Pythagoras’ Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most important results in geometry, linking the three sides of a right-angled triangle. In Key Stage 3 Mathematics, you are expected to know the theorem, use it to find missing side lengths, and apply it to solve problems in two and three dimensions. This article will guide you through the key ideas, worked examples, and common pitfalls, helping you build confidence for your Cambridge curriculum studies.

勾股定理是几何学中最重要的结论之一，它将直角三角形的三条边联系在一起。在初中数学阶段，要求你掌握定理内容，用它来求未知边长，并运用它解决二维和三维空间中的问题。这篇文章将带你梳理核心概念、讲解例题、指出常见错误，帮助你在剑桥课程的学习中建立信心。

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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 hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The hypotenuse is always the longest side and is opposite the right angle.

勾股定理表明，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。斜边总是最长的一条边，并且正对着直角。

This relationship only works for right-angled triangles. If the triangle does not contain a right angle, the theorem cannot be applied directly.

这一关系只适用于直角三角形。如果三角形不含直角，就无法直接应用该定理。

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2. The Formula | 公式表达

The theorem is usually written using the letters a, b and c, where c represents the hypotenuse and a and b are the two shorter sides.

定理通常用字母 a、b 和 c 表示，其中 c 代表斜边，a 和 b 是两条较短的直角边。

a² + b² = c²

If you label a triangle differently, you can still use the formula as long as you correctly identify the hypotenuse.

即使你对三角形的标记方式不同，只要你能正确找出斜边，这个公式依然可以使用。

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3. Finding the Hypotenuse | 求斜边

When you know the lengths of both shorter sides, you can find the hypotenuse by squaring both lengths, adding them, and then taking the square root.

当你知道两条直角边的长度时，可以将它们分别平方再相加，然后开平方来求出斜边。

Example: A right-angled triangle has shorter sides of 5 cm and 12 cm. Then c² = 5² + 12² = 25 + 144 = 169, so c = √169 = 13 cm.

示例：一个直角三角形两条直角边长分别为 5 cm 和 12 cm。那么 c² = 5² + 12² = 25 + 144 = 169，所以 c = √169 = 13 cm。

Always remember to take the square root at the end; many learners forget this step and give the squared value as their answer.

请牢记最后要开平方；许多学生常常忘记这一步，直接把平方值当作答案。

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4. Finding a Shorter Side | 求直角边

To find one of the shorter sides, rearrange the formula. If you know c and one of the shorter sides, say a, then b² = c² – a².

要求其中一条直角边，可以把公式变形。如果已知 c 和一条直角边（比如 a），那么 b² = c² – a²。

b = √(c² – a²)

Example: The hypotenuse is 10 m and one shorter side is 6 m. Find the other side. b² = 10² – 6² = 100 – 36 = 64, so b = √64 = 8 m.

示例：斜边为 10 m，一条直角边为 6 m，求另一条直角边。b² = 10² – 6² = 100 – 36 = 64，因此 b = √64 = 8 m。

Make sure you subtract the squares, not add them, when working backwards to find a shorter side.

在逆向求直角边时，一定要用减法而不是加法来计算平方。

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5. Pythagorean Triples | 勾股数

Some sets of three positive whole numbers satisfy the equation a² + b² = c². These are called Pythagorean triples. The most common examples are (3, 4, 5), (5, 12, 13) and (7, 24, 25).

有些三个一组的正整数满足 a² + b² = c²，它们被称为勾股数。最常见的例子有 (3, 4, 5)、(5, 12, 13) 和 (7, 24, 25)。

If you recognize these triples, you can often solve problems very quickly without using the full calculation. Multiples of these triples, like (6, 8, 10) or (9, 12, 15), also work.

如果你能认出这些勾股数，经常可以无需完整计算就能快速解题。它们的倍数，如 (6, 8, 10) 或 (9, 12, 15)，也同样成立。

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6. Proof of the Theorem (Visual) | 定理的证明（直观）

One famous visual proof shows a square of side (a + b) containing four identical right-angled triangles and a smaller square of side c. By equating areas, you can show that a² + b² = c².

一个著名的直观证明是：画一个边长为 (a + b) 的大正方形，里面包含四个全等的直角三角形和一个边长为 c 的小正方形。通过比较面积，可以推导出 a² + b² = c²。

This proof helps you understand why the theorem works, not just how to use it. It is worth exploring the geometry behind the formula.

这个证明能帮助你理解定理为什么成立，而不仅仅是怎样使用它。深入了解公式背后的几何意义非常值得。

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7. Real-life Applications | 实际应用

Pythagoras’ theorem is not just for textbooks. It is used in navigation, construction, design and even in computer graphics. For example, to find the shortest distance across a rectangular park, you can treat it as the hypotenuse of a right-angled triangle.

勾股定理不仅仅用于课本，它被应用在导航、建筑、设计甚至计算机图形学中。例如，要计算穿过一个长方形公园的最短路径，你可以把它看作是直角三角形的斜边。

When a ladder is placed against a wall, the ground, the wall and the ladder form a right-angled triangle. You can use Pythagoras’ theorem to calculate how high the ladder reaches or how far the foot of the ladder should be from the wall.

当一把梯子靠在墙上时，地面、墙壁和梯子构成一个直角三角形。你可以用勾股定理计算梯子达到的高度，或者梯脚与墙的距离。

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8. Using Pythagoras in 3D | 在三维空间中使用勾股定理

In 3D problems, you often need to find the length of a space diagonal inside a cuboid (rectangular prism). This is done by applying Pythagoras’ theorem twice.

在三维问题中，你经常需要求长方体（长方柱）内部的空间对角线长度。这需要两次应用勾股定理。

First, find the diagonal of the base rectangle using a² + b² = d². Then use this diagonal as one side and the height as the other to find the space diagonal e: e² = d² + height².

先通过 a² + b² = d² 求出底面对角线 d。然后将此对角线作为一条边，高作为另一条边，求空间对角线 e：e² = d² + 高²。

e = √(l² + w² + h²)

Here l, w and h are the length, width and height of the cuboid. This formula saves time once you understand the two-step process.

这里的 l、w 和 h 分别表示长方体的长、宽和高。一旦你理解了这个两步过程，这个公式就能帮你节省时间。

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9. Converse of Pythagoras | 勾股定理的逆定理

The converse of Pythagoras’ theorem states that if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled.

勾股定理的逆定理指出：如果三角形最长边的平方等于另外两边的平方和，那么这个三角形是直角三角形。

This is very useful for checking whether a triangle with given side lengths is right-angled. For example, to test sides 8 cm, 15 cm and 17 cm, check 8² + 15² = 64 + 225 = 289, and 17² = 289, so it is a right-angled triangle.

这对于检验给定边长的三角形是否为直角三角形非常有用。例如，检验边长 8 cm、15 cm 和 17 cm：8² + 15² = 64 + 225 = 289，而 17² = 289，因此它是直角三角形。

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10. Problem Solving Strategies | 解题策略

Before using Pythagoras’ theorem, always sketch a clear diagram and label the sides. Identify which side is unknown and whether you are finding the hypotenuse or a shorter side.

在使用勾股定理之前，一定要画出清晰的示意图并标记边长。确定哪一条边是未知的，以及你是在求斜边还是一条直角边。

Write down the formula in the correct form, substitute the values carefully, and show all your working. Remember to include units in your final answer and check if your answer makes sense (the hypotenuse must be longer than each of the other sides).

写出正确的公式形式，仔细代入数值，并展示出所有计算过程。记得在最终答案里加上单位，并检验答案是否合理（斜边必须比任何一条直角边长）。

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11. Common Mistakes | 常见错误

One of the most frequent mistakes is adding the squares when trying to find a shorter side. Learners often write b² = c² + a² instead of b² = c² – a².

最常见的错误之一是在求直角边时，把平方相加。学生常常写成 b² = c² + a²，而正确形式是 b² = c² – a²。

Another error is forgetting to take the square root at the end, or rounding too early during calculations. Always keep the exact square root until the final step unless the question asks for an approximation.

另一个错误是忘记在最后开平方，或者在计算过程中过早取整。除非题目要求近似值，否则应一直保留精确的平方根到最后一步。

Also, learners may apply the theorem to triangles that are not right-angled. Always check for the right-angle symbol first.

此外，学生也可能把定理用在非直角三角形上。一定要先确认图中是否有直角符号。

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12. Practice Question Walkthrough | 练习讲解

Question: A ship sails 24 km due south and then 7 km due east. How far is it from its starting point, to the nearest km?

问题：一艘船向正南行驶 24 km，然后转向正东行驶 7 km。它离出发点有多远，精确到 km？

The paths south and east form the two shorter sides of a right-angled triangle, and the direct distance back to start is the hypotenuse. So c² = 24² + 7² = 576 + 49 = 625. Then c = √625 = 25 km.

南向和东向的路径构成直角三角形的两条直角边，回到起点的直线距离就是斜边。所以 c² = 24² + 7² = 576 + 49 = 625，然后 c = √625 = 25 km。

The answer is 25 km. This example shows how to model a real-world situation with a right-angled triangle and apply the theorem correctly.

答案是 25 km。这个例题展示了如何用直角三角形来建立实际情境的模型，并正确应用勾股定理。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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