📚 Pythagoras' Theorem for KS3 | 勾股定理（KS3剑桥数学）
Pythagoras' theorem is a cornerstone of Cambridge KS3 mathematics. It gives a direct connection between the three sides of a right-angled triangle, allowing us to calculate an unknown length when two sides are known. The theorem appears in geometry, coordinate work and real-life measurement problems.
勾股定理是剑桥 KS3 数学的基石之一。它建立了直角三角形三条边之间的直接关系，使我们能够在已知两条边时求出未知边。这一定理出现在几何、坐标问题以及现实测量问题中。
1. What Is Pythagoras' Theorem? | 什么是勾股定理？
Pythagoras' theorem states that in a right-angled triangle, 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. The hypotenuse is the side opposite the right angle and is always the longest side.
勾股定理指出，在直角三角形中，以斜边为边长的正方形面积等于以两条直角边为边长的正方形面积之和。斜边是直角所对的边，始终是最长的边。
a² + b² = c²
Here a and b represent the two shorter sides, often called legs, and c represents the hypotenuse.
这里 a 和 b 表示两条较短的边，通常称为直角边，c 表示斜边。
2. Right-Angled Triangles and Labelling | 直角三角形与标记
Before applying the theorem, always identify the right angle and the hypotenuse. The hypotenuse is opposite the right angle. The two remaining sides are perpendicular to each other and can be labelled in either order as a and b.
在应用定理前，首先要确定直角和斜边。斜边位于直角对面。其余两条边互相垂直，可以任意标记为 a 和 b。
In a triangle with vertices A, B and C, if angle C is 90°, then side AB is the hypotenuse. The other sides AC and BC are the legs.
在顶点为 A、B、C 的三角形中，如果角 C 为 90°，那么边 AB 就是斜边。其他两边 AC 和 BC 是直角边。
3. The Formula in Detail | 公式详解
The formula a² + b² = c² means 'a squared plus b squared equals c squared'. Squaring a number means multiplying it by itself. For example, 5² = 5 × 5 = 25.
公式 a² + b² = c² 表示 “a 的平方加 b 的平方等于 c 的平方”。平方是指一个数乘以它自身。例如 5² = 5 × 5 = 25。
To find an unknown side, we often need the inverse operation of squaring, which is taking the square root. The square root of x is written √x. For instance, √25 = 5 because 5² = 25.
要求未知边，我们通常需要平方的逆运算，即开平方根。x 的平方根写作 √x。例如 √25 = 5，因为 5² = 25。
c = √a² + b²
If you are finding the hypotenuse, use the formula above. If you are finding one of the shorter sides, rearrange the formula first.
如果求斜边，使用上面的公式。如果求直角边，则先对公式进行变形。
4. Finding the Hypotenuse | 求斜边
egs of 3 cm and 4 cm. To find the hypotenuse, substitute a = 3 and b = 4 into the formula.
假设一个直角三角形的两条直角边分别为 3 cm 和 4 cm。要求斜边，将 a = 3、b = 4 代入公式。
c² = 3² + 4² = 9 + 16 = 25
Then take the square root: c = √25 = 5 cm. The hypotenuse is 5 cm long.
然后开平方根：c = √25 = 5 cm。斜边长为 5 cm。
Always include the correct unit in your final answer. If the sides were in metres, the hypotenuse would be in metres too.
最终答案要写明正确单位。如果
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