Pythagoras' Theorem | 勾股定理

📚 Pythagoras' Theorem | 勾股定理

Pythagoras' theorem is one of the most famous results in mathematics, linking the three sides of a right-angled triangle. It states that in any right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

勾股定理是数学中最著名的结论之一，它连接了直角三角形的三条边。定理指出：在任何直角三角形中，斜边（直角所对的边）上的正方形面积等于另外两条边上的正方形面积之和。

1. What is Pythagoras' Theorem? | 什么是勾股定理？

ways the longest side, opposite the 90° angle. The theorem only applies to right-angled triangles, so you must first confirm that the triangle has a right angle before using it.

对于一个直角边为 a 和 b、斜边为 c 的直角三角形，该定理可以写成 a² + b² = c²。这里 c 总是最长的边，对着 90° 角。该定理只适用于直角三角形，因此在使用之前你必须先确认三角形有一个直角。

2. The Hypotenuse | 斜边

The hypotenuse is the side opposite the right angle and is always the longest side in a right-angled triangle. Identifying the hypotenuse correctly is essential because the formula isolates c on one side. Even if the triangle is rotated, the hypotenuse remains opposite the right angle and can be recognised by its position facing the square corner mark.

斜边是直角所对的边，始终是直角三角形中最长的边。正确识别斜边至关重要，因为公式中 c 是独立在等式一边的。即使三角形旋转了，斜边仍然位于直角的对面，可以通过它面对直角标记的位置来识别。

3. Using the Formula | 使用公式

quare root. For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 ts and, if the answer is not a perfect square, leave it as a surd or round to the required degree of accuracy.

已知两条直角边求斜边时，先分别平方，再相加，然后取平方根。例如，如果 a = 3 cm，b = 4 cm，那么 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5 cm。请始终记住包含单位，如果答案不是完全平方数，可以保留根号形式或按要求的精确度四舍五入。

When solving problems, it is useful to write the formula first, substitute the known values, and then perform the calculation step by step. This method reduces careless mistakes and helps you check your working later.

解题时，先写出公式，代入已知数值，然后逐步计算是很有帮助的。这种方法可以减少粗心错误，并方便后续检查你的计算过程。

4. Finding a Shorter Side | 求直角边

. For instance, if c = 13 m and b = 5 m, then a² = 13² − 5² = 169 − 25 = 144, so a = 12 m.

如果你知道斜边和一条直角边，可以通过重新整理公式求出缺失的直角边：a² = c² − b² 或 b² = c² − a²。相减后，记得取平方根。例如，如果 c = 13 m，b = 5 m，那么 a² = 13² − 5² = 169 − 25 = 144，所以 a = 12 m。

Be careful to subtract the square of the known leg from the square of the hypotenuse, not the other way round. The hypotenuse is always the largest number, so subtracting correctly ensures a positive result under the square root.

小心要用斜边的平方减去已知直角边的平方，而不是反过来。斜边始终是最大的数，正确相减才能确保平方根下的数是正数。

5. Pythagorean Triples | 勾股数

Some sets of three whole numbers satisfy the Pythagorean equation exactly. Common triples include (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). Recognising these triples can save time in tests, especially when questions involve scaled versions, such as (6, 8, 10) which is a multiple of (3, 4, 5).

有些三个整数的组合恰好满足勾股方程。常见的勾股数包括 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。识别这些数可以在考试中节省时间，尤其是当题目涉及倍数缩放时，例如 (6, 8, 10) 就是 (3, 4, 5) 的倍数。

A Pythagorean triple always consists of two odd numbers and one even number, or all three numbers even. Memorising a few basic triples helps you quickly verify whether a triangle is right-angled when given its side lengths.

勾股数总是由两个奇数和一个偶数组成，或者三个数都是偶数。记住几个基本勾股数可以帮助你根据边长快速验证一个三角形是否是直角三角形。

6. Real-life Applications | 实际应用

Pythagoras' theorem is not restricted to geometry classrooms. Builders use it to ensure corners are square by checking if a triangle with sides 3 m, 4 m and 5 m has a right angle. Navigators apply it to find the shortest distance between two points when travelling across a rectangular grid. Even in computer graphics, the theorem helps calculate distances between pixels.

勾股定理并不局限于几何课堂。建筑工人用它来确保墙角是方的，他们会检查边长分别为 3 米、4 米和 5 米的三角形是否有直角。导航员在矩形网格内行进时，用它来计算两点之间的最短距离。即使在计算机图形学中，该定理也有助于计算像素之间的距离。

Another classic application is a ladder leaning against a wall: you can calculate how high the ladder reaches given its length and the distance from the wall, or find the necessary ladder length for a specific height.

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另一个经典应用是梯子靠在墙上的问题：已知梯子长度和底端离墙的距离，你可以计算梯子能到达多高；或者为特定的高度找到所需的梯子长度。

7. Proof of the Theorem | 定理的证明

There are hundreds of proofs of Pythagoras' gles plus the area of the inner square c². Rearranging the equation leads to a² + b² = c².

勾股定理有数百种证明方法。一种直观的证明使用四个全等的直角三角形拼在一个大正方形内。大正方形的面积 (a+b)² 等于四个三角形的面积加上内部小正方形的面积 c²。重新整理等式即可得到 a² + b² = c²。

Understanding a proof deepens your appreciation of why the theorem works, rather than just memorising the formula. It also prepares you for higher-level mathematical reasoning where constructing logical arguments is key.

理解一个证明能加深你对定理为什么成立的认识，而不仅仅是死记硬背公式。这也为更高层次的数学推理做好了准备，因为在那时构建逻辑论证至关重要。

8. Common Mistakes | 常见错误

+ b² = c². Using the theorem on an acute or obtuse triangle will give an incorrect result.

一个常见的错误是将定理应用于非直角三角形。在使用 a² + b² = c² 之前，一定要寻找直角符号或确认三角形是直角三角形。将定理用于锐角或钝角三角形会得到错误的结果。

Another mistake is confusing which side is the hypotenuse when the triangle is rotated. Remember, the hypotenuse is always opposite the right angle, not necessarily the side that looks longest due to perspective.

另一个错误是当三角形旋转时，混淆哪条边是斜边。请记住，斜边始终位于直角对面，而不一定是由于透视看起来最长的边。

Students sometimes forget to take the square root after adding or subtracting. The equation gives the squared length, so the final step must be to find the square root to obtain the side length.

学生们有时在相加或相减后忘记取平方根。等式给出的是长度的平方，因此最后一步必须求平方根才能得到边长。

9. Practice Problems | 练习题

Try these: (1) A right triangle has legs 6 cm and 8 cm. Find the hypotenuse. (2) The hypotenuse is 17 m and one leg is 8 m. Find the other leg. (3) Is a triangle with sides 9, 12 and 15 right-angled? (4) A ladder 10 m long rests against a wall with its foot 6 m from the wall. How high up the wall does it reach? Work through each step using the formula and check your answers with a calculator if needed.

尝试以下练习：(1) 一个直角三角形的直角边为 6 cm 和 8 cm，求斜边。(2) 斜边是 17 m，一条直角边是 8 m，求另一条直角边。(3) 边长分别为 9、12 和 15 的三角形是直角三角形吗？(4) 一架 10 m 长的梯子靠在墙上，梯脚离墙 6 m，它能达到墙上的多高？每一步都使用公式计算，必要时用计算器检查答案。

= 10² − 6² = 100 − 36 = 64, so height = 8 m. Regular practice builds confidence and speed.

对于 (3)，检验 9² + 12² 是否等于 15²：81 + 144 = 225，而 15² = 225，所以是直角三角形。对于 (4)，高度² = 10² − 6² = 100 − 36 = 64，因此高度 = 8 m。经常练习可以建立信心并提高速度。

10. Summary | 总结

Pythagoras' gles. Always identify the hypotenuse first, use a² + b² = c² or its rearrangement, and don' ve time and apply the theorem to real-world problems involving distance, height and depth.

勾股定理是求解直角三角形中缺失边长的基本工具。始终先确定斜边，使用 a² + b² = c² 或其变形式，并且在最后不要忘记取平方根。识别常见的勾股数以节省时间，并将定理应用于涉及距离、高度和深度的实际问题。

With careful labelling and step-by-step working, you can confidently solve any Pythagoras problem that appears in your Key Stage 3 assessments and beyond.

通过仔细标注和分步计算，你就能自信地解决 Key Stage 3 评估以及更高年级中出现的任何勾股定理问题。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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