Pythagoras’ Theorem and Its Applications | 勾股定理及其应用

📚 Pythagoras’ Theorem and Its Applications | 勾股定理及其应用

Pythagoras’ theorem is one of the most famous results in mathematics. It reveals a fundamental relationship between the three sides of a right‑angled triangle and is used to solve a huge range of problems, from simple missing‑side calculations to real‑life tasks such as navigation, construction and computer graphics.

勾股定理（也称毕达哥拉斯定理）是数学中最著名的定理之一。它揭示了直角三角形三条边之间的基本关系，被广泛用于解决各类问题——从简单的求边长计算，到导航、建筑和计算机图形学等实际任务。

1. What Is Pythagoras’ Theorem? | 什么是勾股定理？

Pythagoras’ theorem states that in any right‑angled triangle, the area of the square drawn on the hypotenuse (the longest side) is equal to the sum of the areas of the squares drawn on the other two sides. In simple terms: if you know the lengths of two sides, you can always find the third.

勾股定理指出：在任何一个直角三角形中，以斜边（最长边）为边长的正方形的面积，等于以两条直角边为边长的两个正方形的面积之和。简单来说，只要知道两条边的长度，你就能求出第三条边。

a² + b² = c²

Here, c always represents the length of the hypotenuse, while a and b represent the lengths of the two shorter sides (the legs). The small raised ‘2’ means the number is squared, i.e. multiplied by itself.

式中，c 始终表示斜边的长度，a 和 b 表示两条直角边的长度。上标“²”表示平方，即该数乘以自身。

---

2. The Right‑Angled Triangle | 直角三角形

Pythagoras’ theorem only works for right‑angled triangles. These triangles contain one angle of exactly 90°. The right angle is often marked with a small square in diagrams. The side directly opposite this right angle is the hypotenuse — it is always the longest side.

勾股定理仅适用于直角三角形。这类三角形包含一个恰好为 90° 的内角，在图中通常用一个小方块标记。直角对面的边就是斜边，它总是最长的一条边。

The two shorter sides that form the right angle are called the ‘legs’ of the triangle. They can be any lengths, but their squares always add up to the square of the hypotenuse.

构成直角的两条较短的边叫作“直角边”（或“股”）。它们的长度可以各不相同，但它们的平方和总是等于斜边的平方。

---

3. Identifying the Hypotenuse | 识别斜边

Before using the formula, it is essential to identify which side is the hypotenuse. Remember: the hypotenuse is always opposite the right angle and is the longest side. If you label a triangle yourself, make sure c is the hypotenuse.

在使用公式之前，必须先确定哪条边是斜边。记住：斜边永远在直角的对面，并且是最长边。如果自己给三角形标注字母，务必让 c 代表斜边。

A common mistake is to label one of the legs as c and then substitute carelessly. Always sketch the triangle and put the right‑angle marker in place before you start.

常见的错误是把某条直角边标成 c，然后随意代入公式。在使用公式前，最好先画出草图并标上直角符号。

---

4. Calculating the Hypotenuse | 计算斜边

When the two shorter sides are known, use the formula c² = a² + b² to find the hypotenuse. For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.

当两条直角边已知时，用公式 c² = a² + b² 求斜边。例如，若 a = 3 cm，b = 4 cm，则 c² = 3² + 4² = 9 + 16 = 25，因此 c = √25 = 5 cm。

Always take the square root of the sum. Many students forget this final step and leave the answer as c² instead of c. Write the full working line by line to avoid dropping marks.

一定要对平方和取平方根。很多同学会忘记这最后一步，把答案写成 c² 而不是 c。建议逐步写出完整过程，以免丢分。

---

5. Calculating a Shorter Side | 计算直角边

If the hypotenuse and one leg are known, rearrange Pythagoras’ theorem to find the missing shorter side. The formula becomes a² = c² − b² (or b² = c² − a²). Then take the square root.

当已知斜边和一条直角边时，可将勾股定理变形，求出缺失的直角边。公式变为 a² = c² − b²（或 b² = c² − a²），然后再开平方。

For instance, if the hypotenuse is 13 m and one leg is 5 m, the missing leg is √(13² − 5²) = √(169 − 25) = √144 = 12 m. This step shows that the legs are shorter than the hypotenuse, which acts as a useful check.

例如，斜边为 13 m，一条直角边为 5 m，则另一条直角边为 √(13² − 5²) = √(169 − 25) = √144 = 12 m。这一步也验证了直角边确实比斜边短，可作为一种简便的检验方法。

---

6. Pythagorean Triples | 勾股数

A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². The most famous triple is (3, 4, 5). Other common triples include (5, 12, 13), (6, 8, 10) and (7, 24, 25). Multiples of these also work, such as (6, 8, 10) being double (3, 4, 5).

勾股数是指满足 a² + b² = c² 的三个正整数。最著名的一组是 (3, 4, 5)。其他常见的勾股数还有 (5, 12, 13)、(6, 8, 10) 和 (7, 24, 25)。它们的倍数同样满足定理，例如 (6, 8, 10) 是 (3, 4, 5) 的两倍。

Recognising triples saves time in exams. If you spot two numbers from a known triple, you can quickly state the third without calculating. For KS3, memorising the (3, 4, 5) and (5, 12, 13) families is especially helpful.

识别勾股数能在考试中节省时间。如果你发现两条边的长度属于某个已知的勾股数组，就可以直接说出第三条边，而无需计算。对 KS3 阶段来说，记住 (3, 4, 5) 和 (5, 12, 13) 及其倍数尤为有用。

---

7. Applying Pythagoras in Real‑World Contexts | 勾股定理的实际应用

Pythagoras’ theorem is not just a classroom exercise. Carpenters use it to ensure corners are square. Navigators find the shortest distance between two points. Builders check whether walls are vertical by measuring diagonals.

勾股定理并非只是课堂上的练习题。木匠用它来确保角落成直角；航海者用它计算两点之间的最短距离；建筑工人通过测量对角线来检验墙壁是否竖直。

For example, a ladder of length 6 m leaning against a wall reaches 5 m up the wall. How far is the foot of the ladder from the wall? Using a² = c² − b² gives a² = 6² − 5² = 36 − 25 = 11, so a = √11 ≈ 3.32 m. This kind of problem appears regularly in KS3 assessments.

例如，一架 6 m 长的梯子斜靠在墙上，顶端距地面 5 m。梯脚离墙多远？利用 a² = c² − b² 可得 a² = 6² − 5² = 36 − 25 = 11，因此 a = √11 ≈ 3.32 m。这类问题经常出现在 KS3 的评估中。

---

8. Distance Between Two Points | 两点之间的距离

Pythagoras’ theorem can be extended to find the distance between two points on a coordinate grid. If the points are (x₁, y₁) and (x₂, y₂), the horizontal and vertical differences form the legs of a right‑angled triangle. The distance d is given by:

勾股定理还可以用来求坐标网格上两点之间的距离。若两点坐标为 (x₁, y₁) 和 (x₂, y₂)，则它们的水平差和垂直差构成直角三角形的两条直角边。距离 d 的公式为：

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

For instance, the distance between (1, 2) and (4, 6) is √[(4 − 1)² + (6 − 2)²] = √(3² + 4²) = √(9 + 16) = √25 = 5 units. This formula is a direct application of a² + b² = c².

例如，点 (1, 2) 与 (4, 6) 间的距离为 √[(4 − 1)² + (6 − 2)²] = √(3² + 4²) = √(9 + 16) = √25 = 5 个单位。该公式直接来自 a² + b² = c²。

---

9. Is a Triangle Right‑Angled? | 判断三角形是否为直角三角形

You can work backwards: if the three sides of a triangle satisfy a² + b² = c², then the triangle must be right‑angled. This test is useful when the angle is not marked. Simply check whether the square of the longest side equals the sum of the squares of the other two sides.

你还可以反过来使用定理：如果三角形的三条边满足 a² + b² = c²，那么这个三角形一定是直角三角形。当题目未标出直角时，这种检验方法非常有用。只需检查最长边的平方是否等于另外两边的平方和。

For a triangle with sides 7 cm, 9 cm and 12 cm, check 7² + 9² = 49 + 81 = 130, but 12² = 144, so 130 ≠ 144. Therefore the triangle is not right‑angled. This method also helps classify triangles as acute, right or obtuse.

对于边长分别为 7 cm、9 cm 和 12 cm 的三角形，检验 7² + 9² = 49 + 81 = 130，而 12² = 144，130 ≠ 144，因此它不是直角三角形。这个方法还可以帮助将三角形分类为锐角、直角或钝角三角形。

---

10. Proving the Theorem Through Area | 用面积法证明定理

At KS3 level, you might be shown a visual proof. One classic method uses four identical right‑angled triangles arranged inside a large square. The area of the large square can be expressed in two ways: as (a + b)² and as c² + 4×(½ab). Equating them leads to a² + b² = c².

在 KS3 阶段，你可能会见到一种直观的证明方法。一种经典的方法是将四个同样的直角三角形摆放在一个大正方形内。大正方形的面积可以用两种方式表达：(a + b)² 以及 c² + 4×(½ab)。将它们等同起来就能得到 a² + b² = c²。

Understanding this proof strengthens your appreciation of why the theorem works, not just how to use it. It also reinforces your skills in algebra and area calculations.

理解这一证明过程可以加深你对定理本质的认识，而不仅仅是会用公式。同时它也能巩固你的代数运算和面积计算能力。

---

11. Common Mistakes to Avoid | 常见错误规避

Several mistakes trip up KS3 learners. First, do not use Pythagoras’ theorem for non‑right‑angled triangles — the cosine rule is needed later. Second, always square before adding; do not add the lengths first and then square. Third, after finding c², do not forget to take the square root. Finally, check that the hypotenuse is the longest side.

有几个错误经常难倒 KS3 阶段的学生。第一，不要在非直角三角形上使用勾股定理——那需要以后学习的余弦定理。第二，一定要先平方再相加，而不是先相加再平方。第三，求出 c² 后别忘了开平方。最后，检查斜边是否确实是最长边。

Another slip is mixing up the sides when rearranging for a shorter side: remember it is c² minus the known leg, not the other way around. Label your diagram clearly and write the equation before substituting numbers.

另一个失误是在求直角边时把边搞混：要记住是用斜边的平方减去已知直角边的平方，而不是反过来。做题时先清楚地标注图形，并在代入数字前写出公式。

---

12. Practice Questions and Summary | 练习题与总结

Try these quick checks: (i) A right‑angled triangle has legs 8 cm and 15 cm. Find the hypotenuse. (ii) A triangle has sides 10 cm, 24 cm and 26 cm. Is it right‑angled? (iii) A ship sails 30 km east then 40 km north. How far is it from its starting point? Answers: (i) 17 cm, (ii) yes, (iii) 50 km.

试试这些快速检测题：(i) 直角三角形两条直角边分别为 8 cm 和 15 cm，求斜边。(ii) 三角形的边长分别为 10 cm、24 cm 和 26 cm，它是直角三角形吗？(iii) 一艘船先向东航行 30 km，再向北航行 40 km，距离出发点多远？答案：(i) 17 cm，(ii) 是，(iii) 50 km。

Pythagoras’ theorem is a pillar of geometry that will appear again and again in mathematics. Master it now by practising a mix of word problems, coordinate questions and reverse checks for right angles. With solid understanding, you will handle KS3 Pythagoras questions confidently and lay a strong foundation for future topics such as trigonometry.

勾股定理是几何学的重要支柱，会在以后的数学学习中反复出现。现在就要通过混合练习（文字题、坐标题及直角逆向检验）来掌握它。有了扎实的理解，你不仅能自信地应对 KS3 的勾股定理题目，还将为未来学习三角函数等内容打下坚实基础。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）