📚 Pythagoras' Theorem Made Easy for KS3 Cambridge Maths | 毕达哥拉斯定理：剑桥KS3数学轻松掌握
Welcome to this KS3 Cambridge Mathematics revision guide. Pythagoras' theorem is one of the most powerful tools for solving problems involving right-angled triangles. You will meet it in geometry, measurement, coordinate work and many real-life contexts. This guide is designed to build your understanding step by step from the basic idea to harder exam-style questions.
欢迎阅读这份 KS3 剑桥数学复习指南。毕达哥拉斯定理是解决直角三角形问题的强大工具之一。你将在几何、测量、坐标以及许多实际情境中用到它。本指南旨在从基本概念到较难的考试题型，逐步建立你的理解。
1. What is Pythagoras' Theorem? | 什么是毕达哥拉斯定理？
Pythagoras' her two sides. This gives the famous relationship a² + b² = c², where c is the hypotenuse.
毕达哥拉斯定理指出：在任何一个直角三角形中，斜边上的正方形面积等于另外两条直角边上的正方形面积之和。由此得到著名关系式 a² + b² = c²，其中 c 是斜边。
a² + b² = c²
The letters a, b and c are not fixed: a and b stand for the two shorter sides, and c always stands for the longest side, which is opposite the right angle. The theorem only works for right-angled triangles, so always check for the right angle before using it.
字母 a、b、c 并不是固定的：a 和 b 表示两条较短的边，c 始终表示最长的边，也就是直角对面的那条边。这个定理只适用于直角三角形，因此在使用之前一定要检查是否有直角。
2. The Hypotenuse and Legs | 斜边与直角边
The hypotenuse is the longest side of a right-angled triangle. It is always opposite the right angle. The other two sides are called the legs, catheti, or simply the shorter sides. Knowing which side is the hypotenuse is the most important first step.
斜边是直角三角形中最长的边，它总是位于直角的对面。另外两条边称为直角边，或者简称较短的边。知道哪条边是斜边是最重要的第一步。
You can locate the hypotenuse by finding the side that does not touch the right angle. In diagrams, the hypotenuse is usually drawn sloping or diagonal, but the key is that it faces the 90° corner. If you label the hypotenuse incorrectly, the whole calculation will be wrong.
你可以通过寻找不与直角相接的那条边来定位斜边。在图中，斜边通常画成倾斜或对角方向，但关键是它正对着 90° 的角。如果把斜边标错了，整个计算都会出错。
3. The Formula a² + b² = c² | 公式 a² + b² = c²
Once you know which side is the hypotenuse, label its length as c. Label the two shorter sides as a and b. The theorem says that adding the squares of a and b gives the square of c. Squaring means multiplying a number by itself.
一旦你知道哪条边是斜边，就把它的长度标为 c。把两条较短的边标为 a 和 b。定理表明：a 与 b 的平方相加等于 c 的平方。平方意味着一个数乘以它本身。
c² = a² + b²
is a direct result of subtracting the square of one leg from the square of the hypotenuse.
你可以把这个公式改写成两种有用的形式：要求直角边，使用 a² = c² - b² 或 b² = c² - a²。这是斜边的平方减去一条直角边的平方所得到的直接结果。
4. Finding the Hypotenuse | 求斜边
le with shorter sides 6 cm and 8 cm. To find the hypotenuse, substitute into a² + b² = c²:
假设一个直角三角形，两条直角边分别为 6 厘米和 8 厘米。要求斜边，代入 a² + b² = c²：
6² + 8² = c² → 36 + 64 = c² → 100 = c²
re root at the end. A common mistake is to stop at c² = 100 and give the answer as 100 cm.
两边同时开平方：c = √100 = 10 厘米。因此斜边长为 10 厘米。始终记住最后要开平方。一个常见错误是算到 c² = 100 后就把答案写成 100 厘米。
5. Finding a Shorter Side | 求直角边
the formula to find the missing leg. For example, suppose c = 13 cm and a = 5 cm. Find b.
当你知道斜边和一条直角边时，可以重新整理公式来求缺失的直角边。例如，假设 c = 13 厘米，a = 5 厘米，求 b。
b² = c² - a² = 13² - 5² = 169 - 25 = 144
hen subtract, and finally take the square root. Doing 13 - 5 = 8 first would be incorrect.
然后 b = √144 = 12 厘米。这是一个经典的 5-12-13 三角形。你必须先平方，再相减，最后开平方。如果先算 13 - 5 = 8 就是错误的。
6. Real-Life Applications | 实际应用
Pythagoras' theorem helps solve practical problems such as finding the length of a ladder leaning against a wall, the diagonal of a rectangular field, or the shortest distance between two points. In these cases, the right angle is often implicit in the situation.
毕达哥拉斯定理有助于解决实际问题，例如求靠墙梯子的长度、矩形场地的对角线长度，或两点之间的最短距离。在这些情况下，直角往往隐含在情境之中。
For a ladder 5 m from a wall reaching 12 m up the wall, the ladder length is √5² + 12² am for word problems. Mark the right angle and label known sides before using the formula.
如果一个梯子底部离墙 5 米，顶端在墙上 12 米高处，那么梯子长度为 √5² + 12² = √169 = 13 米。对于文字题，始终先画出草图。标出直角并标注已知边长，然后再使用公式。
7. Pythagorean Triples | 毕达哥拉斯三元组
4-5 because 3² + 4² = 9 + 16 = 25 = 5². Spotting these triples can save time in questions.
毕达哥拉斯三元组是一组满足 a² + b² = c² 的整数。最小也最著名的是 3-4-5，因为 3² + 4² = 9 + 16 = 25 = 5²。识别这些三元组可以节省做题时间。
Other useful triples include 5-12-13, 7-24-25 and 8-15-17. If you multiply all sides by the same factor, you get another valid triple, for example 6-8-10 is double 3-4-5.
其他常用的三元组包括 5-12-13、7-24-25 和 8-15-17。如果把所有边乘以相同的倍数，你会得到另一个有效的三元组，例如 6-8-10 是 3-4-5 的两倍。
8. Using the Theorem in Coordinate Problems | 坐标系中的应用
You can use Pythagoras' theorem to find the distance between two points on a coordinate grid. Draw a right triangle with the distance as the hypotenuse and the horizontal and vertical changes as the legs. This technique is the foundation of coordinate geometry.
你可以使用毕达哥拉斯定理来求坐标网格上两点之间的距离。画一个直角三角形，将距离作为斜边，水平变化和垂直变化作为直角边。这种方法是坐标几何的基础。
(4, 6), the horizontal leg is 4 - 1 = 3 and the vertical leg is 6 - 2 = 4. The distance is √3² + 4² = √25 = 5. Always subtract the coordinates in the same order to avoid negative lengths.
对于点 (1, 2) 和 (4, 6)，水平直角边为 4 - 1 = 3，垂直直角边为 6 - 2 = 4。距离为 √3² + 4² = √25 = 5。始终以相同顺序相减坐标，以避免出现负长度。
9. Common Mistakes to Avoid | 常见错误
One common error is labelling the longest side incorrectly. Always find the hypotenuse first and label it c before writing any equation. Another mistake is forgetting to square the numbers
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