📚 Pythagoras' Theorem: Finding Missing Sides in Right-Angled Triangles | 勾股定理：求直角三角形中的缺失边长
Pythagoras' theorem is one of the most important ideas in KS3 mathematics. It connects the three sides of a right-angled triangle and allows you to work out an unknown side when you know the other two. This article explains the theorem step by step, with worked examples, common mistakes, real-life applications, and practice questions to help you master the topic.
勾股定理是 KS3 数学中最重要的内容之一。它把直角三角形三条边联系起来，让你在已知两条边的情况下求出未知的一条边。本文将一步步解释这个定理，并配有例题、常见错误、实际应用和练习题，帮助你掌握这个主题。
1. What is a Right-Angled Triangle? | 什么是直角三角形？
A right-angled triangle is a triangle that has one interior angle equal to 90°. The two sides that form the right angle are called the legs or shorter sides. The side opposite the right angle is the longest side, called the hypotenuse.
直角三角形是一个内角等于 90° 的三角形。构成直角的两条边称为直角边或较短边。直角对面的边是最长边，称为斜边。
In diagrams, the right angle is usually marked with a small square. This square tells you immediately that you can use Pythagoras' theorem, but only if the triangle is right-angled.
在图中，直角通常用一个小正方形标记。这个符号立刻告诉你，只要三角形是直角三角形，就可以使用勾股定理。
2. The Hypotenuse | 斜边
The hypotenuse is always the side opposite the right angle. It is always the longest side in a right-angled triangle. You can identify it by looking for the side that does not touch the right angle.
斜边总是直角对面的边。它总是直角三角形中最长的边。你可以通过寻找不接触直角的那条边来识别它。
For example, if the right angle is at vertex C in triangle ABC, then the hypotenuse is the side AB. Labelling the hypotenuse correctly is the first step in every Pythagoras question.
例如，如果直角在三角形 ABC 的顶点 C 处，那么斜边就是边 AB。正确标记斜边是每一道勾股定理题的第一步。
3. Statement of Pythagoras' Theorem | 勾股定理的表述
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
勾股定理指出，在任何直角三角形中，斜边的平方等于另外两条边的平方和。
a² + b² = c²
Here, c represents the hypotenuse, and a and b represent the two shorter sides. You can rearrange this formula to find any missing side.
这里，c 表示斜边，a 和 b 表示两条直角边。你可以变形这个公式来求任何缺失的边。
4. Using the Theorem to Find the Hypotenuse | 用定理求斜边
If you know the lengths of the two shorter sides, you can find the hypotenuse by squaring both shorter sides, adding them, and then taking the square root.
如果知道两条直角边的长度，可以先将两条直角边分别平方、相加，再开平方来求斜边。
c = √a² + b²
cm. First calculate 6² + 8² = 36 + 64 = 100. Then take the square root: c = √100 = 10 cm.
例题：已知 a = 6 cm，b = 8 cm，求 c。先计算 6² + 8² = 36 + 64 = 100。然后开平方：c = √100 = 10 cm。
Always remember to write the units in your final answer. If the lengths are in centimetres, the hypotenuse will also be in centimetres.
请务必在最终答案中写出单位。如果已知长度以厘米为单位，那么斜边也以厘米为单位。
5. Using the Theorem to Find a Shorter Side | 用定理求一条直角边
a to find the missing shorter side: a² = c² − b². Then take the square root of the result.
如果知道斜边和一条直角边，可以变形公式来求缺失的直角边：a² = c² − b²，然后对结果开平方。
a = √c² − b²
n c = 13 cm and a = 5 cm. Calculate b² = 13² − 5² = 169 − 25 = 144. Then b = √144 = 12 cm.
例题：已知 c = 13 cm，a = 5 cm，求 b。计算 b² = 13² − 5² = 169 − 25 = 144。然后 b = √144 = 12 cm。
Note that you subtract the shorter side from the hypotenuse, not add them. This is a very common mistake, so check your formula carefully.
注意是用斜边减去直角边，而不是相加。这是一个非常常见的错误，所以要仔细检查公式。
6. Checking Whether a Triangle is Right-Angled | 判断三角形是否为直角三角形
You can use the converse of Pythagoras' theorem to check whether a triangle is right-angled. If the squares of the two shorter sides add up to the square of the longest side, then the triangle is right-angled.
你可以使用勾股定理的逆定理来判断一个三角形是否为直角三角形。如果两条较短边的平方和等于最长边的平方，那么这个三角形就是直角三角形。
+ 12² = 81 + 144 = 225 and 15² = 225. Since these are equal, the triangle is right-angled.
例题：对于边长 9 cm、12 cm 和 15 cm，检查 9² + 12² = 81 + 144 = 225，而 15² = 225。由于两者相等，该三角形是直角三角形。
If the two values are not equal, the triangle is not right-angled. This technique is useful for testing whether a corner is exactly 90° in construction and design.
如果两个值不相等，则该三角形不是直角三角形。这个方法在建筑和设计中用于检验角落是否恰好为 90° 时非常有用。
7. Pythagorean Triples | 勾股数
. Common examples include (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17). These triples appear often in exam questions.
勾股数是满足 a² + b² = c² 的三个正整数。常见例子包括 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。这些勾股数经常出现在考试题中。
| Shorter side a | Shorter side b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
Recognising these triples can save you time. If you see sides in the ratio 3:4:5, you know the triangle is right-angled and the hypotenuse is the largest number.
识别这些勾股数可以节省时间。如果你看到边长比例为 3:4:5，你就知道该三角形是直角三角形，斜边是最大的那个数。
8. Real-Life Applications | 实际应用
Pythagoras' theorem is used in many real-life situations, such as finding the shortest distance, calculating the length of a ladder needed to reach a wall, and in construction to ensure corners are square.
勾股定理在现实生活中有许多应用，例如求最短距离、计算够到墙壁所需的梯子长度，以及在建筑中确保角落为直角。
. The height it reaches up the wall is found using h² = 5² − 3² = 25 − 9 = 16, so h = 4 m.
例如，一架 5 米长的梯子放在离墙脚 3 米处。它达到墙壁的高度用 h² = 5² − 3² = 25 − 9 = 16 计算，所以 h = 4 米。
Navigation also uses Pythagoras' theorem. If a boat travels 8 km east and then 6 km north, its straight-line distance from the starting point is √8² + 6² = √100 = 10 km.
导航也会用到勾股定理。如果一艘船先向东行驶 8 千米，再向北行驶 6 千米，那么它离起点的直线距离是 √8² + 6² = √100 = 10 千米。
9. Common Mistakes to Avoid | 常见错误
instead of a² + b². These two expressions are not equal because (a + b)² = a² + 2ab + b².
一个常见错误是先加长度再平方，例如写成 (a + b)² 而不是 a² + b²。这两个式子不相等，因为 (a + b)² = a² + 2ab + b²。
Another mistake is confusing which side is the hypotenuse. Always check that the side you call c is opposite the right angle and is the longest side.
另一个错误是分不清哪条边是斜边。一定要检查你标记为 c 的边是否在直角对面，并且是最长边。
Forgetting to take the square root at the end is also common. If c² = 25, then c = √25 = 5, not 25. Write down each step to avoid these errors.
最后忘记开平方也很常见。如果 c² = 25，那么 c = √25 = 5，而不是 25。写下每一步可以避免这些错误。
10. Worked Example: A Ladder Problem | 典型例题：梯子问题
A ladder is 6 m long and rests against a vertical wall. The foot of the ladder is 2 m from the base of the wall. Find the height the ladder reaches up the wall, correct to 2 decimal places if necessary.
一架 6 米长的梯子斜靠在竖直墙壁上，梯脚距墙脚 2 米。求梯子顶端达到的高度，如有需要保留两位小数。
height² + 2² = 6²
t 2² from both sides: height² = 6² − 2² = 36 − 4 = 32. Then take the square root: height = √32 = 5.66 m (to 2 decimal places).
两边同时减去 2²：height² = 6² − 2² = 36 − 4 = 32。然后开平方：height = √32 = 5.66 米（保留两位小数）。
In this example, the ladder, the wall, and the ground form a right-angled triangle. The ladder is the hypotenuse because it is opposite the right angle at the base of the wall.
在这个例子中，梯子、墙壁和地面构成一个直角三角形。梯子是斜边，因为它在墙脚直角对面。
11. Practice Questions and Answers | 练习题与答案
Try these questions to test your understanding. Write down your working for each one before checking the answers.
试做以下题目来检验你的理解。在核对答案前，先写下每道题的计算过程。
- Question 1: Find the hypotenuse of a right-angled triangle with shorter sides 12 cm and 16 cm.
问题 1：求直角边为 12 cm 和 16 cm 的直角三角形的斜边。 - Question 2: Find the missing shorter side when the hypotenuse is 17 cm and one shorter side is 8 cm.
问题 2：已知斜边为 17 cm，一条直角边为 8 cm，求另一条直角边。 - Question 3: Check whether a triangle with sides 10 cm, 24 cm and 26 cm is right-angled.
问题 3：判断边长分别为 10 cm、24 cm 和 26 cm 的三角形是否为直角三角形。 - Question 4: A ship sails 9 km east and then 12 km north. Find its straight-line distance from the starting point.
问题 4：一艘船向东航行 9 千米，再向北航行 12 千米。求它离起点的直线距离。
Answers: 1) 20 cm, 2) 15 cm, 3) Yes, because 10² + 24² = 676 = 26², 4) 15 km.
答案：1) 20 cm，2) 15 cm，3) 是，因为 10² + 24² = 676 = 26²，4) 15 千米。
12. Summary and Key Points | 总结与要点
Pythagoras' where c is the hypotenuse. To find the hypotenuse, add the squares of the shorter sides and then square root. To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse and then square root.
勾股定理只适用于直角三角形。公式是 a² + b² = c²，其中 c 是斜边。求斜边时，先将两条直角边平方相加，再开平方。求直角边时，用斜边平方减去已知直角边的平方，再开平方。
it is right-angled. Practice with Pythagorean triples and real-life problems will make you faster and more confident.
请仔细标记各条边，并写出所有计算步骤。如果一个三角形的三边满足 a² + b² = c²，那么它就是直角三角形。多练习勾股数和实际问题会让你更快、更自信。
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