Pythagoras' Theorem | 毕达哥拉斯定理

📚 Pythagoras’ Theorem | 毕达哥拉斯定理

Pythagoras’ theorem is a fundamental rule in geometry that connects the three sides of a right-angled triangle. At KS3 Cambridge level, you are expected to identify the hypotenuse, apply the formula a² + b² = c², and solve problems in both pure and real-life contexts.

毕达哥拉斯定理是几何中连接直角三角形三条边的基本法则。在 KS3 剑桥阶段，你需要识别斜边、应用公式 a² + b² = c²，并在纯数学和实际情境中解决问题。

1. The Hypotenuse | 认识斜边

In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side because it faces the largest angle, which is 90°.

在直角三角形中，直角所对的边称为斜边。斜边总是最长的边，因为它对着最大的角，即 90°。

The two shorter sides are often labelled a and b. They form the right angle and are sometimes called the legs of the triangle. Getting this labelling right is the first step before using the theorem.

两条较短的边通常标记为 a 和 b。它们构成直角，有时被称为三角形的直角边。正确标记这些边是使用定理前的第一步。

You may see right-angled triangles drawn in different orientations, such as with the hypotenuse at the top or sloping downwards. The key is not the position, but the fact that the hypotenuse is always opposite the marked right angle.

你可能会看到直角三角形以不同方向绘制，例如斜边在顶部或向下倾斜。关键不在于位置，而在于斜边总是与标记的直角相对。

2. Statement of the Theorem | 定理表述

Pythagoras’ theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

毕达哥拉斯定理指出，在任何直角三角形中，斜边的平方等于另外两条边的平方之和。

a² + b² = c²

Here c represents the hypotenuse, while a and b represent the two shorter sides. The formula works only for right-angled triangles, so always check the right angle first.

这里 c 表示斜边，而 a 和 b 表示两条较短的边。该公式只适用于直角三角形，因此一定要先确认直角。

If the triangle is not right-angled, this relationship does not hold. Do not use the formula for acute or obtuse triangles unless you have first split them into right-angled parts.

如果三角形不是直角三角形，这种关系不成立。除非先将锐角三角形或钝角三角形分割成直角三角形，否则不要使用该公式。

3. Finding the Hypotenuse | 求斜边

To find the hypotenuse, substitute the two shorter side lengths into a² + b² = c², add the squares, and then take the square root.

要求斜边，将两条较短的边长代入 a² + b² = c²，求出平方和，然后开平方根。

Example: Find c when a = 6 cm and b = 8 cm.

例子：当 a = 6 cm、b = 8 cm 时，求 c。

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

Always include the correct unit in your final answer. If the square is not perfect, round the square root to a suitable degree of accuracy. For example, if c² = 110, then c = √110, which is approximately 10.5 to 1 decimal place.

最终答案必须包含正确单位。如果平方数不是完全平方数，请将平方根四舍五入到合适的精度。例如，如果 c² = 110，则 c = √110，约为 10.5，保留 1 位小数。

4. Finding a Shorter Side | 求直角边

If the hypotenuse and one shorter side are known, rearrange the formula to find the missing leg. Subtract the square of the known side from the square of the hypotenuse.

如果已知斜边和一条直角边，可以重新整理公式来求缺失的直角边。用斜边的平方减去已知边的平方。

a² = c² – b²

Example: Find b when c = 13 cm and a = 5 cm.

例子：当 c = 13 cm、a = 5 cm 时，求 b。

b² = 13² – 5² = 169 – 25 = 144

b = √144 = 12 cm

This rearranged form is essential because the missing side is not always the hypotenuse. Be careful not to add when you should subtract, as this is a very common error.

这种重新整理的形式非常重要，因为缺失的边并不总是斜边。注意不要在该用减法时用了加法，这是一个非常常见的错误。

5. Checking for Right-Angled Triangles | 判断直角三角形

The converse of Pythagoras’ theorem can be used to test whether a triangle is right-angled. If the three side lengths satisfy a² + b² = c², where c is the longest side, the triangle must be right-angled.

毕达哥拉斯定理的逆定理可用于检验一个三角形是否为直角三角形。如果三条边长满足 a² + b² = c²，其中 c 是最长边，则该三角形一定是直角三角形。

Example: Do sides 9 cm, 12 cm and 15 cm form a right-angled triangle?

例子：边长 9 cm、12 cm 和 15 cm 能否构成直角三角形？

9² + 12² = 81 + 144 = 225 = 15²

Since the equation holds, the triangle is right-angled. If the two sides do not add up to the square of the longest side, the triangle is not right-angled.

由于等式成立，该三角形是直角三角形。如果两边平方之和不等于最长边的平方，则该三角形不是直角三角形。

This check is useful in construction, navigation and proof questions where you may need to justify that an angle is exactly 90°.

这种检验在建筑、航海和证明题中非常有用，在这些题目中你可能需要证明某个角恰好是 90°。

6. Pythagorean Triples | 毕达哥拉斯三元组

A Pythagorean triple is a set of three positive integers that satisfy a² +

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Pythagoras’ Theorem | 毕达哥拉斯定理