📚 Pythagoras' Theorem: Finding Missing Sides | 勾股定理：求缺失边长
Pythagoras' se using the formula a² + b² = c², and apply it to a variety of Cambridge-style questions.
勾股定理是 KS3 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，帮助我们解决涉及长度、距离和实际测量的问题。本文将逐步解释这个定理，练习使用公式 a² + b² = c²，并将其应用到各种剑桥风格的题目中。
1. What is Pythagoras' Theorem? | 什么是勾股定理？
Pythagoras' theorem applies only to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
勾股定理只适用于直角三角形。它指出：斜边的平方等于另外两条边的平方和。
a² + b² = c²
Here, c represents the hypotenuse, which is the longest side opposite the right angle. The letters a and b stand for the two shorter sides, often called the legs.
这里，c 代表斜边，也就是直角对面的最长边。字母 a 和 b 代表两条较短的边，通常称为直角边。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is always the side opposite the right angle. It is also the longest side of a right-angled triangle.
斜边总是直角对面的边。它也是直角三角形中最长的一条边。
For example, in a triangle with sides 6 cm, 8 cm, and 10 cm, the 10 cm side must be the hypotenuse because it is opposite the largest angle (90°) and is longer than 6 cm and 8 cm.
例如，在一个边长为 6 cm、8 cm 和 10 cm 的三角形中，10 cm 的边一定就是斜边，因为它对着最大的角（90°），并且比 6 cm 和 8 cm 都长。
Labelling the hypotenuse as c is a good habit before using the formula.
在使用公式之前，养成把斜边标记为 c 的好习惯。
3. The Formula in Action: Finding the Hypotenuse | 公式应用：求斜边
shorter sides of 3 cm and 4 cm. To find the hypotenuse c, we substitute into a² + b² = c².
假设一个直角三角形的两条直角边分别为 3 cm 和 4 cm。为了求斜边 c，我们代入 a² + b² = c²。
Step 1: square each shorter side: 3² = 9 and 4² = 16.
第一步：将每条直角边平方：3² = 9，4² = 16。
Step 2: add the squares: 9 + 16 = 25.
第二步：把平方相加：9 + 16 = 25。
Step 3: take the positive square root: √25 = 5.
第三步：取正的平方根：√25 = 5。
c = √3² + 4² = √25 = 5 cm
The hypotenuse is 5 cm. This is the classic 3-4-5 triangle.
斜边为 5 cm。这就是经典的 3-4-5 三角形。
4. Finding a Shorter Side | 求一条直角边
and one shorter side. To find the other shorter side, rearrange the formula: a² = c² - b².
有时你已知斜边和一条直角边。为了求另一条直角边，需要调整公式：a² = c² - b²。
Example: a ladder of length 10 m leans against a wall. The foot of the ladder is 6 m from the wall. Find the height reached on the wall.
例子：一架长 10 m 的梯子斜靠在墙上。梯子脚离墙 6 m。求梯子在墙上达到的高度。
e c = 10 m, b = 6 m, so the height a is found by a² = 10² - 6² = 100 - 36 = 64, giving a = √64 = 8 m.
这里 c = 10 m，b = 6 m，所以高度 a 由 a² = 10² - 6² = 100 - 36 = 64 求得，因此 a = √64 = 8 m。
a = √c² - b² = √10² - 6² = 8 m
Always subtract the square of the known shorter side from the square of the hypotenuse, never the other way round.
一定要用斜边的平方减去已知直角边的平方，千万不要反过来减。
5. Pythagorean Triples | 勾股数
Some sets of whole numbers fit the theorem exactly. These are called Pythagorean triples.
有些整数组成的集合刚好满足勾股定理。这些数称为勾股数。
Common Pythagorean triples:
常见勾股数：
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 8 | 15 | 17 |
Recognising these triples saves time in exams. For example, if you see sides 5, 12, and 13, you can immediately say the triangle is right-angled.
在考试中识别这些勾股数可以节省时间。例如，如果你看到边长为 5、12 和 13，就可以立刻判断这个三角形是直角三角形。
6. Using the Theorem in Real-Life Problems | 实际问题中的应用
Many Cambridge KS3 questions present Pythagoras in context: a ladder, a sail, a diagonal of a rectangle, or a journey north then east.
许多剑桥 KS3 题目会在实际情境中考查勾股定理：梯子、船帆、矩形的对角线，或者先向北再向东的行程。
Example: A boat sails 9 km north and then 12 km east. How far is it from its starting point?
例子：一艘船向北航行 9 km，再向东航行 12 km。它离出发点有多远？
The two legs are 9 km and
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