Pythagoras’ Theorem for Cambridge KS3 | 剑桥KS3毕达哥拉斯定理精讲

📚 Pythagoras’ Theorem for Cambridge KS3 | 剑桥KS3毕达哥拉斯定理精讲

Pythagoras’ Theorem is a fundamental concept in geometry that describes the relationship between the sides of a right-angled triangle. It is an essential topic in the Cambridge Lower Secondary Mathematics curriculum (Key Stage 3) and lays the foundation for more advanced trigonometry and geometry. This article will guide you through the theorem, its proof, applications, and common exam questions, helping you build confidence and accuracy.

毕达哥拉斯定理（勾股定理）是几何学中描述直角三角形三边关系的基本定理，也是剑桥初中数学课程（KS3阶段）的核心内容，为后续的三角学和几何学习打下基础。本文将带你全面掌握该定理的证明、应用及常见考试题型，帮助你建立信心并提高解题准确性。

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1. Right-Angled Triangles | 直角三角形

A right-angled triangle is a triangle in which one of the angles measures exactly 90°. The side opposite the right angle is called the hypotenuse — it is always the longest side. The other two sides are referred to as legs or shorter sides. In diagrams, the right angle is often marked with a small square.

直角三角形是指其中一个角恰好为90°的三角形。直角所对的边称为斜边（hypotenuse），它总是最长的一条边。另外两条边称为直角边（或短边）。在图形中，直角通常用一个小方格来标注。

To apply Pythagoras’ Theorem correctly, you must first identify the hypotenuse and the two legs. The hypotenuse is crucial because the theorem relates its square to the sum of the squares of the other two sides.

要正确应用毕达哥拉斯定理，你必须首先识别出斜边和两条直角边。斜边至关重要，因为定理将它的平方与另两条边的平方和联系起来。

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2. Statement of the Theorem | 定理陈述

Pythagoras’ Theorem states: In any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as:

毕达哥拉斯定理指出：在任何直角三角形中，斜边长度（c）的平方等于另两条直角边长度（a 和 b）的平方和。可以表示为：

a² + b² = c²

Where ‘c’ represents the length of the hypotenuse, and ‘a’ and ‘b’ represent the lengths of the other two sides. The variables can be swapped, but the key is that the side labelled ‘c’ must be the hypotenuse.

其中 c 代表斜边长度，a 和 b 代表另外两条边的长度。变量可以互换，但标为 c 的边必须是斜边。

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3. Visual Proof | 面积证明

A classic visual proof involves drawing squares 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, with sides 3, 4, and 5: 3² + 4² = 9 + 16 = 25, which is 5².

一个经典的直观证明是在直角三角形的每条边上各画一个正方形。斜边上的正方形面积等于另两条边上正方形面积之和。例如，边长为 3、4、5 的三角形：3² + 4² = 9 + 16 = 25，即 5²。

This area relationship is often illustrated with grids or rearranging shapes to show that the two smaller squares can be cut and reassembled into the larger square. This demonstration helps learners understand why the theorem works without relying solely on algebra.

这种面积关系常通过网格或图形的重新排列来展示，两个较小的正方形可以切割并重新拼成较大的正方形。这种演示有助于学生理解定理为何成立，而不仅仅是依靠代数证明。

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4. Finding the Hypotenuse | 求斜边

When you know the lengths of the two legs (a and b), you can calculate the hypotenuse (c) using the formula c = √(a² + b²). For instance, if a = 6 cm and b = 8 cm, then c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

当你已知两条直角边的长度（a 和 b），可以用公式 c = √(a² + b²) 来计算斜边（c）。例如，若 a = 6 cm，b = 8 cm，则 c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm。

Always remember to take the square root at the end and include the correct units. The hypotenuse will always be larger than either leg, which provides a quick check for your answer.

切记最后要取平方根，并注明正确的单位。斜边的长度总是大于任何一条直角边，这个性质可以用来快速检查答案。

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5. Finding a Shorter Side | 求直角边

To find a missing shorter side when you know the hypotenuse and one leg, rearrange the formula: a = √(c² – b²). For example, if the hypotenuse is 13 cm and one leg is 5 cm, the missing leg a = √(13² – 5²) = √(169 – 25) = √144 = 12 cm.

当已知斜边和一条直角边求另一条直角边时，可调整公式：a = √(c² – b²)。例如，斜边为 13 cm，一条直角边为 5 cm，则缺失的直角边 a = √(13² – 5²) = √(169 – 25) = √144 = 12 cm。

Be careful with subtraction: always subtract the square of the known leg from the square of the hypotenuse, not the other way around. This ensures you get a positive number under the root.

注意减法顺序：始终用斜边的平方减去已知直角边的平方，而不要反过来减，以确保根号内的数字为正。

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6. Pythagorean Triples | 毕达哥拉斯三元组

A Pythagorean triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². The most common triple is (3, 4, 5). Others include (5, 12, 13), (7, 24, 25), and (8, 15, 17). Multiples of these triples also work, e.g., (6, 8, 10) from (3, 4, 5).

毕达哥拉斯三元组是指满足 a² + b² = c² 的三个正整数 (a, b, c)。最常见的一组是 (3, 4, 5)。其他还有 (5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。这些三元组的倍数同样适用，例如由(3, 4, 5)得到的(6, 8, 10)。

Recognising triples can save time in exams. If you spot that two sides of a right triangle form part of a triple multiple, you can quickly determine the third side without calculation.

识别三元组可以在考试中节省时间。如果你发现直角三角形的两条边是某个三元组倍数的组成部分，你可以不经过计算就直接得出第三条边的长度。

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7. The Converse of Pythagoras | 定理的逆定理

The converse states: If the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is right-angled. For example, a triangle with sides 9, 12, 15: 9² + 12² = 81 + 144 = 225 = 15², so it contains a right angle opposite the side of length 15.

逆定理指出：如果三角形最长边的平方等于另外两边平方之和，那么这个三角形是直角三角形。例如，边长为 9, 12, 15 的三角形：9² + 12² = 81 + 144 = 225 = 15²，因此长度为 15 的边所对的角是直角。

This is useful for proving whether a given triangle is right-angled or not. In construction and design, you can use this property to verify that corners are perfectly square (e.g., using 3-4-5 triangle measurements).

这可以用来证明一个给定的三角形是否为直角三角形。在建筑和设计中，你可以运用这一性质来检查角落是否完全为直角（例如利用 3-4-5 三角形的测量方法）。

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8. Real-World Applications | 实际应用

Pythagoras’ Theorem appears in many real-life contexts: calculating the length of a ladder needed to reach a certain height when leaning against a wall; finding the shortest distance across a park; determining the diagonal size of a TV screen; or measuring distances on maps.

毕达哥拉斯定理在许多实际情境中都有应用：计算靠在墙上的梯子达到某一高度所需的长度；找出穿过公园的最短距离；确定电视屏幕的对角线尺寸；或在地图上测量距离。

In navigation, the theorem is used to find straight-line distances between two points when horizontal and vertical distances are known. It also underpins the distance formula in coordinate geometry.

在导航中，当已知水平与垂直距离时，该定理可用于求两点之间的直线距离。

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