Pythagoras' Theorem Explained | 毕达哥拉斯定理详解

📚 Pythagoras’ Theorem Explained | 毕达哥拉斯定理详解

Pythagoras’ Theorem is one of the most famous results in geometry, linking the three sides of a right-angled triangle. For any right-angled triangle, the area of the square on the hypotenuse (the longest side, opposite the right angle) is exactly equal to the sum of the areas of the squares on the other two sides. This relationship allows us to find a missing side length when two sides are known, and it underpins much of trigonometry, coordinate geometry, and real-world measurement.

毕达哥拉斯定理是几何学中最著名的结论之一，它揭示了直角三角形三条边之间的关系。对于任意直角三角形，斜边（最长边，直角对边）上的正方形面积恰好等于另外两条边上的正方形面积之和。利用这一关系，我们可以在已知两条边的情况下求出第三条边的长度，它也是三角函数、坐标几何和实际测量中许多内容的基础。

1. The Statement of the Theorem | 定理的表述

If a right-angled triangle has shorter sides of lengths a and b, and a hypotenuse of length c, then a² + b² = c². This is often remembered as ‘the square on the hypotenuse equals the sum of the squares on the other two sides’. The letters can be swapped as long as c always represents the hypotenuse, the side opposite the right angle.

如果一个直角三角形的两条直角边长度分别为 a 和 b，斜边长度为 c，那么 a² + b² = c²。这常被记为“斜边的平方等于两直角边的平方和”。只要 c 始终代表斜边（直角对边），字母的位置可以交换。

a² + b² = c²

2. Identifying the Hypotenuse | 识别斜边

The hypotenuse is always the longest side of a right-angled triangle and is always opposite the right angle. You can spot it quickly by locating the right angle box in a diagram – the side that does not touch that box is the hypotenuse. In word problems, the hypotenuse is often the direct distance between two points when the other sides run horizontally and vertically.

斜边始终是直角三角形中最长的边，并且总是位于直角的对面。你可以通过在图示中找出直角方框来快速定位斜边——不与这个方框相接的边就是斜边。在文字题中，当其他两边分别沿水平和垂直方向时，斜边通常是两点之间的直线距离。

3. Rearranging the Formula to Find a Missing Leg | 公式变形求直角边

To find a shorter side (leg), subtract the square of the known shorter side from the square of the hypotenuse, then take the square root: a = √(c² − b²) or b = √(c² − a²). Always label sides clearly before substituting, and remember that the squared hypotenuse must be larger than the squared leg to get a real answer.

求直角边时，用斜边的平方减去已知直角边的平方，然后开平方根：a = √(c² − b²) 或 b = √(c² − a²)。在代入数值之前，务必清晰地标注各边，并记住斜边的平方必须大于直角边的平方，才能得到实数解。

leg = √(hypotenuse² − known leg²)

4. Worked Example: Finding the Hypotenuse | 计算示例：求斜边

A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
Using a² + b² = c² → 6² + 8² = c² → 36 + 64 = c² → c² = 100 → c = √100 = 10 cm. The hypotenuse is 10 cm. Note that we only take the positive root because length cannot be negative.

一个直角三角形的两条直角边分别为 6 厘米和 8 厘米。求斜边长度。
使用 a² + b² = c² → 6² + 8² = c² → 36 + 64 = c² → c² = 100 → c = √100 = 10 厘米。斜边长为 10 厘米。注意我们只取正平方根，因为长度不能为负数。

5. Worked Example: Finding a Missing Leg | 计算示例：求直角边

A right-angled triangle has a hypotenuse of 13 m and one leg of 5 m. Find the other leg.
Let the missing leg be b, so 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = √144 = 12 m. The missing leg is 12 m. This 5-12-13 triple is a common Pythagorean triple worth remembering.

一个直角三角形的斜边长为 13 米，一条直角边长为 5 米。求另一条直角边。
设未知直角边为 b，则有 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = √144 = 12 米。未知直角边长为 12 米。这组 5-12-13 是值得记住的常见勾股数。

6. Pythagorean Triples | 勾股数

A Pythagorean triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and their multiples, such as (6, 8, 10) from doubling (3, 4, 5). Recognising these triples can speed up problem solving without needing to calculate square roots each time.

勾股数是由三个正整数 (a, b, c) 组成的数组，满足 a² + b² = c²。常见的例子包括 (3, 4, 5)、(5, 12, 13)、(8, 15, 17)，以及它们的倍数，例如将 (3, 4, 5) 翻倍得到的 (6, 8, 10)。识别这些勾股数可以加快解题速度，无需每次都计算平方根。

Triple (3, 4, 5)	3² + 4² = 5² → 9+16=25
(5, 12, 13)	25 + 144 = 169
(8, 15, 17)	64 + 225 = 289

7. Verifying a Right Angle: The Converse | 逆定理：验证直角

The converse of Pythagoras’ Theorem states that if the sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle is right-angled with the right angle opposite side c. This is a powerful tool for checking whether a triangle is right-angled without measuring an angle. For example, a triangle with sides 9, 12, 15 has 9² + 12² = 81 + 144 = 225 = 15², so it is a right-angled triangle.

毕达哥拉斯定理的逆定理指出，如果一个三角形的三条边满足 a² + b² = c²（其中 c 是最长边），那么这个三角形是直角三角形，且直角位于 c 的对边。这是检查三角形是否为直角三角形的一个强有力工具，无需测量角度。例如，三边为 9、12、15 的三角形，有 9² + 12² = 81 + 144 = 225 = 15²，因此它是直角三角形。

8. Applications in Real Life | 实际生活中的应用

Pythagoras’ Theorem is used in construction to ensure walls are perpendicular, in navigation to find the shortest distance between two points, in computer graphics to calculate pixel distances, and in sports to measure straight-line throws or kicks. Any situation where two distances are at right angles can be modelled with a right-angled triangle and solved using the theorem.

毕达哥拉斯定理在建筑施工中用于确保墙体垂直，在导航中用于求两点间的最短距离，在计算机图形学中用于计算像素距离，在体育运动中也用于测量直线投掷或踢击的距离。任何两条距离相互垂直的情形都可以用直角三角形来建模，并用该定理求解。

9. Solving Problems with Multiple Steps | 多步问题求解

Some problems require applying the theorem twice, for instance when a diagonal is inside a rectangle, which can be split into two right-angled triangles. You might first find the diagonal of one face of a cuboid, and then use that diagonal with another edge to find the space diagonal. Always draw a clear diagram and label all known lengths and right angles before starting.

有些问题需要两次运用定理，例如求一个矩形内的对角线时，可以将矩形分割为两个直角三角形。你可能需要先求出长方体一个面的对角线，然后用这条对角线与另一条棱来求空间对角线。在解题前，一定要画出清晰的示意图并标出所有已知长度和直角。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One frequent mistake is mixing up which side is the hypotenuse: always locate the right angle first. Another is forgetting to take the square root at the end, leaving the answer as c² rather than c. Also, when subtracting, ensure that the known values are placed correctly: a = √(c² − b²), not √(b² − c²) which would give an error if c is the longest side. Practise with both numeric and algebraic examples to build fluency.

一个常见错误是混淆哪条边是斜边：始终要先确定直角的位置。另一个错误是忘记最后开平方根，将答案保留为 c² 而不是 c。此外，在进行减法时，要确保已知值代入正确：a = √(c² − b²)，而不是 √(b² − c²)，因为如果 c 是最长边，后者会出错。通过数值和代数例题的练习来熟练掌握。

11. Linking Pythagoras to Trigonometry | 与三角函数的联系

Pythagoras’ Theorem is the foundation of the trigonometric identity sin²θ + cos²θ = 1 for a right-angled triangle. If we label the sides relative to an angle θ, the legs can be expressed using sine and cosine, and substituting them into a² + b² = c² yields the identity. This connection becomes essential in higher-level mathematics, but the core geometric idea remains the same.

毕达哥拉斯定理是直角三角形中三角恒等式 sin²θ + cos²θ = 1 的基础。如果我们用相对角 θ 来标记各边，直角边可以用正弦和余弦表示，将它们代入 a² + b² = c² 即可得到该恒等式。这种联系在更高层次的数学中至关重要，但核心的几何思想保持不变。

12. Practice Questions | 练习题

1. A ladder 5 m long leans against a wall. The foot of the ladder is 1.5 m away from the wall. How high up the wall does the ladder reach?
2. Find the distance between the points (2, 3) and (6, 7) on a coordinate grid, using Pythagoras’ Theorem.
3. Is a triangle with sides 7, 8, 11 right-angled? Justify your answer.

1. 一把 5 米长的梯子斜靠在墙上，梯脚距墙 1.5 米。梯子顶端离地面有多高？
2. 利用毕达哥拉斯定理求坐标网格中点 (2, 3) 和 (6, 7) 之间的距离。
3. 三边长分别为 7、8、11 的三角形是否为直角三角形？请说明理由。

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Pythagoras’ Theorem Explained | 毕达哥拉斯定理详解