📚 Pythagoras' Theorem | 毕达哥拉斯定理
Pythagoras' theorem is one of the most important results in KS3 Cambridge mathematics. It connects the three sides of a right-angled triangle and allows you to calculate an unknown side length when the other two sides are known.
毕达哥拉斯定理是 KS3 剑桥数学中最重要的定理之一。它把直角三角形的三条边联系起来，使你能在已知另外两条边的情况下计算未知边的长度。
1. What is Pythagoras' theorem? | 什么是毕达哥拉斯定理？
Pythagoras' theorem describes a special relationship between the sides of a right-angled triangle. It states that the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides.
毕达哥拉斯定理描述了直角三角形三条边之间的特殊关系。它指出，以斜边为边画出的正方形面积等于以另外两条边为边画出的正方形面积之和。
a² + b² = c²
Here, c is the hypotenuse, while a and b are the two shorter sides called the legs.
这里，c 是斜边，而 a 和 b 是两条较短的边，称为直角边。
2. Identifying the hypotenuse | 识别斜边
Always start by locating the right angle. The hypotenuse is the side directly opposite the right angle. It is also the longest side in any right-angled triangle.
解题时先找到直角。斜边就是正对直角的边，也是任何直角三角形中最长的边。
For example, in a triangle with sides 5 cm, 12 cm and 13 cm, the hypotenuse must be 13 cm because 13 is the largest number and is opposite the 90° angle.
例如，在一个边长分别为 5 cm、12 cm 和 13 cm 的三角形中，斜边一定是 13 cm，因为 13 是最大的数，并且对着 90° 的角。
Before using the formula, always check that you have correctly labelled the hypotenuse as c. The other two sides can be labelled a and b in either order.
在使用公式之前，一定要检查你是否把斜边正确标为 c。另外两条边可以随意标为 a 和 b，顺序不限。
3. The formula and its meaning | 公式及其含义
ections. You can find the hypotenuse if you know both shorter sides, or find a shorter side if you know the hypotenuse and one other side.
公式 c² = a² + b² 可以双向使用。如果知道两条直角边，可以求斜边；如果知道斜边和一条直角边，也可以求另一条直角边。
The squares are important: you must square each length before adding or subtracting, then take the square root of the final result.
平方运算非常重要：你必须先把每条边平方，再进行加法或减法，最后对结果取平方根。
| Quantity | Meaning | Unit |
|---|---|---|
| a² | Square of one shorter side | cm², m², etc. |
| b² | Square of the other shorter side | cm², m², etc. |
| c² | Square of the hypotenuse | cm², m², etc. |
The side length itself has the original unit, such as cm or m, because you take the square root at the end.
边长本身的单位就是原来的单位，如 cm 或 m，因为最后你取了平方根。
4. Finding the hypotenuse | 求斜边
e: A right-angled triangle has shorter sides a = 6 cm and b = 8 cm. Find the hypotenuse c.
例题：一个直角三角形的两条直角边分别为 a = 6 cm 和 b = 8 cm。求斜边 c。
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
Always take the positive square root because a length cannot be negative.
一定要取正的平方根，因为长度不能为负。
Another example: If a = 5 m and b = 12 m, then c² = 25 + 144 = 169, so c = √169 = 13 m.
另一个例子：如果 a = 5 m，b = 12 m，那么 c² = 25 + 144 = 169，所以 c = √169 = 13 m。
5. Finding a shorter side | 求直角边
b². Subtract the square of the known leg from the square of the hypotenuse, then take the square root.
当已知斜边和一条直角边时，把公式变形为 a² = c² − b²。用斜边的平方减去已知直角边的平方，然后再取平方根。
ght-angled triangle has hypotenuse c = 13 cm and one leg b = 5 cm. Find the missing leg a.
例题：一个直角三角形的斜边 c = 13 cm，一条直角边 b = 5 cm。求另一条直角边 a。
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 cm
Be careful: when finding a shorter side you must subtract, not add. This is one of the most common errors in KS3 tests.
注意：求直角边时必须用减法，而不是加法。这是 KS3 考试中最常见的错误之一。
6. Checking for a right-angled triangle | 判断是否为直角三角形
The converse of Pythagoras' ngle satisfy a² + b² = c², where c is the longest side, then the triangle is right-angled.
毕达哥拉斯定理的逆定理指出，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。
Example: Are sides 9 cm, 12 cm and 15 cm the sides of a right-angled triangle? Check: 9² + 12² = 81 + 144 = 225, and 15² = 225, so the answer is yes.
例子：边长为 9 cm、12 cm 和 15 cm 的三角形是直角三角形吗？检验：9² + 12² = 81 + 144 = 225，而 15² = 225，所以答案是肯定的。
m, 7 cm and 10 cm do not form a right-angled triangle because 6² + 7² = 85, but 10² = 100.
如果等式不成立，那么这个三角形就不是直角三角形。例如，6 cm、7 cm 和 10 cm 不能组成直角三角形，因为 6² + 7² = 85，而 10² = 100。
7. Pythagorean triples | 毕达哥拉斯三元组
bers that satisfy a² + b² = c². Recognising common triples can save time in KS3 questions.
毕达哥拉斯三元组是满足 a² + b² = c² 的三个正整数组合。认识常见的三元组可以让你在 KS3 考试中节省时间。
| a | b | c | Check |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144
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