Pythagoras' Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most important rules in IGCSE Mathematics. It connects the three sides of a right-angled triangle and appears in algebra, geometry, trigonometry and coordinate problems.

勾股定理是IGCSE数学中最重要的定理之一。它把一个直角三角形的三条边联系在一起，广泛出现在代数、几何、三角函数和坐标问题中。

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

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

在直角三角形中，直角所对的边称为斜边。勾股定理指出：斜边的平方等于两条直角边的平方和。

a² + b² = c²

Here c is the hypotenuse, while a and b are the two shorter sides.

其中 c 是斜边，a 和 b 是两条较短的直角边。

2. The Formula in Words | 公式的文字表述

The theorem can also be written as: hypotenuse squared = square of one leg + square of the other leg.

这个定理也可以写成：斜边的平方 = 一条直角边的平方 + 另一条直角边的平方。

This version helps you decide which side is the hypotenuse before doing any calculation.

这种写法能帮助你在计算之前先判断哪条边是斜边。

3. Finding the Hypotenuse | 求斜边

If you are given the two shorter sides, substitute them into the formula and solve for c.

如果已知两条直角边，把它们代入公式，求出 c 即可。

c² = a² + b², so c = √(a² + b²)

Example: A right-angled triangle has sides of length 3 cm and 4 cm. Find the hypotenuse.

例：一个直角三角形的两条直角边分别为 3 cm 和 4 cm，求斜边。

c² = 3² + 4² = 9 + 16 = 25

c = √25 = 5 cm

Always remember to take the square root at the end; squaring alone is not the answer.

最后一定要开平方；只算出平方数并不是答案。

4. Finding a Shorter Side | 求直角边

When the hypotenuse and one leg are given, rearrange the equation to find the missing leg.

当已知斜边和一条直角边时，要重新整理方程，求出另一条直角边。

a² = c² − b², so a = √(c² − b²)

Example: A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.

例：一个直角三角形的斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。

a² = 13² − 5² = 169 − 25 = 144

a = √144 = 12 cm

Subtraction must be done before taking the square root.

要先做减法，再开平方。

5. Pythagorean Triples | 勾股数

A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². Recognising common triples can save time in non-calculator papers.

勾股数是一组满足 a² + b² = c² 的三个整数。熟悉常见的勾股数能在不使用计算器的卷子中节省时间。

3, 4, 5	5, 12, 13	7, 24, 25
8, 15, 17	9, 40, 41	12, 35, 37

Multiples of these triples are also triples. For example, 6, 8, 10 is a multiple of 3, 4, 5.

这些勾股数的倍数仍然是勾股数。例如 6、8、10 是 3、4、5 的倍数。

6. The Converse of the Theorem | 勾股定理的逆定理

The converse says: if three sides satisfy a² + b² = c², then the triangle is right-angled.

逆定理的内容是：如果三角形的三条边满足 a² + b² = c²，那么这个三角形是直角三角形。

Example: Does a triangle with sides 6 cm, 8 cm and 10 cm contain a right angle?

例：一个三角形的三边为 6 cm、8 cm 和 10 cm，它是否包含直角？

6² + 8² = 36 + 64 = 100 = 10²

Since the equation holds, the triangle is right-angled, with the 10 cm side as the hypotenuse.

因为等式成立，所以这个三角形是直角三角形，10 cm 的边是斜边。

7. Applications in 2D Coordinates | 在二维坐标中的应用

Pythagoras’ theorem can find the distance between two points on a coordinate plane.

勾股定理可以用来求坐标平面中两点之间的距离。

For points (x₁, y₁) and (x₂, y₂), the distance d satisfies:

对于点 (x₁, y₁) 和 (x₂, y₂)，距离 d 满足：

d² = (x₂ − x₁)² + (y₂ − y₁)²

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Example: Find the distance between (1, 2) and (4, 6).

例：求点 (1, 2) 和点 (4, 6) 之间的距离。

d² = (4 − 1)² + (6 − 2)² = 3² + 4² = 25

d = √25 = 5

8. 3D Problems | 三维问题

In 3D shapes such as cuboids, Pythagoras’ theorem can be applied in two stages to find a space diagonal.

在长方体等三维图形中，可以通过两步使用勾股定理来求空间对角线。

For a cuboid with length l, width w and height h, the space diagonal d is:

对于长 l、宽 w、高 h 的长方体，空间对角线 d 为：

d² = l² + w² + h²

Example: A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of a space diagonal.

例：一个长方体的尺寸为 3 cm、4 cm、12 cm，求空间对角线的长度。

d² = 3² + 4² + 12² = 9 + 16 + 144 = 169

d = √169 = 13 cm

In longer questions, always show the intermediate right-angled triangle you are working with.

在较长的题目中，一定要写出你正在使用的中间直角三角形。

9. Word Problems | 应用题

Many exam problems place Pythagoras’ theorem in a real-life context such as ladders, travels or maps.

许多考试题会把勾股定理放在实际背景中，例如梯子、行程或地图。

Example: A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?

例：一把 5 m 长的梯子靠在竖直墙上，梯脚离墙脚 3 m。梯子能到达墙上多高？

height² = 5² − 3² = 25 − 9 = 16

height = √16 = 4 m

Read the question carefully to decide whether the unknown side is the hypotenuse or a shorter side.

仔细读题，判断未知边是斜边还是直角边。

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

Here are the most common mistakes and the tips you need to avoid them.

以下是最常见的错误，以及帮助你避免这些错误的考试技巧。

Do not use Pythagoras’ theorem on a triangle that is not right-angled.
不要在非直角三角形中使用勾股定理。
Do not confuse the hypotenuse with a leg. The hypotenuse is always opposite the right angle and is the longest side.
不要把斜边和直角边混淆。斜边始终在直角对面，是最长的边。
Do not forget to take the square root at the end.
最后不要忘记开平方。
Do not round too early in multi-step calculations.
在多步骤计算中不要过早四舍五入。
Always label your diagram clearly and write the equation before substituting values.
在图中清楚标记，先写出方程再代入数值。

In the Edexcel IGCSE exam, method marks are important. Show the squared values and the final square root step clearly.

在 Edexcel IGCSE 考试中，方法分很重要。要清楚写出平方值以及最后的开平方步骤。

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Pythagoras’ Theorem | 勾股定理