📚 Pythagoras and Trigonometry in Right-Angled Triangles | 直角三角形中的毕达哥拉斯定理与三角学
This revision guide covers the IGCSE Mathematics topic of Pythagoras' theorem and right-angled triangle trigonometry. You will learn how to calculate missing sides, find unknown angles, solve real-world problems and avoid common errors.
本复习指南涵盖 IGCSE 数学中毕达哥拉斯定理与直角三角形三角学这一主题。你将学习如何计算缺失边长、求未知角、解决实际问题，并避免常见错误。
1. Right-Angled Triangle Basics | 直角三角形基础
A right-angled triangle contains one angle of exactly 90°, called the right angle. The side opposite this angle is the hypotenuse, which is always the longest side. The other two sides are called legs or shorter sides. In IGCSE problems, you will often need to identify the hypotenuse before applying Pythagoras or trigonometric ratios.
直角三角形包含一个恰好为 90° 的角，称为直角。直角所对的边是斜边，它总是最长的边。另外两条边称为直角边或较短边。在 IGCSE 题目中，在应用毕达哥拉斯定理或三角比之前，通常需要先识别斜边。
When labelling a triangle for trigonometry, choose one acute angle as the reference angle θ. The side next to θ but not the hypotenuse is the adjacent side; the side across from θ is the opposite side. These labels change if you choose the other acute angle.
在为三角学标记三角形时，选择一个锐角作为参考角 θ。与 θ 相邻但不是斜边的边是邻边；θ 所对的边是对边。如果选择另一个锐角，这些标记会相应改变。
2. Pythagoras' Theorem | 毕达哥拉斯定理
Pythagoras' f the hypotenuse has length c and the other sides have lengths a and b, then a² + b² = c².
毕达哥拉斯定理指出，在任何直角三角形中，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。用代数形式表示，如果斜边长为 c，另两条边长为 a 和 b，则 a² + b² = c²。
a² + b² = c²
This formula only applies to right-angled triangles. If the triangle is not right-angled, do not use Pythagoras' theorem. You must first confirm that the angle between sides a and b is 90°.
该公式只适用于直角三角形。如果三角形不是直角三角形，不要使用毕达哥拉斯定理。必须首先确认边 a 和 b 之间的夹角为 90°。
3. Finding the Hypotenuse | 求斜边
if the legs measure 6 cm and 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
要求斜边，将两条较短边代入 a² + b² = c²，然后对结果取平方根。例如，如果两条直角边分别为 6 cm 和 8 cm，则 c² = 6² + 8² = 36 + 64 = 100，因此 c = √100 = 10 cm。
Common Pythagorean triples include 3-4-5, 5-12-13, 7-24-25 and 8-15-17. Any multiple of these also works, such as 6-8-10 or 9-12-15. Recognising these triples can save time in exams.
常见的毕达哥拉斯三元组包括 3-4-5、5-12-
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