Pythagoras’ Theorem: Right-Angled Triangles and Applications | 毕达哥拉斯定理：直角三角形及其应用

📚 Pythagoras’ Theorem: Right-Angled Triangles and Applications | 毕达哥拉斯定理：直角三角形及其应用

Pythagoras’ theorem is one of the most famous results in geometry, linking the sides of a right-angled triangle. It is named after the ancient Greek mathematician Pythagoras. This theorem is not only essential for KS3 mathematics but also forms the foundation for many topics in higher-level maths and science. In this article, we will explore the statement, proof, applications, and common pitfalls of Pythagoras’ theorem, ensuring you can confidently solve related problems.

毕达哥拉斯定理是几何学中最著名的结论之一，它建立了直角三角形三边之间的关系。该定理以古希腊数学家毕达哥拉斯的名字命名。它不仅对KS3阶段的数学至关重要，也是高阶数学和科学中诸多课题的基础。本文将深入探讨该定理的表述、证明、应用及常见错误，帮助你自信解决相关问题。

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1. The Statement of the Theorem | 定理陈述

In any right-angled triangle, the area of the square drawn on the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the areas of the squares drawn on the other two sides.

在任何直角三角形中，以斜边（最长边，直角对边）为边长的正方形面积，等于以另外两条边为边长的两个正方形面积之和。

If we label the lengths of the two shorter sides (legs) as a and b, and the hypotenuse as c, the theorem is expressed algebraically as:

如果我们把两个较短边（直角边）的长度标记为 a 和 b，斜边标记为 c，则该定理可用代数式表示为：

a² + b² = c²

This equation holds true only for right-angled triangles. It allows us to find an unknown side length when the other two are known.

这个等式仅在直角三角形中成立。它让我们能够在已知两边的情况下求出第三边的长度。

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2. Understanding the Hypotenuse and Legs | 理解斜边与直角边

The hypotenuse is the side opposite the right angle and is always the longest side of the triangle. The other two sides are called legs or catheti. It is crucial to identify the hypotenuse correctly before applying the theorem.

斜边是直角的对边，永远是三角形中最长的一条边。另外两条边称为直角边。在应用定理之前，正确识别斜边至关重要。

For example, in a triangle with sides 3 cm, 4 cm, and 5 cm, the side of length 5 cm must be the hypotenuse because 5 is the largest number. The right angle lies between the sides of length 3 cm and 4 cm.

例如，在一个边长分别为3厘米、4厘米和5厘米的三角形中，5厘米的那条边一定是斜边，因为5最大。直角位于3厘米和4厘米这两条边之间。

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3. The Formula and its Rearrangement | 公式及其变形

The standard formula c² = a² + b² can be rearranged to find a shorter side when the hypotenuse and one leg are known. By subtracting b² from both sides, we get a² = c² − b², and similarly b² = c² − a².

标准公式 c² = a² + b² 可以变形，用于当已知斜边和一条直角边时求另一条直角边。两边减去 b² 可得 a² = c² − b²，同理 b² = c² − a²。

Then we take the square root to find the length: a = √(c² − b²). Remember that length is always positive, so we use the principal (positive) square root.

然后取平方根得到边长：a = √(c² − b²)。记住长度总是正数，所以我们使用算术平方根（正数平方根）。

When solving problems, always write down the formula you are using and substitute carefully. Do not round intermediate values; round only the final answer as required.

解题时，务必将所用的公式写下来并仔细代入。不要对中间值进行四舍五入，只按要求把最终答案四舍五入。

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4. Calculating the Hypotenuse | 计算斜边

To find the hypotenuse, given legs a and b, use c = √(a² + b²). For instance, if a = 6 cm and b = 8 cm, then c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

已知直角边 a 和 b 求斜边，使用 c = √(a² + b²)。例如，若 a = 6 cm，b = 8 cm，则 c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm。

It is useful to check your answer: in any right triangle, the hypotenuse must be longer than either leg, but shorter than the sum of the legs. Here 10 is greater than 6 and 8 but less than 14.

检查答案很有用：在任何直角三角形中，斜边必须比任意一条直角边长，但比两直角边之和短。此处10既大于6和8，又小于14。

Sometimes the squares produce numbers that are not perfect squares. For example, legs 2 cm and 3 cm give c = √(4 + 9) = √13. Leave your answer in surd form or round to a given number of decimal places (e.g., √13 ≈ 3.61 cm).

有时平方后得到的数不是完全平方数。例如，直角边2 cm和3 cm给出 c = √(4 + 9) = √13。答案可保留根号形式，或按给定的小数位数四舍五入（如 √13 ≈ 3.61 cm）。

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5. Calculating a Shorter Side | 计算直角边

When the hypotenuse and one leg are known, find the missing leg using a = √(c² − b²). Suppose c = 13 cm and b = 12 cm. Then a = √(13² − 12²) = √(169 − 144) = √25 = 5 cm.

当斜边和一条直角边已知时，使用 a = √(c² − b²) 求未知直角边。假设 c = 13 cm，b = 12 cm。则 a = √(13² − 12²) = √(169 − 144) = √25 = 5 cm。

Be careful with the order: subtract the square of the known leg from the square of the hypotenuse, never the other way around, otherwise you might get a negative number under the square root, which is impossible for a real length.

注意顺序：用斜边的平方减去已知直角边的平方，切勿反过来，否则平方根下的数可能为负，这对实际长度来说是不可能的。

Again, if the result is not a perfect square, leave it as a surd or approximate. For instance, if c = 10 cm and b = 7 cm, then a = √(100 − 49) = √51 ≈ 7.14 cm.

同样，若结果不是完全平方数，可保留为根式或求近似值。例如，若 c = 10 cm，b = 7 cm，则 a = √(100 − 49) = √51 ≈ 7.14 cm。

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6. Proving the Theorem (Visual Approach) | 定理证明（直观方法）

There are hundreds of proofs of Pythagoras’ theorem. One classic proof uses a large square of side length (a + b) containing four identical right triangles arranged around a smaller square of side length c. By comparing areas, we can show that (a + b)² = 4 × (½ab) + c², which simplifies to a² + b² = c².

毕达哥拉斯定理有数百种证明方法。一种经典的证明方法使用一个边长为 (a + b) 的大正方形，内含四个完全相同的直角三角形，它们围绕着一个边长为 c 的小正方形排列。通过比较面积，我们可以证明 (a + b)² = 4 × (½ab) + c²，化简后得到 a² + b² = c²。

Another visual proof involves drawing squares on each side of a right triangle and then showing that the two smaller squares can be cut and rearranged to exactly cover the largest square. This proof highlights the geometric meaning of the theorem.

另一种直观证明是在直角三角形的每条边上各画一个正方形，然后证明两个较小的正方形可以被剪切并重新排列，恰好覆盖最大的正方形。这一证明突出了定理的几何意义。

For KS3 level, it is not necessary to memorise the proof steps, but understanding that the theorem is not just a formula but a statement

Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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