📚 Mastering Pythagoras’ Theorem | 掌握勾股定理
Pythagoras’ theorem is one of the most famous results in mathematics. It provides a simple relationship between the side lengths of a right-angled triangle. Whether you are building a ramp, navigating using coordinates, or designing a roof, this theorem is an essential tool. In this article, we will explore its statement, how to use it to find missing sides, and its real-life applications, all aligned with the Cambridge Lower Secondary mathematics curriculum.
勾股定理是数学中最著名的定理之一。它给出了直角三角形三条边之间的一个简单关系。无论是搭建斜坡、用坐标导航，还是设计屋顶，这个定理都是一个必不可少的工具。在本文中，我们将讲解其表述、如何用它求未知边长以及它在现实生活中的应用，所有内容均贴合剑桥初中数学课程。
1. Introduction to Right-Angled Triangles | 直角三角形的介绍
A right-angled triangle is a triangle that has one angle exactly equal to 90°. The side opposite this right angle is the longest side, called the hypotenuse. The other two sides are referred to as the legs of the triangle. In everyday life, right angles appear in corners of books, screens, and sports fields.
直角三角形是有一个角恰好为90°的三角形。直角所对的边是最长边，称为斜边。另外两条边叫做直角边。在日常生活中，书本、屏幕和运动场的角落中都存在着直角。
2. 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. This relationship allows us to calculate an unknown side if the other two sides are known. The theorem is named after the ancient Greek mathematician Pythagoras.
勾股定理指出，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。如果我们知道了其中两条边的长度，这种关系就能帮我们计算出未知边。这个定理以古希腊数学家毕达哥拉斯的名字命名。
3. The Theorem Formula | 定理公式
If the legs have lengths a and b, and the hypotenuse has length c, then the theorem is written as:
如果直角边长为 a 和 b，斜边长为 c，那么定理可以写成：
a² + b² = c²
The letters a and b can be swapped since addition is commutative, but c must always represent the hypotenuse. To find the hypotenuse, you take the square root of both sides: c = √(a² + b²). To find a leg, you rearrange: a² = c² – b², then a = √(c² – b²).
由于加法满足交换律，a 和 b 可以互换，但 c 必须始终代表斜边。要求斜边，可以在两边开平方：c = √(a² + b²)。要求一条直角边，则需要移项：a² = c² – b²，然后 a = √(c² – b²)。
4. Identifying the Hypotenuse | 识别斜边
The hypotenuse is always opposite the right angle. It is the longest side in the triangle. When a triangle is drawn with the right-angle box in a corner, the side directly facing that box is the hypotenuse. Misidentifying the hypotenuse leads to incorrect calculations, so always locate the 90° angle first.
斜边始终位于直角的对边。它是三角形中最长的边。当三角形画有直角标记时，正对这个标记的边就是斜边。一旦认错斜边，就会导致计算错误，因此一定要先找到90°角。
5. Calculating the Hypotenuse | 计算斜边长度
Example: A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse, c.
例题：一个直角三角形的直角边长分别为6 cm和8 cm。求斜边长 c。
Step 1: Write the formula: a² + b² = c².
步骤1：写出公式：a² + b² = c²。
Step 2: Substitute the known values: 6² + 8² = c² → 36 + 64 = c² → 100 = c².
步骤2：代入已知值：6² + 8² = c² → 36 + 64 = c² → 100 = c²。
Step 3: Take the square root: c = √100 = 10 cm. The hypotenuse is 10 cm.
步骤3：开平方：c = √100 = 10 cm。斜边长为10 cm。
6. Calculating a Shorter Side | 计算短边长度
Example: The hypotenuse of a right-angled triangle is 13 m, and one leg is 5 m. Find the length of the other leg.
例题：一直角三角形的斜边长为13 m，一条直角边为5 m。求另一条直角边的长度。
Write the rearranged formula: b² = c² – a².
写出变形公式：b² = c² – a²。
Substitute: b² = 13² – 5² = 169 – 25 = 144.
代入：b² = 13² – 5² = 169 – 25 = 144。
Take the square root: b = √144 = 12 m. The missing side is 12 m.
开平方：b = √144 = 12 m。缺失的边长为12 m。
Always check that your answer makes sense – the hypotenuse must be the longest side.
务必检查答案是否合理——斜边必须是最长的边。
7. Real-Life Applications | 勾股定理的实际应用
Pythagoras’ theorem is used whenever we deal with lengths that form a right angle. For example, to determine the length of a ladder needed to reach a certain height on a wall, the wall and the ground form a right angle. If a ladder stands 3 m from a wall and reaches 4 m high, the ladder length is the hypotenuse: √(3² + 4²) = 5 m.
当我们处理成直角关系的长度时，就可以使用勾股定理。例如，要计算梯子需要多长才能靠墙达到一定高度，墙和地面构成一个直角。如果梯脚距墙3 m，梯顶高4 m，梯子长度就是斜边：√(3² + 4²) = 5 m。
Other applications include calculating the shortest distance between two points on a coordinate grid, determining diagonal screen sizes, and planning routes in navigation.
其他应用还包括计算坐标网格上两点之间的最短距离、确定屏幕对角尺寸，以及规划导航路线等。
8. Pythagorean Triples | 勾股数
A Pythagorean triple is a set of three positive whole numbers (a, b, c) that satisfy a² + b² = c². Recognizing these triples can save time. Common triples are (3, 4, 5), (5, 12, 13), and (6, 8, 10).
勾股数是指满足 a² + b² = c² 的三个正整数 (a, b, c)。能认出这些数组可以节省时间。常见的勾股数有 (3, 4, 5)、(5, 12, 13) 和 (6, 8, 10)。
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
If you multiply a triple by a common factor, the result is also a Pythagorean triple. For example, multiplying (3, 4, 5) by 2 gives (6, 8, 10).
如果将一组勾股数同时乘以一个公因数，得到的数组同样是勾股数。比如，把 (3, 4, 5) 乘以2，就得到 (6, 8, 10)。
9. Checking for a Right Angle | 检验直角
The converse of Pythagoras’ theorem is also true: if the sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle is right-angled. This is used by builders to ensure corners are exactly 90°, using the 3-4-5 method.
勾股定理的逆定理同样成立：如果三角形的三条边满足 a² + b² = c²（其中 c 为最长边），那么这个三角形是直角三角形。建筑工人常用3-4-5的方法来确保墙角恰好为90°。
10. Common Mistakes to Avoid | 常见错误
Do not use the hypotenuse as a leg. Always label the hypotenuse first. Another mistake is forgetting to take the square root at the end – c² is not the same as c. Also, remember that lengths must be positive, so the square root yields a positive value.
不要把斜边当作直角边用。首先标记好斜边。另一个常见错误是忘记在最后开平方——c² 与 c 是不一样的。还要注意，边长必须为正，因此开平方只取正值。
Some students misapply the formula by adding the given side lengths before squaring, e.g. calculating (3 + 4)² instead of 3² + 4². Always square first, then add.
有些学生在使用公式时会先加再平方，例如计算 (3 + 4)² 而不是 3² + 4²。一定要先平方，再相加。
11. Summary | 总结
Pythagoras’ theorem is a powerful tool for solving right-angled triangle problems. Remember: a² + b² = c², with c as the hypotenuse. Use it to find a missing side, verify right angles, and solve practical problems. Practise with a variety of questions, including word problems and coordinate geometry, to build confidence.
勾股定理是解决直角三角形问题的有力工具。记住：a² + b² = c²，其中 c 是斜边。用它求未知边长、检验直角，并解决实际问题。请通过各类题目进行练习，包括文字题和坐标几何题，以增强信心。
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