144 = 12 cm

Many mistakes happen when students forget to square the numbers or subtract in the wrong order. Always write down c² − a² (hypotenuse squared minus known leg squared) to avoid getting a negative number under the square root.

很多错误是因为学生忘记平方或减的顺序弄反了。一定要写成 c² − a²（斜边的平方减去已知直角边的平方），以确保根号内的数不会是负数。

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5. The Converse of Pythagoras’ Theorem | 勾股定理的逆定理

The converse of Pythagoras’ theorem can be used to prove whether a triangle is right-angled. If the three given side lengths satisfy a² + b² = c² (where c is the longest side), then the triangle is right-angled, and the right angle lies opposite side c.

勾股定理的逆定理可用于证明一个三角形是否为直角三角形。如果给定的三条边长满足 a² + b² = c²（c 为最长边），那么这个三角形就是直角三角形，且直角在 c 的对边。

Example: A triangle has sides of length 9 cm, 12 cm and 15 cm. Check: 9² + 12² = 81 + 144 = 225, and 15² = 225. Since both sides of the equation match, the triangle is right-angled with the right angle opposite the 15 cm side.

例如：一个三角形边长分别为 9 cm、12 cm 和 15 cm。验证：9² + 12² = 81 + 144 = 225，而 15² = 225。等式两边相等，因此该三角形是直角三角形，直角在 15 cm 边所对的角处。

This test is extremely useful when working with coordinates or when checking whether a given shape contains a right angle, such as a ladder leaning against a wall or a bookshelf standing perpendicular to the floor.

这一检验在坐标几何中或判断图形是否包含直角时非常有用，例如梯子靠墙、书架垂直于地面等情况。

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6. Pythagorean Triples | 勾股数

A Pythagorean triple consists of three positive integers a, b and c that satisfy a² + b² = c². The most well-known triple is (3, 4, 5). Other common triples include (5, 12, 13), (6, 8, 10), (8, 15, 17) and (7, 24, 25). Any scalar multiple of a triple, such as 2×(3,4,5) = (6,8,10), is also a valid triple.

勾股数是指满足 a² + b² = c² 的三个正整数 a、b、c。最著名的勾股数是 (3,4,5)。其他常见的还有 (5,12,13)、(6,8,10)、(8,15,17) 和 (7,24,25)。勾股数的任意整数倍，例如 2×(3,4,5) = (6,8,10)，也是一个有效的勾股数。

a	b	c
3	4	5
5	12	13
6	8	10
8	15	17
7	24	25

Recognising these triples can save you a lot of time in exams because you can jump straight to the answer without calculating square roots, as long as you have verified the triple matches the given sides.

熟悉这些勾股数能让你在考试中节省大量时间，因为一旦验证题目给出的边长符合某个勾股数，就可以直接写出答案而无需进行平方根计算。

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7. Applying Pythagoras’ Theorem to Real-Life Problems | 勾股定理的实际应用

Real-life situations often involve hidden right-angled triangles. A ladder leaning against a vertical wall, a diagonal path across a rectangular field, or the distance between two points on a map can all be modelled with right-angled triangles. In such cases, drawing a clear diagram and labelling the known sides is the key first step.

实际问题中经常隐藏着直角三角形。梯子靠在竖直墙壁上、穿过矩形场地的对角线路径、地图上两点间的距离等，都可以用直角三角形来建模。遇到这类问题，第一步要画一个清晰的示意图并标出已知边。

Example: A ladder of length 5 m leans against a wall. The foot of the ladder is 2 m away from the wall. How high up the wall does the ladder reach? Let the height be h. Then h² + 2² = 5², so h² = 25 − 4 = 21, giving h = √21 ≈ 4.58 m (to 3 significant figures). The answer must be less than the ladder length, which makes sense.

例如：一把 5 m 长的梯子靠在墙上，梯脚离墙 2 m。问梯子顶端距地面多高？设高度为 h，则有 h² + 2² = 5²，因此 h² = 25 − 4 = 21，h = √21 ≈ 4.58 m（保留三位有效数字）。答案小于梯子长度，这很合理。

Always check that your interpretation of the problem creates a right-angled triangle. If a wall is vertical and the ground horizontal, the angle between them is 90°, so the ladder, wall and ground form a right-angled triangle.

务必验证你对问题的理解确实形成了一个直角三角形。如果墙是竖直的且地面是水平的，墙与地面之间的夹角就是 90°，因此梯子、墙和地面构成直角三角形。

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8. Using Pythagoras’ Theorem in Coordinate Geometry | 坐标几何中的勾股定理

The distance between two points (x₁, y₁) and (x₂, y₂) on a coordinate grid can be found using Pythagoras’ theorem. The horizontal difference is Δx = x₂ − x₁ and the vertical difference is Δy = y₂ − y₁. These form the legs of a right-angled triangle, and the straight-line distance d is the hypotenuse:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

在坐标网格上，两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离可以用勾股定理求得。水平方向差值为 Δx = x₂ − x₁，垂直方向差值为 Δy = y₂ − y₁。它们构成直角三角形的两条直角边，直线距离 d 就是斜边：

d = √((x₂ − x₁)² + (y₂ − y₁)²)

For example, the distance between (1, 2) and (4, 6) is √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5 units. This formula, often called the distance formula, is a direct application of Pythagoras’ theorem and is used extensively in KS3 and beyond.

例如，(1,2) 和 (4,6) 之间的距离为 √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5 个单位。这个公式通常称为距离公式，是勾股定理的直接应用，在 KS3 及更高年级中广泛使用。

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9. Common Mistakes to Avoid | 常见错误

One of the most frequent errors is confusing when to add and when to subtract. Remember: to find the hypotenuse, add squares; to find a shorter side, subtract the square of the known leg from the square of the hypotenuse. Always ask yourself: ‘Am I looking for the longest side or one of the shorter sides?’

最常见的错误之一是混淆何时相加、何时相减。记住：求斜边时，把平方相加；求直角边时，用斜边的平方减去已知直角边的平方。每次都问自己：“我是在求最长边还是一条直角边？”

Another typical mistake is forgetting to take the square root at the end. Writing c² = a² + b² without evaluating c = √(a² + b²) will lose marks. Also, pupils sometimes square a side length incorrectly – pay close attention to units and to squaring fractions or decimals.

另一个典型错误是忘记最后开平方。只写了 c² = a² + b² 而没有计算出 c = √(a² + b²) 会失分。此外，有些学生平方计算时出错——要特别注意单位以及分数或小数的平方。

Using Pythagoras’ theorem on non-right-angled triangles is a serious mistake. Always first confirm that the triangle contains a 90° angle. When a diagram is not provided, sketch your own and mark the right angle clearly.

在非直角三角形上使用勾股定理是严重错误。一定要先确认三角形含有 90° 角。如果题目没有配图，自己画图并清楚地标出直角。

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10. Proof by Rearrangement | 重组证明

One classic proof of the theorem uses area. Imagine drawing a square with side length (a + b) and then placing four identical right-angled triangles with legs a and b inside it. The central region forms a square of side c. The total area of the large square can be expressed as (a + b)² or as the sum of the areas of the four triangles and the inner square: 4 × (½ab) + c². Equating these gives:

(a + b)² = 2ab + c²

Expanding the left side: a² + 2ab + b² = 2ab + c². Cancelling 2ab from both sides leaves a² + b² = c². This visual proof is often taught in KS3 to help students see why the theorem works, not just remember the formula.

一种经典证明利用面积。想象一个边长为 (a + b) 的正方形，内部放入四个直角边分别为 a 和 b 的全等直角三角形，中间会留出一个边长为 c 的正方形。大正方形的总面积可以表示为 (a + b)²，也可以表示为四个三角形的面积加上内部正方形面积：4 × (½ab) + c²。令两者相等得：

(a + b)² = 2ab + c²

左边展开：a² + 2ab + b² = 2ab + c²。两边约去 2ab，得到 a² + b² = c²。这个直观证明在 KS3 教学中经常使用，帮助学生理解定理为什么成立，而不仅仅是记住公式。

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11. Practice Questions and Examples | 练习题与示例

Try these examples to consolidate your understanding:

• Find the hypotenuse of a right-angled triangle with legs 9 cm and 12 cm. Answer: √(9² + 12²) = √225 = 15 cm.
• Find the missing shorter side if the hypotenuse is 25 cm and one leg is 24 cm. Answer: √(25² − 24²) = √(625 − 576) = √49 = 7 cm.
• A rectangle has length 8 cm and width 6 cm. Calculate the length of its diagonal. Answer: diagonal = √(8² + 6²) = 10 cm.
• Is a triangle with sides 7 cm, 10 cm and 12 cm right-angled? Check: 7² + 10² = 49 + 100 = 149, while 12² = 144. Not equal, so not right-angled.
• A 13 m rope is stretched from the top of a vertical pole to a point on the ground 5 m from the base of the pole. Find the height of the pole. Answer: h = √(13² − 5²) = √144 = 12 m.

通过这些例子巩固你的理解：

• 求直角边分别为 9 cm 和 12 cm 的直角三角形的斜边。答案：√(9² + 12²) = √225 = 15 cm。
• 若斜边为 25 cm，一条直角边为 24 cm，求缺失的直角边。答案：√(25² − 24²) = √(625 − 576) = √49 = 7 cm。
• 一个矩形长 8 cm、宽 6 cm，计算其对角线长度。答案：对角线 = √(8² + 6²) = 10 cm。
• 边长分别为 7 cm、10 cm 和 12 cm 的三角形是直角三角形吗？检验：7² + 10² = 49 + 100 = 149，而 12² = 144。不相等，因此不是直角三角形。
• 一根 13 m 的绳子从竖直杆顶端拉到地面上距杆脚 5 m 处。求杆的高度。答案：h = √(13² − 5²) = √144 = 12 m。

When solving these, always write down the formula first, substitute carefully and then simplify. Show your working step by step to earn full marks in exams.

解答这些题目时，务必先写出公式，仔细代入再化简，一步一步展示解题过程，以在考试中拿到满分。

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12. Summary | 总结

Pythagoras’ theorem is an essential relationship for right-angled triangles: the square on the hypotenuse equals the sum of the squares on the other two sides (a² + b² = c²). It can be used to find a missing hypotenuse (add) or a missing leg (subtract). The converse allows us to test whether a triangle is right-angled. Pythagorean triples and the distance formula are powerful shortcuts and extensions. Mastering the correct setup, squaring, subtraction order and final square root is the key to success.

勾股定理是直角三角形的一个重要关系：斜边的平方等于两条直角边的平方和 (a² + b² = c²)。它可以用来求缺失的斜边（相加）或缺失的直角边（相减）。逆定理可以用来检验三角形是否为直角三角形。勾股数和距离公式是强大的快捷方法与扩展。掌握正确的列式、平方、减法顺序和最后的开平方，是成功的关键。

Always draw a diagram, label sides clearly, and check your answer makes sense. With practice, Pythagoras’ theorem becomes a reliable friend in your geometry toolkit.

务必画出示意图，清楚标记各边，并检查答案是否合理。通过练习，勾股定理将成为你几何工具箱中值得信赖的好帮手。

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