📚 Pythagoras' Theorem | 勾股定理
Pythagoras' theorem is one of the most famous results in mathematics. It describes the special relationship between the three sides of a right-angled triangle. Once you understand this rule, you can work out the length of any missing side in such a triangle, as long as you know the other two sides. This concept is not just a classroom exercise – it is used in construction, navigation, interior design and many areas of everyday life. By the end of this guide, you will be able to state the theorem, apply it to find missing sides, check whether a triangle is right-angled, and solve real-world problems with confidence.
勾股定理是数学中最著名的结论之一。它描述了直角三角形三条边之间的特殊关系。一旦你理解了这个规则，只要知道另外两条边的长度，就可以求出直角三角形中任意一条未知边的长度。这个概念绝不仅仅是课堂练习——它在建筑、导航、室内设计和日常生活的许多领域都有应用。在学完本指南后，你将能够陈述这一定理、用它求未知边、判断一个三角形是否为直角三角形，并自信地解决实际问题。
1. What is Pythagoras' theorem? | 什么是勾股定理？
, the theorem is written as a² + b² = c². This rule only works for right-angled triangles.
在任何直角三角形中，最长边的平方等于另外两条边的平方之和。最长的那条边称为斜边，它总是直角的对边。如果我们把三条边分别标记为 a、b 和 c，其中 c 是斜边，那么定理可以写作 a² + b² = c²。这个规则仅适用于直角三角形。
Pythagoras' theorem: a² + b² = c²
勾股定理：a² + b² = c²
2. Identifying the hypotenuse | 识别斜边
The hypotenuse is the side that faces the right angle. It is always the longest side in the triangle, no matter how the triangle is rotated. When you start a problem, find the right angle first, then locate the side directly across from it – that is your hypotenuse. The other two sides are called the legs, and they form the right angle.
斜边是正对着直角的边。无论三角形如何旋转，它始终是最长的边。开始解题时，先找出直角，然后找到正对着直角的边——那就是斜边。另外两条边称为直角边，它们组成了直角。
- Look for the square symbol marking 90°.
- 寻找标记 90° 的方块符号。
- The hypotenuse is opposite this angle.
- 斜边就在这个角的对面。
3. Finding the hypotenuse | 求斜边的长度
he lengths of the two shorter sides, you can find the hypotenuse by using the equation c = √a² + b². First, square both known lengths, add them together, then take the square root of the total. Remember to include units in your final answer.
当你知道两条较短边的长度时，可以用公式 c = √a² + b² 求出斜边。先把两条已知边的长度分别平方，加起来，再对总和开平方根。请记得在最终答案中加上单位。
Example: If a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.
例子：如果 a = 3 cm，b = 4 cm，那么 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5 cm。
4. Finding a shorter side | 求一条直角边的长度
To find a missing shorter side, rearrange the formula. If you need side a, use a = √c² - b². Square the hypotenuse, subtract the square of the known leg, and then find the square root of the result. This works exactly the same way for side b.
要求一条未知的直角边，需要重新整理公式。如果你要求边 a，可以用 a = √c² - b²。把斜边平方，减去已知直角边的平方，然后再对结果开平方根。对边 b 来说，方法完全一样。
Example: c = 10 m, b = 6 m → a² = 10² - 6² = 100 - 36 = 64, a = √64 = 8 m.
例子：c = 10 m，b = 6 m → a² = 10² - 6² = 100 - 36 = 64，a = √64 = 8 m。
5. Checking for a right angle | 检查直角
The converse of Pythagoras' e longest side, then the triangle must contain a right angle opposite side c. This is a powerful tool for verifying right angles without using a protractor.
勾股定理的逆定理可以帮助你判断一个三角形是否为直角三角形。如果三条边满足 a² + b² = c²，且 c 为最长边，那么这个三角形必定在边 c 的对角处包含一个直角。这是不用量角器就能验证直角的强大工具。
es 5 cm, 12 cm, 13 cm → 5² + 12² = 25 + 144 = 169, and 13² = 169, so a right angle exists.
例子：三边为 5 cm、12 cm、13 cm → 5² + 12² = 25 + 144 = 169，而 13² = 169，因此存在直角。
6. Pythagorean triples | 勾股数
Some sets of whole numbers fit Pythagoras' theorem perfectly. These are called Pythagorean triples. Common examples are (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). Multiples of these triples also work, such as (6, 8, 10) which is double the first triple. Recognising triples can save time in problems.
有些整数组完美符合勾股定理，它们被称为勾股数。常见的例子有 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。这些勾股数的倍数也成立，比如 (6, 8, 10) 就是第一组的两倍。识别勾股数可以在解题时节省时间。
| Triple (3, 4, 5) | 3² + 4² = 5² |
| 勾股数 (3, 4, 5) | 3² + 4² = 5² |
| Triple (5, 12, 13) | 5² + 12² = 13² |
| 勾股数 (5, 12, 13) | 5² + 12² = 13² |
7. Applying Pythagoras in practical problems | 勾股定理在实际问题中的应用
Word problems often involve ladders leaning against walls, rectangles, or distances on a map. Start by sketching a right-angled triangle from the information given, label the sides you know, and decide which side is missing. Then apply the appropriate formula. Always check that your answer makes sense in the context.
应用题经常涉及靠墙的梯子、长方形或地图上的距离。一开始要根据所给信息画出一个直角三角形，标出已知边，并确定哪条边是未知的，然后套用相应的公式。永远要检查答案在上下文里是否合理。
nst a wall. The foot of the ladder is 2 m from the wall. How high up does the ladder reach? Answer: height = √5² - 2² = √25 - 4 = √21 ≈ 4.58 m.
例子：一架 5 米长的梯子靠在墙上，梯脚距墙 2 米。梯子能达到多高？答案：高度 = √5² - 2² = √25 - 4 = √21 ≈ 4.58 m。
8. Using Pythagoras in coordinate geometry | 在坐标几何中使用勾股定理
), the distance d = √[(x₂ - x₁)² + (y₂ - y₁)²]. This is a direct application of Pythagoras' theorem.
网格上两点之间的距离可以通过构建直角三角形来求得。水平距离和垂直距离就是直角边，而直线距离就是斜边。如果两点为 (x₁, y₁) 和 (x₂, y₂)，那么距离 d = √[(x₂ - x₁)² + (y₂ - y₁)²]。这就是勾股定理的直接应用。
Example: (1, 2) to (4, 6) → d = √[(4-1)² + (6-2)²] = √3² + 4² = √25 = 5 units.
例子：从 (1, 2) 到 (4, 6) → d = √[(4-1)² + (6-2)²] = √3² + 4² = √25 = 5 个单位。
9. Solving problems with diagonals of rectangles | 解长方形对角线问题
diagonal of a rectangle splits it into two right-angled triangles. You can use Pythagoras' l theorem to find the diagonal when you know the length and width. For a rectangle with sides l and w, the diagonal d = √l² + w². This idea is also used for television and computer screen sizes.
长方形的对角线将它分成两个直角三角形。知道长和宽时，你可以用勾股定理求出对角线的长度。对于边长为 l 和 w 的长方形，对角线 d = √l² + w²。这一思路同样用于电视和电脑屏幕的尺寸计算。
Example: A rectangle is 8 cm by 6 cm. Diagonal = √8² + 6² = √64 + 36 = √100 = 10 cm.
例子：一个长方形长 8 cm、宽 6 cm，对角线 = √8² + 6² = √64 + 36 = √100 = 10 cm。
10. Common mistakes and how to avoid them | 常见错误及如何避免
A frequent error is adding squares when you should subtract, or forgetting to take the square root at the end. Always identify the hypotenuse first: if you are finding it, add and then square root; if you are finding a shorter side, subtract and then square root. Another mistake is misidentifying the right angle or using the theorem on a non-right triangle.
一个常见错误是在应该减法时却做了加法，或者忘记最后开平方根。一定要先识别斜边：如果是求斜边，就先加再开平方根；如果是求直角边，就先减再开平方根。另一个错误是错误识别直角，或者在非直角三角形上使用该定理。
- Always draw a diagram.
- 永远要画图。
- Label the hypotenuse as c.
- 将斜边标记为 c。
- Write the formula a² + b² = c² before substituting.
- 在代入前先写出公式 a² + b² = c²。
11. Practice questions to test yourself | 自测练习题
Try these questions to strengthen your skills. 1) A triangle has sides 9 cm and 12 cm. Find the hypotenuse. 2) The hypotenuse is 15 cm and one leg is 9 cm. Find the other leg. 3) Determine if a triangle with sides 8 m, 10 m, 12 m is right-angled. 4) A ship sails 30 km east and then 40 km north. How far is it from the starting point? Check your answers using the methods above.
试试这些题目来强化技能。1）一个三角形的两条边分别为 9 cm 和 12 cm，求斜边。2）斜边为 15 cm，一条直角边为 9 cm，求另一条直角边。3）判断一个边长分别为 8 m、10 m、12 m 的三角形是否为直角三角形。4）一艘船向东行驶 30 km，然后向北行驶 40 km，它离出发点有多远？用上述方法核对你的答案。
12. Summary and key points | 总结与要点
Pythagoras' ypotenuse. Always identify the hypotenuse correctly before applying the formula. Remember the converse can be used to test for a right angle. Practise with both numerical problems and word problems. The theorem opens the door to trigonometry and more advanced geometry, so building a strong foundation now is essential.
勾股定理指出，在直角三角形中，a² + b² = c²，其中 c 是斜边。在运用公式之前，务必正确识别斜边。记住，逆定理可以用来检测直角。既要练习数字题，也要练习应用题。勾股定理为三角学和更高级的几何打开了大门，所以现在打下扎实的基础至关重要。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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