📚 Pythagoras’ Theorem: Finding Missing Sides in Right-Angled Triangles | 毕达哥拉斯定理：求直角三角形中的未知边
This revision article targets a key skill from Cambridge KS3 Mathematics, often tested in resource p246_1: using Pythagoras’ theorem to find unknown sides in right-angled triangles. The theorem is one of the most useful tools in geometry and appears frequently in tests, real-life problems and later IGCSE work.
这篇复习文章针对 Cambridge KS3 数学中常见于资源 p246_1 的重要技能：运用毕达哥拉斯定理求直角三角形中的未知边。该定理是几何中最有用的工具之一，经常出现在测试、实际问题以及未来的 IGCSE 学习中。
1. Right-Angled Triangles and the Hypotenuse | 直角三角形与斜边
A right-angled triangle has one angle equal to 90°. The side directly opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are called the shorter sides or legs.
直角三角形有一个等于 90° 的角。直角的对边总是最长边，称为斜边。另外两条边称为较短的边或直角边。
Before using Pythagoras’ theorem, you must be able to identify the hypotenuse confidently. It is never next to the right angle, and it is always longer than either of the other two sides.
在使用毕达哥拉斯定理之前，你必须能够准确地识别斜边。斜边绝不紧邻直角，且总是长于另外两条边中的任意一条。
2. The Theorem in Symbols | 定理的符号表达
If a and b represent the two shorter sides and c represents the hypotenuse, then Pythagoras’ theorem states the following relationship:
如果用 a 和 b 表示两条直角边，用 c 表示斜边，那么毕达哥拉斯定理表示如下关系：
a² + b² = c²
This equation tells us that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. The relationship only works for right-angled triangles.
这个等式告诉我们，两条直角边的平方和等于斜边的平方。这种关系只在直角三角形中成立。
3. Finding the Hypotenuse | 求斜边
When you are given the two shorter sides, you can find the hypotenuse by squaring both shorter sides, adding the results, and then taking the square root.
当已知两条直角边时，你可以先将两条直角边分别平方，相加后再开平方，从而求出斜边。
Worked example: Find the hypotenuse when the shorter sides are 6 cm and 8 cm.
例题：当直角边分别为 6 cm 和 8 cm 时，求斜边。
c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm
The hypotenuse is exactly 10 cm. Notice that the final step is to take the square root, not to leave the answer as 100.
斜边正好是 10 cm。请注意最后一步是开平方，而不是把答案写成 100。
4. Finding a Shorter Side | 求直角边
If you know the hypotenuse and one shorter side, rearrange the theorem to subtract instead of add. For example, if you know c and a, then the missing side b is found by:
如果已知斜边和一条直角边，则需要将定理变形为减法而不是加法。例如，已知 c 和 a，那么未知边 b 可由下式求得：
b = √(c² – a²)
Worked example: The hypotenuse is 13 cm and one shorter side is 5 cm. Find the other shorter side.
例题：斜边为 13 cm，一条直角边为 5 cm。求另一条直角边。
b = √(13² – 5²) = √(169 – 25) = √144 = 12 cm
The missing shorter side is 12 cm. Subtracting is essential here because the hypotenuse is the largest quantity.
未知直角边为 12 cm。这里必须用减法，因为斜边是最大的量。
5. Checking for a Right Angle | 判断直角
Pythagoras’ theorem can also be used to test whether a triangle is right-angled. If the three sides satisfy the equation a² + b² = c², where c is the longest side, then the triangle is right-angled.
毕达哥拉斯定理还可以用来检验一个三角形是否为直角三角形。如果三条边满足方程 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: Consider sides 7 cm, 24 cm and 25 cm. Check the longest side, 25 cm, against the other two.
例子：考虑三边 7 cm、24 cm 和 25 cm。用最长边 25 cm 来检验另外两边。
7² + 24² = 49 + 576 = 625 = 25²
Since both sides of the equation are equal, the triangle is right-angled. If the two sides are not equal, the triangle is not right-angled.
由于等式两边相等，该三角形是直角三角形。如果两边不相等，则该三角形不是直角三角形。
6. Pythagorean Triples | 毕达哥拉斯三元组
Some whole-number sets satisfy Pythagoras’ theorem exactly. These sets are called Pythagorean triples. Recognising them can save time in KS3 tests and mental calculations.
一些整数组恰好满足毕达哥拉斯定理。这些整数组称为毕达哥拉斯三元组。识别它们可以在 KS3 测试和心算中节省时间。
| Triple 三元组 | Check 检验 |
| 3, 4, 5 | 3² + 4² = 9 + 16 = 25 = 5² |
| 5, 12, 13 | 5² + 12² = 25 + 144 = 169 = 13² |
| 7, 24, 25 | 7² + 24² = 49 + 576 = 625 = 25² |
If you see two sides from a triple, you can often write down the third side without recalculating everything from scratch.
如果你看到三元组中的两条边，通常可以直接写出第三条边，而无需从头重新计算。
7. Applying Pythagoras in Real Life | 实际应用
The theorem is useful for distances, construction, navigation and any situation where a right-angled triangle can be drawn. A ladder leaning against a wall is a classic example.
该定理在距离、建筑、导航以及任何可以画出直角三角形的情形中都非常有用。梯子靠墙是一个典型的例子。
Example: A 5 m ladder is placed 3 m away from a wall. Find the height reached on the wall.
例子：一把 5 m 的梯子底端离墙 3 m，求梯子顶端在墙上的高度。
h = √(5² – 3²) = √(25 – 9) = √16 = 4 m
The ladder reaches a height of 4 m on the wall. The ladder itself is the hypotenuse because it is opposite the right angle between the wall and the ground.
梯子在墙上的高度为 4 m。梯子本身是斜边，因为它是墙与地面所成直角的对边。
8. Common Errors and How to Avoid Them | 常见错误及避免方法
Many students lose marks because they place the hypotenuse in the wrong position. Remember that c must always be the longest side, directly opposite the right angle.
许多学生因为将斜边放错位置而失分。记住 c 必须始终是最长边，并且是直角的对边。
- Forgetting to take the square root at the end. 最后忘记开平方。
- Adding instead of subtracting when finding a shorter side. 求直角边时误用加法而非减法。
- Applying the theorem to triangles that do not have a right angle. 将定理用于没有直角的三角形。
- Not labelling sides clearly before substituting values. 在代入数值之前没有清楚地标注各边。
Label the sides first, write down the correct formula, substitute carefully and then calculate step by step.
先标注各边，写下正确的公式，仔细代入，然后一步一步计算。
9. Practice Questions | 练习题
Try these questions to check your understanding. Work them out fully before looking at the answers.
尝试以下问题来检查你的理解。请先完整计算，再看答案。
- Find the hypotenuse when the shorter sides are 9 cm and 12 cm. 当直角边为 9 cm 和 12 cm 时，求斜边。
- Find the missing shorter side when the hypotenuse is 17 cm and one leg is 8 cm. 当斜边为 17 cm，一条直角边为 8 cm 时，求未知直角边。
- Decide whether a triangle with sides 10, 24, 26 is right-angled. 判断三边为 10、24、26 的三角形是否为直角三角形。
Check your answers: the first two answers are both 15 cm, and the third triangle is right-angled because 10² + 24² = 26².
答案：前两题的答案都是 15 cm，第三个三角形是直角三角形，因为 10² + 24² = 26²。
10. Summary and Key Points | 总结与要点
Pythagoras’ theorem connects the three sides of any right-angled triangle. Use c = √(a² + b²) for the hypotenuse, and use a = √(c² – b²) for a shorter side.
毕达哥拉斯定理联系了任意直角三角形的三条边。求斜边用 c = √(a² + b²)，求直角边用 a = √(c² – b²)。
- Only use the theorem in right-angled triangles. 仅在直角三角形中使用该定理。
- The hypotenuse is the longest side, opposite the 90° angle. 斜边是最长边，直角的对边。
- Square, add or subtract, then take the square root. 先平方，再相加或相减，最后开平方。
- Check your answer by substituting all three sides back into a² + b² = c². 将三条边代回 a² + b² = c² 来检验答案。
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