📚 IGCSE Maths Teacher’s Book Focus: Pythagoras’ Theorem | IGCSE 数学教师用书专题：勾股定理
Pythagoras’ theorem is one of the most important results in IGCSE Mathematics. It links the three sides of a right-angled triangle and is tested frequently in both core and extended papers. This revision guide explains the theorem, shows how to use it to find missing sides, and develops the skills needed for real-life, coordinate and 3D problems.
勾股定理是 IGCSE 数学中最重要的结论之一。它把直角三角形的三条边联系起来，在核心卷和扩展卷中都很常见。本复习指南将解释该定理，展示如何用它求未知边，并培养处理实际生活、坐标和三维问题的技能。
1. Statement of Pythagoras’ Theorem | 勾股定理的表述
In any right-angled triangle, the side opposite the right angle is called the hypotenuse. Pythagoras’ theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
在任何直角三角形中，与直角相对的一边称为斜边。勾股定理指出，斜边的平方等于另外两条边的平方和。
c² = a² + b²
This formula only works when the triangle contains a 90° angle. It is not valid for acute or obtuse triangles.
该公式只在三角形含有 90° 角时成立，对锐角三角形或钝角三角形不适用。
In the formula, a and b usually stand for the two shorter sides, while c stands for the hypotenuse. You can rename the sides, but the hypotenuse must always be alone on one side of the equation.
在公式中，a 和 b 通常表示两条较短的边，而 c 表示斜边。你可以重新命名各边，但斜边必须始终单独位于等式的一侧。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is always the longest side. It is found directly opposite the right angle, never adjacent to it.
斜边总是最长的一边。它位于直角的对面，绝不与直角相邻。
In diagrams, look for the small square marking the right angle; the side across from that square is the hypotenuse. This helps you avoid mixing up the sides before substituting into the formula.
在图中，找到标记直角的小方块，该方块对面的边就是斜边。这有助于你在代入公式前避免把边弄混。
A common misunderstanding is to assume that a vertical side must be the hypotenuse. Only the side opposite the right angle can be the hypotenuse, regardless of how the triangle is drawn.
一个常见的误解是认为竖直的边一定是斜边。只有直角对面的边才能是斜边，与三角形的绘制方向无关。
3. Finding the Hypotenuse | 求斜边
To find the hypotenuse, substitute the two shorter sides into c = √(a² + b²). Square both lengths first, add them, and then take the square root.
求斜边时，把两条较短的边代入 c = √(a² + b²)。先平方各边长度，相加后再开平方。
c = √(a² + b²)
Example: if a = 6 cm and b = 8 cm, then c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
例：若 a = 6 厘米，b = 8 厘米，则 c = √(6² + 8²) = √(36 + 64) = √100 = 10 厘米。
Remember that the hypotenuse must be longer than either of the two shorter sides. If your answer is not the longest side, check your substitution and your calculation.
记住斜边必须比两条直角边中的任何一条都长。如果你的答案不是最长边，请检查代入和计算。
4. Finding a Shorter Side | 求直角边
When the hypotenuse and one shorter side are known, rearrange the formula: a² = c² – b², so a = √(c² – b²). This time you subtract the squares rather than adding them.
当已知斜边和一条直角边时，改写公式：a² = c² – b²，因此 a = √(c² – b²)。这一次你是在做平方相减，而不是相加。
a = √(c² – b²)
Example: if c = 13 cm and b = 5 cm, then a = √(13² – 5²) = √(169 – 25) = √144 = 12 cm.
例：若 c = 13 厘米，b = 5 厘米，则 a = √(13² – 5²) = √(169 – 25) = √144 = 12 厘米。
Always check that the answer is shorter than the hypotenuse; if it is longer, the sides have been mixed up. In a right-angled triangle, the hypotenuse is always the largest value.
务必检查答案是否短于斜边；如果比斜边长，就说明边代错了。在直角三角形中，斜边总是最大的值。
5. Pythagorean Triples | 勾股数
A set of three positive integers that satisfies a² + b² = c² is called a Pythagorean triple. These triples appear regularly in IGCSE questions, especially in non-calculator papers.
满足 a² + b² = c² 的三个正整数称为勾股数。这些数组在 IGCSE 题目中经常出现，尤其是在不使用计算器的试卷中。
The most common triples are (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). Any multiple of a triple, such as (6, 8, 10), also works.
最常见的勾股数有 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。勾股数的任意倍数，如 (6, 8, 10)，也成立。
| Triple | 勾股数 | Check | 验证 |
|---|---|
| 3, 4, 5 | 3²
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