📚 IGCSE Maths G-2: Pythagoras' Theorem and Trigonometry | IGCSE 数学 G-2：勾股定理与三角函数比
Pythagoras' theorem and trigonometry are two of the most powerful tools in IGCSE mathematics. They connect the side lengths and angles of right-angled triangles, and they appear in topics as varied as coordinate geometry, bearings, vectors and 3D shapes. This article explains the core ideas, common methods and exam strategies needed for the G-2 topic.
勾股定理与三角学是 IGCSE 数学中两个极为重要的工具，它们把直角三角形的边长与角联系起来，并广泛出现在坐标几何、方位角、向量和三维图形等专题中。本文解释 G-2 主题所需的核心概念、常用方法和考试策略。
1. Right-Angled Triangles and Labelling | 直角三角形与边角标记
A right-angled triangle has one angle of exactly 90°. The longest side is always opposite the right angle and is called the hypotenuse. The other two sides are usually labelled 'opposite' and 'adjacent' relative to a chosen acute angle θ.
直角三角形有一个恰好为 90° 的角。最长边总是对着直角，称为斜边。另外两条边相对于选定的锐角 θ 通常标记为“对边”和“邻边”。
The labels change when the chosen angle changes, so always draw or mark the angle before writing anything else. In a diagram, you could use capital letters for vertices and lower-case letters for the sides opposite those vertices.
当所选的角改变时，边的标记也会随之改变，因此一定要先画出或标出该角，再写其他内容。在图中，可以用大写字母表示顶点，用小写字母表示这些顶点所对的边。
- Hypotenuse | 斜边：the side opposite the right angle, always the longest side.
- Opposite | 对边：the side opposite the chosen angle θ.
- Adjacent | 邻边：the side next to the chosen angle θ, but not the hypotenuse.
2. Pythagoras' Theorem: The Core Rule | 勾股定理：核心规则
For any right-angled triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. In symbols, if c is the hypotenuse and a, b are the two shorter sides:
对于任何直角三角形，斜边上的正方形面积等于另外两条边上的正方形面积之和。用符号表示，如果 c 是斜边，a 和 b 是两条直角边，则：
a² + b² = c²
This theorem only works when the triangle has a right angle. You cannot use it on acute or obtuse triangles unless you first create a right-angled triangle using a perpendicular line.
该定理只适用于有直角的三角形。对于锐角三角形或钝角三角形，不能直接使用；除非先作一条垂线构造出直角三角形。
For example, if a = 3 and b = 4, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. This is the classic 3-4-5 triangle.
例如，如果 a = 3，b = 4，则 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5。这就是经典的 3-4-5 三角形。
3. Finding the Hypotenuse | 求斜边
To find the hypotenuse, square the two known shorter sides, add them, and then take the square root. The calculation follows directly from the theorem:
求斜边时，先把两条已知直角边分别平方，再相加，最后取平方根。计算直接来自定理：
c = √a² + b²
For example, if a = 6 cm and b = 8 cm, then c = √6² + 8² = √36 + 64 = √100 = 10 cm. The hypotenuse is 10 cm.
例如，如果 a = 6 cm，b = 8 cm，则 c = √6² + 8² = √36 + 64 = √100 = 10 cm。斜边为 10 cm。
Always check that your final hypotenuse is longer than both of the shorter sides. If it is not, you have probably made an arithmetic or labelling error.
始终检查最终求得的斜边是否大于两条直角边。如果不是，很可能出现了计算或标记错误。
4. Finding a Shorter Side | 求直角边
To find a shorter side, rearrange Pythagoras' theorem so that the shorter side is the subject. If a is the unknown shorter side, then:
求直角边时，要把勾股定理变形，使所求直角边成为主项。如果 a 是未知直角边，则：
a = √c² − b²
For example, if the hypotenuse c = 13 m and one shorter side b = 5 m, then a = √13² − 5² = √169 − 25 = √144 = 12 m.
例如，如果斜边 c = 13 m，一条直角边 b = 5 m，则 a = √13² − 5² = √169 − 25 = √144 = 12 m。
Notice that you subtract the squares, not add them. Subtracting the wrong way will give an invalid square root, so always subtract the known shorter side squared from the hypotenuse squared.
注意这里要把两个平方相减，而不是相加。减错方向会导致根号下出现负数，所以一定要用斜边的平方减去已知直角边的平方。
5. The Converse of Pythagoras' Theorem | 勾股定理的逆定理
side, then the triangle is right-angled, and the right angle is opposite the longest side.
如果一个三角形的三条边满足 a² + b² = c²，其中 c 是最长边，那么这个三角形就是直角三角形，且直角对着最长边。
This result is useful for checking whether a corner is exactly 90°, for example in construction, navigation or coordinate geometry. You only need the three side lengths.
这个结论在检查某个角是否恰好为 90° 时非常有用，例如在建筑、导航或坐标几何中。只需要三条边的长度即可判断。
des 5, 12 and 13 satisfy 5² + 12² = 25 + 144 = 169 = 13², so the triangle is right-angled.
例如，边长 5、12 和 13 满足 5² + 12² = 25 + 144 = 169 = 13²，所以该三角形是直角三角形。
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