Pythagoras’ Theorem: Finding Missing Sides and Checking Right Angles | 勾股定理：求未知边与验证直角

📚 Pythagoras’ Theorem: Finding Missing Sides and Checking Right Angles | 勾股定理：求未知边与验证直角

Pythagoras’ theorem is a central topic in the Cambridge KS3 mathematics curriculum. It links algebra with geometry and gives us a way to calculate unknown lengths in right-angled triangles. This guide explains the rule clearly, shows step-by-step methods, and helps you avoid the most common mistakes in tests.

勾股定理是剑桥KS3数学课程的核心主题。它把代数与几何联系起来，为我们提供了一种计算直角三角形未知边长的方法。本指南清晰地解释这一定理，展示分步方法，并帮助你避免考试中最常见的错误。

1. The Right-Angled Triangle and Its Sides | 直角三角形及其边

A right-angled triangle is a triangle with one interior angle exactly equal to 90°. The longest side is always opposite this right angle and is called the hypotenuse.

直角三角形是一个内角恰好等于90°的三角形。最长边总是位于直角的对面，称为斜边。

The two sides that form the right angle are known as the shorter sides or legs. In most questions they are labelled a and b, while the hypotenuse is labelled c. When you draw a diagram, mark the right angle first and then place c opposite it.

构成直角的两条边称为较短边或直角边。在大多数题目中，它们被标记为 a 和 b，斜边标记为 c。画图时，先标出直角，然后把 c 放在直角的对面。

Always identify the hypotenuse before substituting values into the formula. A common mistake is to place a shorter side in the position of c, which leads to an incorrect result.

在代入公式之前，务必先确定斜边。一个常见错误是把较短边放到 c 的位置，这会导致结果错误。

2. The Rule: a² + b² = c² | 定理：a² + b² = c²

Pythagoras’ theorem states that in any right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

勾股定理指出：在任意直角三角形中，两条较短边的平方和等于斜边的平方。

a² + b² = c²

Here a and b are the lengths of the shorter sides, and c is the length of the hypotenuse. The theorem only works when the triangle contains a right angle. If a triangle is not right-angled, this relationship does not hold.

这里 a 和 b 是两条较短边的长度，

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