📚 Pythagoras' Theorem: Right-Angled Triangles and Its Converse | 勾股定理：直角三角形及其逆定理
Pythagoras' theorem is a central topic in the Cambridge KS3 mathematics curriculum, and it returns again and again in IGCSE and A Level work. It applies only to right-angled triangles and gives a relationship between the lengths of the three sides: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Its converse lets us work in the opposite direction - from three side lengths back to a right angle - which turns the theorem into a powerful testing and proving tool as well as a calculating one.
勾股定理是剑桥 KS3 数学课程的核心主题，并在 IGCSE 与 A Level 的学习中反复出现。它只适用于直角三角形，并给出三条边长之间的关系：斜边的平方等于另外两条边的平方和。它的逆定理则让我们反向推理——从三条边长判断是否存在直角——因此它不仅是计算工具，更是判断与证明的有力工具。
1. What is Pythagoras' theorem? | 什么是勾股定理？
In any right-angled triangle, the side opposite the right angle is called the hypotenuse. If the two shorter sides have lengths a and b, and the hypotenuse has length c, then the theorem states the following relationship.
在任何直角三角形中，与直角相对的边叫做斜边。如果两条较短边的长度为 a 和 b，斜边为 c，那么定理表述如下关系。
c² = a² + b²
This formula can be rearranged to find any one of the three sides, provided the other two lengths are known. The theorem is named after the Greek mathematician Pythagoras, although the same relationship was already recorded by Babylonian and Chinese mathematicians more than a thousand years earlier.
这个公式可以变形来求三条边中的任意一条，只要另外两条边已知。该定理以古希腊数学家毕达哥拉斯命名，不过同样的关系早在千余年前就已被巴比伦与中国数学家记录下来。
Two conditions must be satisfied before the theorem may be used. First, the triangle must contain a right angle of exactly 90°. Second, you must correctly identify which side is the hypotenuse, because the hypotenuse is the only side that appears alone on one side of the equation.
使用该定理必须满足两个条件。第一，三角形中必须含有恰好 90° 的直角。第二，你必须正确识别哪条边是斜边，因为斜边是方程中唯一单独出现在一边的量。
2. The hypotenuse and the shorter sides | 斜边与较短直角边
The hypotenuse is always the longest side of a right-angled triangle. It is opposite the largest angle, which is the right angle of 90°. In a diagram the right angle is normally marked with a small square, and the hypotenuse is simply the side that the square does not touch.
斜边总是直角三角形中最长的边。它对着最大的角，即 90° 的直角。在图形中，直角通常用小方块标记，而斜边就是那个小方块没有碰到的那条边。
ce. In the formula c² = a² + b², c must be the hypotenuse, while a and b are the two legs.
两条较短的边通常称为直角边。它们交于直角，标注的顺序可以互换。在公式 c² = a² + b² 中，c 必须是斜边，a 和 b 是两条直角边。
| Term 术语 | Description 说明 |
| Hypotenuse 斜边 | The longest side; it lies opposite the 90° angle 最长边，位于 90° 直角对面 |
| Legs 直角边 | The two shorter sides; they meet at the right angle 两条较短边，交于直角 |
| Right angle 直角 | A 90° angle, usually marked with a small square 90° 角，通常用小方块标记 |
A useful check before calculating is to look at your answer: the hypotenuse you work out must be longer than each of the two legs. If it is not, you have almost certainly substituted the numbers in the wrong place.
计算前有一个实用检查：你算出的斜边必须比两条直角边都长。如果不是，几乎可以肯定你把数字代错了位置。
3. Finding the hypotenuse | 求斜边
To find the hypotenuse, substitute the two shorter sides into the formula, add the squares, and then take the square root of the result. Because the square root is applied to the total, the rearranged form is written as follows.
要求斜边，把两条较短边代入公式，先求平方和，再对结果取平方根。由于平方根作用于整个和，变形后的形式写作如下。
c = √a² + b²
m. This is the smallest Pythagorean triple and it appears in examination questions very frequently.
例如，若 a = 6 cm、b = 8 cm，则 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。这是最小的勾股数组合，在考试题中出现频率极高。
13 m. Again the answer is a whole number, because 5, 12, 13 is another well-known triple.
再看一例：若 a = 5 m、b = 12 m，则 c² = 5² + 12² = 25 + 144 = 169，所以 c = √169 = 13 m。答案同样是整数，因为 5、12、13 是另一组著名的勾股数。
xactly the same way. If a = 3.2 cm and b = 4.7 cm, then c² = 10.24 + 22.09 = 32.33, so c = √32 at the intermediate step; keep the full calculator display and round only at the very end.
小数的情况完全相同。若 a = 3.2 cm、b = 4.7 cm，则 c² = 10.24 + 22.09 = 32.33，所以 c = √32.33 = 5.686… ≈ 5.69 cm。不要在中间步骤取近似值，应保留计算器的完整显示，只在最后一步四舍五入。
bers and square each one. Take the square root and give the answer with units, rounded to the accuracy requested.
每道题都可以遵循三步法：先写出公式 c² = a² + b²；再代入数字并分别平方；最后取平方根，写出带单位的答案，并按题目要求的精度四舍五入。
4. Finding a shorter side | 求一条直角边
When the hypotenuse and one leg are known, the theorem must be rearranged so that the unknown leg is the subject. Subtracting the known leg from the hypotenuse gives the following two equivalent forms.
当已知斜边和一条直角边时，需要把定理变形，使未知的直角边成为主元。从斜边中减去已知直角边，可得到下面两种等价形式。
a² = c² − b² a = √c² − b²
ice that the subtraction is essential: adding 169 and 25 would give 194, which is far too large to be a leg of this triangle.
例如，若斜边 c = 13 cm，一条直角边 b = 5 cm，则 a² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 cm。注意这里的减法是关键：若把 169 与 25 相加得到 194，对这个三角形来说作为直角边显然过大。
base of the wall. The height reached is found from 5² − 3² = 25 − 9 = 16, so the height is √16 = 4 m. The ladder is the hypotenuse, the wall and the ground form the two legs.
一道经典应用题也用到同一思路。一架长 5 m 的梯子斜靠在竖直墙面上，梯脚离墙脚 3 m。梯子顶端的高度由 5² − 3² = 25 − 9 = 16 求得，即高度为 √16 = 4 m。梯子是斜边，墙和地面构成两条直角边。
The most common error in this type of question is to add the two squares instead of subtracting them. A quick mental check works well: the unknown leg must be shorter than the hypotenuse, and the hypotenuse must always sit alone on one side of the equation.
这类题目最常见的错误是把两个平方相加而不是相减。一个快速的判断方法很有用：未知的直角边必须短于斜边，而斜边在方程中必须始终单独位于一边。
5. Pythagorean triples | 勾股数与毕达哥拉斯三元组
e 9 + 16 = 25. Learning a handful of triples by heart saves valuable time in examinations.
勾股数（毕达哥拉斯三元组）是满足 a² + b² = c² 的三个正整数 a、b、c。最小也最著名的一组是 3、4、5，因为 9 + 16 = 25。记住几组常见勾股数，能在考试中节省宝贵时间。
