Pythagoras' Theorem | 毕达哥拉斯定理

📚 Pythagoras’ Theorem | 毕达哥拉斯定理

Welcome to this Cambridge KS3 Mathematics revision guide on Pythagoras’ Theorem. This topic appears in the geometry section of the Cambridge Lower Secondary curriculum and is essential for solving problems involving right-angled triangles. In this article, you will learn the theorem, how to apply the formula c² = a² + b², and how to avoid common mistakes. The explanations are given in both English and Chinese so that you can check your understanding at every step.

欢迎阅读本剑桥 KS3 数学复习指南，主题是毕达哥拉斯定理。该主题属于剑桥初中课程几何部分，是解决直角三角形问题的基础。在本文中，你将学习这一定理、如何运用公式 c² = a² + b²，以及如何避免常见错误。每个步骤都提供英文和中文对照解释，方便你随时检查自己的理解。

1. What is Pythagoras’ Theorem? | 什么是毕达哥拉斯定理？

Pythagoras’ Theorem describes a special relationship between the three sides of a right-angled triangle. It states that in any right-angled triangle, the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides. This rule was known in several ancient cultures, but it is named after the Greek mathematician Pythagoras because his school produced one of the earliest recorded proofs.

毕达哥拉斯定理描述了直角三角形三条边之间的特殊关系。它指出，在任意直角三角形中，最长边上的正方形面积等于另外两条边上的正方形面积之和。这一规律在多个古代文明中都曾被认识，但因其学派留下了最早的证明记录之一，所以以希腊数学家毕达哥拉斯的名字命名。

You will use this theorem when you know two sides of a right-angled triangle and need to find the third side. It is not a formula for all triangles, so you must first check that the triangle contains a right angle.

当你已知直角三角形的两条边并需要求第三条边时，就会用到这个定理。它不是适用于所有三角形的公式，因此你必须先确认三角形中有一个直角。

2. The Hypotenuse | 斜边

The longest side of a right-angled triangle is called the hypotenuse. The hypotenuse is always opposite the right angle and is labelled as c in the standard formula. The two shorter sides are usually labelled a and b, and it does not matter which one is which because addition is commutative.

直角三角形的最长边叫做斜边。斜边总是对着直角，在标准公式中标记为 c。两条较短的边通常标记为 a 和 b，哪条是 a 哪条是 b 并不重要，因为加法满足交换律。

A useful way to identify the hypotenuse is to locate the right angle first. The side that does not touch the right angle at either end is the hypotenuse. It is also the side opposite the largest angle, since the right angle is always the largest angle in a right-angled triangle.

找出斜边的一个有效方法是先找到直角。两端都不与直角接触的那条边就是斜边。它也是最大角所对的边，因为直角始终是直角三角形中最大的角。

3. The Formula c² = a² + b² | 公式 c² = a² + b²

The theorem can be written as:

c² = a² + b²

Here c is the length of the hypotenuse, while a and b are the lengths of the two shorter sides. This formula only works for right-angled triangles. When you square each side length, you are working with areas, which is why the theorem is often illustrated with squares drawn on each side.

该定理可以写成：

c² = a² + b²

其中 c 是斜边的长度，a 和 b 是两条较短边的长度。这个公式只适用于直角三角形。当你对每条边长取平方时，实际上是在处理面积，因此这个定理常用画在三条边上的正方形来说明。

Remember that c² means c × c, not c × 2. Many early mistakes come from confusing squaring with doubling. For example, if a = 3 and b = 4, you must calculate 3² + 4² = 9 + 16 = 25, not 3 + 4 = 7.

请记住，c² 表示 c × c，而不是 c × 2。许多初学时的错误都来自把平方和乘以 2 混淆。例如，如果 a = 3、b = 4，你必须计算 3² + 4² = 9 + 16 = 25，而不是 3 + 4 = 7。

4. Finding the Hypotenuse | 求斜边

If you know the two shorter sides, substitute them into the formula and solve for c. For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25. Taking the square root gives c = √25 = 5 cm.

如果已知两条较短边，把它们代入公式并求出 c。例如，如果 a = 3 cm、b = 4 cm，那么 c² = 3² + 4² = 9 + 16 = 25。取平方根得到 c = √25 = 5 cm。

Follow these steps every time: first square a, then square b, add the results, and finally take the square root of the total. Do not forget the final square root step, because leaving the answer as c² = 25 is not the same as finding c.

每次都按以下步骤进行：先求 a 的平方，再求 b 的平方，把结果相加，最后对总和开平方根。不要忘记最后开平方的步骤，因为把答案写成 c² = 25 并不等于求出 c。

Step 1: Write the formula c² = a² + b².
第 1 步：写出公式 c² = a² + b²。
Step 2: Substitute the known shorter sides.
第 2 步：代入已知的较短边。
Step 3: Square each number and add.
第 3 步：对每个数平方并相加。
Step 4: Take the square root to find c.
第 4 步：开平方根求出 c。

5. Finding a Shorter Side | 求直角边

If you know the hypotenuse and one shorter side, rearrange the formula before substituting. For example, if c = 13 cm and b = 5 cm, then a² = c² − b² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 cm.

如果已知斜边和一条直角边，先变形公式再代入。例如，如果 c = 13 cm、b = 5 cm，那么 a² = c² − b² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 cm。

The rearranged form is very important. Since c is always the longest side, the expression c² − b² will always be positive when b is a shorter side. If you get a negative number under the square root, it usually means you have labelled the hypotenuse incorrectly.

这种变形形式非常重要。由于 c 始终是最长边，当 b 为直角边时，表达式 c² − b² 总是正数。如果平方根下出现了负数，通常说明你把斜边标错了。

Always write the rearranged formula as a² = c² − b² first. This keeps your working clear and reduces the chance of accidentally writing a² = b² − c², which would give a negative result.

一定要先把变形后的公式写成 a² = c² − b²。这样能让解题过程更清晰，减少误写成 a² = b² − c² 而导致结果为负的可能性。

6. Pythagorean Triples | 毕达哥拉斯三元组

Some sets of three whole numbers satisfy the rule exactly, such as 3, 4, 5 and 5, 12, 13. These are called Pythagorean triples. Recognising them can speed up checking because multiples like 6, 8, 10 also work.

有些三个整数组成的集合恰好满足这一规则，例如 3、4、5 和 5、12、13。这些数被称为毕达哥拉斯三元组。识别它们可以加快检验速度，因为诸如 6、8、10 这样的倍数也成立。

You can test a triple by substituting the largest number as c. For example, for 8, 15, 17, check whether 8² + 15² = 17². Since 64 + 225 = 289 and 17² = 289, the triple is valid.

你可以把最大的数作为 c 来检验一个三元组。例如，对于 8、15、17，检查 8² + 15² 是否等于 17²。因为 64 + 225 = 289，而 17² = 289，所以这个三元组成立。

Pythagorean Triple	Check
3, 4, 5	3² + 4² = 9 + 16 = 25 = 5²
6, 8, 10	6² + 8² = 36 + 64 = 100 = 10²
5, 12, 13	5² + 12² = 25 + 144 = 169 = 13²
8, 15, 17	8² + 15² = 64 + 225 = 289 = 17²

7. Real-Life Applications | 实际应用

Pythagoras’ Theorem is used to find distances that cannot be measured directly, such as the diagonal of a rectangular field, the length of a ladder leaning against a wall, or the shortest distance between two points on a map. In construction and navigation, it helps workers check for right angles and calculate exact lengths.

毕达哥拉斯定理用于求无法直接测量的距离，例如矩形场地的对角线、靠在墙上的梯子长度，或地图上两点之间的最短距离。在建筑和导航中，它帮助工作人员检查直角并计算精确长度。

For example, if a rectangular room is 6 m long and 8 m wide, the diagonal across the floor can be found by treating the length and width as the two shorter sides. The diagonal is then √(6² + 8²) = √100 = 10 m.

例如，如果一个长方形房间长 6 m、宽 8 m，可以把长和宽看作两条较短边来求地面对角线。对角线就是 √(6² + 8²) = √100 = 10 m。

This method works because the diagonal of a rectangle divides it into two congruent right-angled triangles. In each triangle, the diagonal is the hypotenuse.

这个方法之所以有效，是因为长方形的对角线把它分成两个全等的直角三角形。在每个三角形中，对角线都是斜边。

8. Common Mistakes | 常见错误

Students often forget to take the square root when finding a side, or they add the hypotenuse to a shorter side instead of using squares. Another common error is applying the theorem to non-right-angled triangles. Always identify the hypotenuse first and check the right angle before calculating.

学生在求边时经常忘记开平方根，或者把斜边与直角边直接相加而不是使用平方。另一个常见错误是把定理用在非直角三角形上。计算前一定要先找出斜边并确认直角。

One way to avoid the square root error is to write the final answer in two stages: first show c² = 25, then show c = √25 = 5. This makes it clear that you have taken the square root.

避免忘记开平方的一个方法是把最终答案分成两步写：先写 c² = 25，再写 c = √25 = 5。这样就能清楚地表明你已经取了平方根。

Also, do not round too early. If you round a side length before squaring or before taking a square root, the final answer may be inaccurate. Keep exact values such as √13 until the very end, then round if the question asks for a decimal answer.

此外，不要过早取近似值。如果你在平方或开平方之前就把边长四舍五入，最终答案可能不准确。保留诸如 √13 这样的准确值到最后，如果题目要求小数答案，再在最后进行四舍五入。

9. Worked Example 1 | 例题 1

Problem: A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

题目：一个直角三角形的两条直角边分别为 6 cm 和 8 cm，求斜边。

Solution: Write the formula c² = a² + b². Substitute a = 6 and b = 8: c² = 6² + 8² = 36 + 64 = 100. Take the square root: c = √100 = 10 cm.

解答：写出公式 c² = a² + b²。代入 a = 6 和 b = 8：c² = 6² + 8² = 36 + 64 = 100。取平方根：c = √100 = 10 cm。

This triangle is a multiple of the 3, 4, 5 triple, which gives a quick check. Since 6 = 2 × 3, 8 = 2 × 4 and 10 = 2 × 5, the answer is consistent with a known Pythagorean triple.

这个三角形是 3、4、5 三元组的倍数，因此可以快速检验。因为 6 = 2 × 3，8 = 2 × 4，10 = 2 × 5，所以答案与已知的毕达哥拉斯三元组一致。

10. Worked Example 2 | 例题 2

Problem: A 17 m ladder reaches a window 15 m above the ground. How far is the base of the ladder from the wall?

题目：一架 17 m 的梯子靠在墙上，顶端离地面 15 m。梯子底部离墙多远？

Solution: Let the distance be x. The ladder is the hypotenuse, so x² + 15² = 17². This gives x² = 289 − 225 = 64, so x = √64 = 8 m.

解答：设距离为 x，梯子是斜边，所以 x² + 15² = 17²。由此得到 x² = 289 − 225 = 64，因此 x = √64 = 8 m。

Notice that we subtracted the square of the known side from the square of the hypotenuse. We did not write x = 17 − 15, because the relationship is between squares, not between the original lengths.

注意，我们是用斜边的平方减去已知直角边的平方。不要写成 x = 17 − 15，因为定理描述的是平方之间的关系，而不是原始长度之间的直接加减。

11. Practice Checklist | 练习清单

Use this checklist to test yourself before an assessment:

在评估前用下面的清单来检验自己：

Can I label the hypotenuse correctly?
我能正确标出斜边吗？
Can I write the formula c² = a² + b² from memory?
我能默写公式 c² = a² + b² 吗？
Can I solve for a shorter side by rearranging?
我能通过变形求直角边吗？
Can I recognise common Pythagorean triples?
我能识别常见的毕达哥拉斯三元组吗？
Can I apply the theorem to real-life problems involving diagonals and ladders?
我能把定理应用到涉及对角线和梯子的实际问题中吗？
Do I remember to take the square root at the end?
我是否记得最后要开平方根？

If you can answer yes to all of these, you are ready to move on to more challenging questions, including those where you must first draw or visualise the right-angled triangle.

如果你对以上所有问题都能回答“是”，那么你就准备好挑战更难的题目了，包括那些需要先画出或想象出直角三角形的题目。

12. Summary | 总结

Pythagoras’ Theorem connects the sides of a right-angled triangle through c² = a² + b². Practise with different values, remember to take square roots, and always check that the triangle has a right angle. Mastery of this topic will support later work in trigonometry and coordinate geometry.

毕达哥拉斯定理通过 c² = a² + b² 将直角三角形的三边联系起来。用不同的数值进行练习，记得开平方根，并始终检查三角形是否有直角。掌握这一主题将为之后的三角学和坐标几何学习打下基础。

When you encounter a problem, start by labelling the hypotenuse as c and the other sides as a and b. Then choose the correct form of the formula: use c² = a² + b² to find the hypotenuse, or a² = c² − b² to find a shorter side. Finally, take the square root and check that your answer is reasonable.

遇到问题时，先把斜边标为 c，其他边标为 a 和 b。然后选择正确的公式形式：求斜边用 c² = a² + b²，求直角边用 a² = c² − b²。最后开平方根并检查答案是否合理。

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Pythagoras’ Theorem | 毕达哥拉斯定理