Pythagoras’ Theorem — 勾股定理 | KS3 Cambridge Mathematics

Introduction to Pythagoras’ Theorem—勾股定理简介

Pythagoras’ Theorem is one of the most famous and useful results in all of mathematics. Named after the ancient Greek mathematician Pythagoras, this theorem describes a special relationship between the three sides of a right-angled triangle. It is a cornerstone of geometry that you will encounter throughout KS3 and beyond, from GCSE to A-Level mathematics and even in physics and engineering.

勾股定理是数学中最著名且最实用的定理之一。它得名于古希腊数学家毕达哥拉斯，描述的是直角三角形三条边之间的一种特殊关系。它是几何学的基石，贯穿 KS3 阶段及以后的学习，从 GCSE 到 A-Level 数学，甚至在物理和工程中都会用到。

What Is a Right-Angled Triangle?—什么是直角三角形？

Before we explore the theorem itself, let us make sure we understand what a right-angled triangle is. A right-angled triangle is a triangle that has one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse — it is always the longest side of the triangle. The other two sides are called the legs, and they form the right angle.

在探索定理本身之前，我们首先要理解什么是直角三角形。直角三角形是指有一个角等于 90 度的三角形。直角所对的边称为斜边，它总是三角形中最长的边。另外两条边称为直角边，它们构成直角。

Stating the Theorem—定理的表述

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. If we label the hypotenuse as c and the other two sides as a and b, then the theorem can be written as: c squared equals a squared plus b squared.

勾股定理指出：在任何一个直角三角形中，斜边长度的平方等于另外两条直角边长度的平方之和。如果我们将斜边标记为 c，另外两条直角边标记为 a 和 b，那么定理可以写成：c 的平方等于 a 的平方加 b 的平方。

Visual Proof with Squares—正方形的可视化证明

One of the clearest ways to understand Pythagoras’ Theorem is through a geometric proof using squares. Imagine drawing a square on each side of a right-angled triangle. The area of the square on the hypotenuse equals the combined area of the squares on the other two sides. For example, in the classic 3-4-5 triangle, the square on side 3 has area 9, the square on side 4 has area 16, and the square on the hypotenuse has area 25. And indeed, 9 plus 16 equals 25.

理解勾股定理最直观的方法之一是通过正方形的几何证明。想象在直角三角形的每一条边上画一个正方形。斜边上的正方形面积等于另外两条直角边上正方形面积之和。例如，在经典的 3-4-5 三角形中，边长 3 上的正方形面积为 9，边长 4 上的正方形面积为 16，斜边上的正方形面积为 25。确实，9 加 16 等于 25。

Finding the Hypotenuse—求斜边的长度

To find the length of the hypotenuse, we take the square root of the sum of the squares of the other two sides. For instance, if a right-angled triangle has legs of length 6 cm and 8 cm, then the hypotenuse length is the square root of 6 squared plus 8 squared, which equals the square root of 36 plus 64, which equals the square root of 100, which is 10 cm.

要求斜边的长度，我们取另外两条直角边长度的平方之和的平方根。例如，如果一个直角三角形的直角边长度分别为 6 cm 和 8 cm，那么斜边的长度等于 6 的平方加 8 的平方的平方根，即 36 加 64 的平方根，也就是 100 的平方根，等于 10 cm。

Finding a Shorter Side—求直角边的长度

Pythagoras’ Theorem can also be used to find the length of one of the shorter sides if we know the hypotenuse and the other shorter side. We simply rearrange the formula: a squared equals c squared minus b squared. For example, if the hypotenuse is 13 cm and one leg is 5 cm, the other leg squared equals 13 squared minus 5 squared, which equals 169 minus 25, giving 144. So the missing side is 12 cm.

勾股定理也可以反过来用于求直角边的长度，只要我们已知斜边和另一条直角边的长度。我们只需重新排列公式：a 的平方等于 c 的平方减去 b 的平方。例如，如果斜边长 13 cm，一条直角边长 5 cm，那么另一条直角边的平方等于 13 的平方减去 5 的平方，即 169 减 25，得到 144。因此缺失的边长为 12 cm。

Pythagorean Triples—勾股数

Some sets of three whole numbers satisfy Pythagoras’ Theorem perfectly. These are called Pythagorean triples. The most famous one is 3, 4, 5. Other examples include 5, 12, 13 and 8, 15, 17 and 7, 24, 25. These triples are useful because they give you right-angled triangles with whole-number side lengths, making calculations much easier. They often appear in KS3 exam questions, so it is worth memorising a few of them.

有些三个整数的组合完美地满足勾股定理。这些被称为勾股数。最著名的一组是 3, 4, 5。其他例子包括 5, 12, 13 以及 8, 15, 17 以及 7, 24, 25。这些勾股数非常有用，因为它们能给出边长为整数的直角三角形，使计算变得简单得多。它们经常出现在 KS3 的考试题目中，所以值得记住几组。

Real-Life Applications—实际生活中的应用

Pythagoras’ Theorem is not just an abstract mathematical idea; it has countless real-world applications. Builders use it to check that walls are perfectly perpendicular to each other. Surveyors use it to calculate distances across uneven terrain. Navigators use it to find the shortest distance between two points. Even in sports, like calculating the diagonal of a football pitch or the shortest throw from the outfield in cricket, Pythagoras’ Theorem finds practical use.

勾股定理不仅是一个抽象的数学概念，它在现实世界中有着无数的应用。建筑工人用它来检查墙壁是否完全垂直。测量员用它来计算崎岖地形中的距离。导航员用它来找到两点之间的最短距离。即使在体育运动中，比如计算足球场的对角线或板球中外场的最短投掷距离，勾股定理也有着实际用途。

Applying Pythagoras in 3D—三维空间中的勾股定理应用

Pythagoras’ Theorem extends naturally into three dimensions. To find the space diagonal of a rectangular box, we apply the theorem twice. First, find the diagonal of the base using the length and width. Then, use that diagonal and the height of the box to find the space diagonal. This technique is especially useful for KS3 students preparing for more advanced geometry in later years.

勾股定理自然地延伸到三维空间。要求长方体盒子的空间对角线，我们应用两次定理。首先，用长度和宽度求出底面的对角线。然后，用该对角线和盒子的高度求出空间对角线。这种方法对于为高年级更深入的几何学做准备的 KS3 学生特别有用。

Common Mistakes to Avoid—常见错误及避免方法

When using Pythagoras’ Theorem, students often make a few common errors. First, they forget that the theorem only works for right-angled triangles. Always check that there is a right angle before applying it. Second, they sometimes add the hypotenuse to one of the legs instead of rearranging correctly. Remember that to find a shorter side, you subtract, not add. Third, they may forget to take the square root at the end and leave the answer as a squared value. Always finish by taking the square root.

在使用勾股定理时，学生经常会犯几个常见错误。首先，他们忘记了这一定理仅适用于直角三角形。在应用之前，一定要确认存在直角。其次，他们有时会错误地将斜边与直角边相加，而不是正确地重新排列公式。记住，求直角边时应该相减而不是相加。第三，他们可能忘记最后取平方根，将答案保留为平方值。一定要最后取平方根。

Practice Problems with Solutions—练习题及解答

Let us work through some practice problems together. Problem 1: A right-angled triangle has legs of 9 cm and 12 cm. Find the hypotenuse. Solution: 9 squared plus 12 squared equals 81 plus 144 equals 225. Square root of 225 is 15, so the hypotenuse is 15 cm.

让我们一起做一些练习题。题目 1：一个直角三角形的直角边分别为 9 cm 和 12 cm。求斜边的长度。解答：9 的平方加 12 的平方等于 81 加 144 等于 225。225 的平方根是 15，所以斜边长为 15 cm。

Problem 2: The hypotenuse of a right-angled triangle is 17 cm, and one leg is 8 cm. Find the other leg. Solution: The unknown leg squared equals 17 squared minus 8 squared, which equals 289 minus 64, giving 225. Square root of 225 is 15, so the missing leg is 15 cm. Notice that 8, 15, 17 is a Pythagorean triple.

题目 2：直角三角形的斜边长为 17 cm，一条直角边长为 8 cm。求另一条直角边的长度。解答：未知直角边的平方等于 17 的平方减去 8 的平方，即 289 减 64，得到 225。225 的平方根是 15，所以缺失的直角边长为 15 cm。注意，8, 15, 17 是一个勾股数。

Problem 3: A ladder 5 metres long leans against a vertical wall. The foot of the ladder is 3 metres from the wall. How high up the wall does the ladder reach? Solution: This forms a right-angled triangle with the ladder as the hypotenuse. Height squared equals 5 squared minus 3 squared, which equals 25 minus 9, giving 16. Square root of 16 is 4, so the ladder reaches 4 metres up the wall.

题目 3：一架 5 米长的梯子靠在竖直的墙上。梯子的底部距离墙壁 3 米。梯子在墙上能达到多高？解答：这构成一个直角三角形，梯子为斜边。高度平方等于 5 的平方减去 3 的平方，即 25 减 9，得到 16。16 的平方根是 4，所以梯子能达到 4 米高。

The History of Pythagoras’ Theorem—勾股定理的历史

Although the theorem is named after Pythagoras, who lived around 570 to 495 BCE, the relationship between the sides of a right-angled triangle was known to earlier civilisations. Babylonian clay tablets dating from around 1800 BCE show evidence of Pythagorean triples being used in practical calculations. Ancient Indian mathematicians also described the theorem in the Sulba Sutras. In China, the theorem was known as the Gougu Theorem, recorded in the ancient mathematical text Zhoubi Suanjing. However, Pythagoras is credited with providing the first formal proof of the theorem in the Greek tradition of deductive mathematics.

虽然该定理以毕达哥拉斯命名，他大约生活在公元前 570 年至公元前 495 年，但直角三角形边长之间的关系在此之前就已经被更早的文明所知晓。公元前 1800 年左右制作的巴比伦泥板显示了勾股数在实际计算中的应用。古印度数学家也在 Sulba Sutras 中描述了这一定理。在中国，该定理被称为勾股定理，记载于古代数学文献《周髀算经》中。然而，毕达哥拉斯被认为是在希腊演绎数学传统中首次给出了定理的形式化证明。

Coordinate Geometry and the Distance Formula—坐标几何与距离公式

Pythagoras’ Theorem is the foundation of the distance formula used in coordinate geometry. To find the distance between two points on a coordinate grid, we can construct a right-angled triangle whose legs are parallel to the axes. The horizontal leg is the difference in x-coordinates, the vertical leg is the difference in y-coordinates, and the distance between the points is the hypotenuse. This gives us the formula: distance equals the square root of the square of the difference in x plus the square of the difference in y.

勾股定理是坐标几何中距离公式的基础。要求坐标网格上两点之间的距离，我们可以构造一个直角边与坐标轴平行的直角三角形。水平直角边是 x 坐标之差，竖直直角边是 y 坐标之差，两点之间的距离就是斜边的长度。由此我们得到公式：距离等于 x 坐标差值的平方加 y 坐标差值的平方之和的平方根。

Proof by Rearrangement—通过重组进行证明

There are over 350 known proofs of Pythagoras’ Theorem, making it one of the most-proved theorems in mathematics. One elegant proof uses rearrangement. Place four identical right-angled triangles inside a large square. When arranged one way, the empty space forms a square on the hypotenuse. When rearranged differently, the empty spaces form two squares on the legs. Since the total area of the large square is the same in both arrangements, the area of the square on the hypotenuse must equal the sum of the areas of the squares on the legs.

勾股定理有超过 350 种已知的证明方法，使其成为数学中证明最多的定理之一。一种优雅的证明方法使用了重组。将四个相同的直角三角形放入一个大正方形中。当以一种方式排列时，空白空间在斜边上形成一个正方形。当以另一种方式重新排列时，空白空间在两条直角边上形成两个正方形。由于两种排列方式中大正方形的总面积是相同的，因此斜边上正方形的面积必须等于两条直角边上正方形面积之和。

Pythagoras and Trigonometry—勾股定理与三角学

Pythagoras’ Theorem is deeply connected to trigonometry. In a right-angled triangle, the sine, cosine, and tangent ratios all depend on the relationship between the sides. In fact, the most fundamental identity in trigonometry, that sine squared plus cosine squared equals one for any angle, is a direct consequence of Pythagoras’ Theorem applied to the unit circle. Understanding Pythagoras’ Theorem is therefore essential preparation for GCSE trigonometry.

勾股定理与三角学有着深刻的联系。在一个直角三角形中，正弦、余弦和正切比都取决于边与边之间的关系。事实上，三角学中最基本的恒等式 – 对于任何角度，正弦平方加余弦平方等于 1 – 正是将勾股定理应用于单位圆的直接结果。因此，理解勾股定理是 GCSE 三角学的必要准备。

Isosceles Right Triangles and Special Angles—等腰直角三角形与特殊角

A particularly important special case is the isosceles right triangle, where the two legs are equal in length. If each leg has length 1 unit, then by Pythagoras’ Theorem, the hypotenuse has length equal to the square root of 2. This is a famous irrational number, approximately equal to 1.414. This triangle also has angles of 45, 45, and 90 degrees, making it a key standard triangle in trigonometry. The ratio of sides is 1 to 1 to the square root of 2.

一个特别重要的特例是等腰直角三角形，其中两条直角边长度相等。如果每条直角边长度为 1 个单位，那么根据勾股定理，斜边长度等于根号 2。这是一个著名的无理数，大约等于 1.414。这个三角形也有 45 度、45 度和 90 度的角，使其成为三角学中的关键标准三角形。边长的比例为 1 比 1 比根号 2。

Applications in Construction and Design—建筑与设计中的应用

Builders have used the 3-4-5 triangle for thousands of years to create perfect right angles. By measuring 3 units along one direction and 4 units along a perpendicular direction, the diagonal between these points should be exactly 5 units if the angle is truly 90 degrees. This technique, sometimes called the Egyptian rope-stretchers method, is still used on construction sites today. Architects also rely on Pythagoras’ Theorem when designing roof pitches, staircases, and foundations.

几千年来，建筑工人一直使用 3-4-5 三角形来创建完美的直角。沿一个方向量取 3 个单位，沿垂直方向量取 4 个单位，如果夹角正好是 90 度，那么这两点之间的斜边长度应该恰好是 5 个单位。这种方法有时被称为埃及拉绳法，至今仍在建筑工地上使用。建筑师在设计屋顶坡度、楼梯和地基时也依赖于勾股定理。

Checking Your Understanding—检验你的理解

Here is a quick self-assessment to test your understanding. Try these questions without looking at the solutions, then check your answers. Question 1: Is a triangle with sides 6, 8, and 11 a right-angled triangle? Answer: 6 squared plus 8 squared equals 100, but 11 squared equals 121. Since 100 is not equal to 121, this is not a right-angled triangle. Question 2: A rectangular field measures 24 m by 7 m. What is the distance from one corner to the opposite corner? Answer: The diagonal equals the square root of 24 squared plus 7 squared, which is the square root of 576 plus 49, giving the square root of 625, which is 25 m.

这里有一个快速自测来检验你的理解。先不看答案尝试以下问题，然后再核对。问题 1：边长为 6、8、11 的三角形是直角三角形吗？答案：6 的平方加 8 的平方等于 100，但 11 的平方等于 121。由于 100 不等于 121，这不是直角三角形。问题 2：一个长方形场地长 24 m、宽 7 m。从一个角到对角的距离是多少？答案：对角线等于 24 的平方加 7 的平方的平方根，即 576 加 49 的平方根，得到 625 的平方根，等于 25 m。

Pythagoras and Irrational Numbers—勾股定理与无理数

One of the most profound discoveries linked to Pythagoras’ Theorem is the existence of irrational numbers. The ancient Greeks were shocked to discover that the hypotenuse of an isosceles right triangle with legs of length 1 is the square root of 2, a number that cannot be expressed as a simple fraction. Legend has it that the Pythagorean who revealed this secret was drowned at sea. Understanding that some lengths produce irrational results is an important conceptual step for KS3 students moving toward more advanced mathematics.

与勾股定理相关的最深刻发现之一是无理数的存在。古希腊人震惊地发现，直角边长度为 1 的等腰直角三角形的斜边长度是根号 2，这是一个无法用简单分数表示的数。传说中，泄露这一秘密的毕达哥拉斯学派成员在海上被淹死了。理解一些长度会产生无理结果是 KS3 学生迈向更高级数学的重要概念性一步。

Using Pythagoras to Classify Triangles—用勾股定理对三角形进行分类

Pythagoras’ Theorem can also help us determine whether a triangle is acute, right-angled, or obtuse. For a triangle with sides a, b, and c where c is the longest side, if c squared equals a squared plus b squared, the triangle is right-angled. If c squared is less than a squared plus b squared, the triangle is acute. If c squared is greater than a squared plus b squared, the triangle is obtuse. This is known as the converse of Pythagoras’ Theorem, and it is a useful tool for triangle analysis.

勾股定理还可以帮助我们判断一个三角形是锐角三角形、直角三角形还是钝角三角形。对于一个边长为 a、b、c 的三角形，其中 c 是最长边，如果 c 的平方等于 a 的平方加 b 的平方，则该三角形是直角三角形。如果 c 的平方小于 a 的平方加 b 的平方，则该三角形是锐角三角形。如果 c 的平方大于 a 的平方加 b 的平方，则该三角形是钝角三角形。这被称为勾股定理的逆定理，是三角形分析的一个有用工具。

Pythagoras in Composite Shapes—组合图形中的勾股定理

In KS3 and GCSE exams, Pythagoras’ Theorem often appears in questions involving composite shapes. For example, you might need to find the height of an isosceles triangle by dropping a perpendicular from the apex to the base, creating two right-angled triangles. Or you might need to find the diagonal of a rectangle, or the edge of a kite. The key is to identify right-angled triangles within the larger shape and isolate them. Drawing a clear diagram and labelling all known lengths is the first and most important step.

在 KS3 和 GCSE 考试中，勾股定理经常出现在涉及组合图形的问题中。例如，你可能需要通过从顶点向底边作垂线来求等腰三角形的高，从而构造出两个直角三角形。或者你可能需要求矩形的对角线，或者风筝的边长。关键是要在更大的图形中识别出直角三角形并将它们隔离出来。绘制清晰的图示并标注所有已知长度是第一步也是最重要的一步。

Pythagoras in Three Dimensions: The Box Diagonal—三维空间中的勾股定理：盒子的对角线

Let us explore the 3D case in more detail. Consider a rectangular box with width w, depth d, and height h. To find the space diagonal, which is the longest straight line you can draw inside the box from one corner to the opposite corner, we use Pythagoras’ Theorem twice. First, find the diagonal of the base: the square root of w squared plus d squared. Then use that diagonal and the height: distance equals the square root of the base diagonal squared plus h squared. Combining these gives us the direct formula: distance equals the square root of w squared plus d squared plus h squared.

让我们更详细地探讨三维情况。考虑一个宽为 w、深为 d、高为 h 的长方体盒子。要求空间对角线，即从盒子一个角到对角的可画出最长直线，我们使用两次勾股定理。首先，求底面的对角线：w 的平方加 d 的平方的平方根。然后用该对角线和高度：距离等于底面对角线的平方加 h 的平方的平方根。将两者结合起来，我们得到直接公式：距离等于 w 的平方加 d 的平方加 h 的平方的平方根。

Exam Technique for Pythagoras Questions—勾股定理的考试技巧

When tackling Pythagoras questions in exams, follow a structured approach. Step 1: Read the question carefully and identify where the right angle is. Step 2: Label the sides clearly as a, b, or c. Step 3: Write down the formula. Step 4: Substitute the known values. Step 5: Solve for the unknown side. Step 6: Check that your answer is reasonable. For example, the hypotenuse must be longer than either leg. Step 7: State your answer with correct units. Marks are often awarded for showing your working clearly, even if the final answer is wrong.

在考试中解答勾股定理问题时，要遵循结构化的方法。第一步：仔细阅读题目，确定直角所在的位置。第二步：将三条边清楚地标记为 a、b 或 c。第三步：写出公式。第四步：代入已知值。第五步：求解未知边的长度。第六步：检查答案是否合理。例如，斜边必须长于任何一条直角边。第七步：陈述答案并附上正确的单位。即使最终答案错误，清晰地展示解题过程通常也能获得步骤分数。

Word Problems Involving Pythagoras—涉及勾股定理的文字题

Word problems test your ability to translate a real-world situation into a mathematical model. For example: a ship sails 12 km east and then 9 km north. How far is it from its starting point? This forms a right-angled triangle with legs 12 km and 9 km. The distance is the square root of 144 plus 81, which is the square root of 225, which is 15 km. Another example: a television screen measures 80 cm wide and 60 cm tall. What is the screen size measured diagonally? The diagonal is the square root of 6400 plus 3600, which is the square root of 10000, which is 100 cm.

文字题考察你将现实情况转化为数学模型的能力。例如：一艘船向东航行 12 km，然后向北航行 9 km。它离出发点有多远？这构成一个直角边分别为 12 km 和 9 km 的直角三角形。距离是 144 加 81 的平方根，即 225 的平方根，等于 15 km。另一个例子：电视屏幕宽 80 cm，高 60 cm。对角线测量的屏幕尺寸是多少？对角线是 6400 加 3600 的平方根，即 10000 的平方根，等于 100 cm。

Summary—总结

Pythagoras’ Theorem is a fundamental result in geometry that states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It allows us to calculate unknown side lengths and has widespread applications in mathematics, science, and everyday life. Mastering this theorem in KS3 provides a strong foundation for all future geometry studies, including trigonometry at GCSE and A-Level.

勾股定理是几何学中一个基础性结论，它指出在直角三角形中，斜边的平方等于两条直角边的平方之和。它使我们能够计算未知的边长，并在数学、科学和日常生活中有着广泛的应用。在 KS3 阶段掌握这一定理，为今后的所有几何学学习奠定坚实基础，包括 GCSE 和 A-Level 的三角学内容。

Remember the key formula: the square of the hypotenuse equals the sum of the squares of the legs. Know your Pythagorean triples such as 3-4-5 and 5-12-13. Always check that the triangle contains a right angle before applying the theorem, and always take the square root as your final step. With these principles in mind, you will find Pythagoras’ Theorem both powerful and elegant.

记住关键公式：斜边的平方等于直角边的平方之和。知道常见的勾股数，如 3-4-5 和 5-12-13。在应用定理之前，始终检查三角形是否包含直角，并始终将取平方根作为最后一步。牢记这些原则，你会发现勾股定理既强大又优雅。

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