📚 Pythagoras and Trigonometry in Right-Angled Triangles | 直角三角形中的毕达哥拉斯定理与三角学
Pythagoras’ theorem and right-angled triangle trigonometry are two of the most powerful tools in KS3 geometry. They allow you to calculate missing sides and angles, solve real-world problems and prepare for Cambridge Checkpoint and IGCSE work. This article revises the key rules, shows worked examples and gives exam tips for mixed problems.
毕达哥拉斯定理和直角三角形三角学是 KS3 几何中最有力的两大工具。它们能帮你计算缺失的边和角，解决实际问题，并为 Cambridge Checkpoint 和 IGCSE 学习做好准备。本文复习关键规则，给出例题，并提供综合题的考试技巧。
1. Naming the Sides of a Right-Angled Triangle | 直角三角形的边命名
In a right-angled triangle, the longest side is opposite the right angle and is called the hypotenuse. It is always opposite the 90° angle.
在直角三角形中，最长的边对着直角，称为斜边。它总是位于 90° 角的对面。
The other two sides are usually labelled as ‘opposite’ and ‘adjacent’ relative to a chosen acute angle θ. The opposite side is the side facing angle θ, while the adjacent side is the side next to θ, but not the hypotenuse.
另外两条边通常相对于选定的锐角 θ 标记为 ‘对边’ 和 ‘邻边’。对边是面向角 θ 的边，而邻边是紧靠 θ 的边，但不是斜边。
Always draw and label a triangle before starting a calculation. A clear diagram reduces mistakes where students mix up the opposite and adjacent sides.
在开始计算前一定要画图并标注三角形。清晰的图形可以减少学生混淆对边和邻边的错误。
2. Pythagoras’ Theorem: Statement and Formula | 毕达哥拉斯定理：表述与公式
For any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This relationship only works for right-angled triangles, so always check for a right angle before using it.
对于任意直角三角形，斜边的平方等于另外两条边的平方和。这个关系只适用于直角三角形，所以在使用前一定要确认是否有一个直角。
a² + b² = c²
In the formula, c represents the hypotenuse, while a and b represent the two shorter sides. The formula connects the three side lengths and is the foundation for distance problems in coordinates and real-life contexts.
在公式中，c 表示斜边，a 和 b 表示两条直角边。这个公式把三条边的长度联系起来，是坐标和实际生活中距离问题的基础。
3. Finding the Hypotenuse | 求斜边
If the two shorter sides are known, substitute them into a² + b² = c² and then take the square root to find c.
如果已知两条直角边，把它们代入 a² + b² = c²，然后取平方根求出 c。
c = √(a² + b²)
Worked example: Find c when a = 6 cm and b = 8 cm. First c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
例题：当 a = 6 厘米、b = 8 厘米时，求 c。先算 c² = 6² + 8² = 36 + 64 = 100，因此 c = √100 = 10 厘米。
Remember to include the correct units and do not forget the square root step. A common error is to add the two side lengths instead of squaring them first.
记住要带上正确单位，也不要忘记开平方这一步。一个常见错误是直接把两条边相加，而不是先平方再相加。
4. Finding a Shorter Side | 求一条直角边
To find a shorter side, rearrange the formula by subtracting the known shorter side squared from the hypotenuse squared.
要求一条直角边，需要把公式变形，用斜边的平方减去已知直角边的平方。
a = √(c² – b²)
Worked example: If the hypotenuse c = 13 cm and one side b = 5 cm, then a² = 13² – 5² = 169 – 25 = 144, so a = √144 = 12 cm.
例题：如果斜边 c = 13 厘米，一条直角边 b = 5 厘米，那么 a² = 13² – 5² = 169 – 25 = 144，所以 a = √144 = 12 厘米。
A common mistake is to add the two given sides when finding a shorter side. Remember that finding a shorter side always involves subtracting squares.
一个常见错误是在求直角边时把两条已知边相加。记住求直角边时总是要对平方做减法。
5. Pythagorean Triples and Converse | 毕达哥拉斯三元组与逆定理
A Pythagorean triple is a set of three integers that satisfy a² + b² = c². Well-known examples include 3, 4, 5 and 5, 12, 13.
毕达哥拉斯三元组是指一组满足 a² + b² = c² 的整数。著名的例子包括 3、4、5 和 5、12、13。
| Triple | Check |
|---|---|
| 3, 4, 5 | 9 + 16 = 25 |
| 5, 12, 13 | 25 + 144 = 169 |
| 6, 8, 10 | 36 + 64 = 100 |
| 7, 24, 25 | 49 + 576 = 625 |
The converse of Pythagoras’ theorem says that if a² + b² = c² for the three side lengths of a triangle, then the triangle is right-angled. This gives a quick way to test whether a triangle contains a 90° angle.
毕达哥拉斯定理的逆定理指出，如果一个三角形的三条边长满足 a² + b² = c²，那么这个三角形就是直角三角形。这提供了一种快速检验三角形是否含有 90° 角的方法。
6. Introduction to Sine, Cosine and Tangent | 正弦、余弦与正切简介
In any right-angled triangle, the three trigonometric ratios relate an acute angle to the ratios of two sides. These ratios depend only on the angle θ, not on the overall size of the triangle.
在任意直角三角形中，三个三角比将一个锐角与两条边的比值联系起来。这些比值只取决于角 θ，而与三角形的整体大小无关。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
A helpful memory aid is ‘SOH CAH TOA’: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
一个有用的记忆口诀是 ‘SOH CAH TOA’：正弦是对边除以斜边，余弦是邻边除以斜边，正切是对边除以邻边。
Always write the sine, cosine or tangent of the angle as a ratio of lengths. For example, sin 30° = 0.5 means that in a right-angled triangle with a 30° angle, the opposite side is half the hypotenuse.
一定要把角的正弦、余弦或正切写成边长之比。例如 sin 30° = 0.5 表示在一个含 30° 角的直角三角形中，对边是斜边的一半。
7. Choosing the Correct Trig Ratio | 选择正确的三角比
First label the sides as O for opposite, H for hypotenuse and A for adjacent relative to the given or required angle. Then identify which two sides are involved in the problem.
首先相对于已知角或要求的角，将边标记为 O 对边、H 斜边和 A 邻边。然后确定题目涉及哪两条边。
If you know or need O and H, use sine. If A and H are involved, use cosine. If O and A are involved, use tangent.
如果已知或需要 O 和 H，使用正弦。如果涉及 A 和 H，使用余弦。如果涉及 O 和 A，使用正切。
| Sides involved | 涉及的边 | Ratio to use | 使用的三角比 |
|---|---|
| O and H | sin θ |
| A and H | cos θ |
| O and A | tan θ |
Highlight or underline the two relevant sides in a problem. This visual step helps avoid choosing the wrong ratio under exam pressure.
在题目中把有关的两条边高亮或画线。这个可视化步骤有助于在考试压力下避免选错三角比。
8. Finding Missing Sides by Trigonometry | 用三角函数求缺失边
Write the ratio equation, substitute the known side and angle, then solve for the unknown side. Use the sine, cosine or tangent button on your calculator, not the inverse button, when finding a side.
写出比例方程，代入已知边和角度，然后解出未知边。求边时使用计算器上的正弦、余弦或正切键，而不是反函数键。
Worked example: Find x when θ = 35° and the hypotenuse is 12 cm, with x opposite θ. Use sin 35° = x ÷ 12, so x = 12 × sin 35° ≈ 6.88 cm.
例题：当 θ = 35°，斜边为 12 厘米，且 x 是 θ 的对边时，求 x。使用 sin 35° = x ÷ 12，所以 x = 12 × sin 35° ≈ 6.88 厘米。
Worked example: Find y when θ = 50° and the adjacent side is 8 m, with y opposite θ. Use tan 50° = y ÷ 8, so y = 8 × tan 50° ≈ 9.53 m.
例题：当 θ = 50°，邻边为 8 米，且 y 是 θ 的对边时，求 y。使用 tan 50° = y ÷ 8，所以 y = 8 × tan 50° ≈ 9.53 米。
Round final answers to a sensible degree of accuracy, usually 3 significant figures unless the question says otherwise. Never round too early during the working.
最终答案要取合理的精确度，除非题目另有说明，通常保留 3 位有效数字。计算过程中不要过早取整。
9. Finding Missing Angles | 求缺失角
To find an angle, use the inverse trigonometric functions sin⁻¹, cos⁻¹ and tan⁻¹ on your calculator. These functions convert a side ratio back into an angle measure.
要求角度，使用计算器上的反三角函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。这些函数将边长之比转换回角度值。
Worked example: If the opposite side is 4 cm and the hypotenuse is 5 cm, then sin θ = 4 ÷ 5 = 0.8, so θ = sin⁻¹(0.8) ≈ 53.1°.
例题：如果对边为 4 厘米，斜边为 5 厘米，那么 sin θ = 4 ÷ 5 = 0.8，所以 θ = sin⁻¹(0.8) ≈ 53.1°。
Make sure your calculator is in degree mode when working with degree measures. If it is in radian mode, your answers will look completely different and will be marked wrong.
使用角度制时，确保计算器处于度数模式。如果处于弧度模式，答案会完全不同，并且会被判为错误。
Always check that the angle answer is reasonable. The largest angle in a right-angled triangle must be the right angle, so the two acute angles should each be between 0° and 90°.
一定要检查角度答案是否合理。直角三角形中最大的角一定是直角，所以两个锐角都应在 0° 到 90° 之间。
10. Mixed Applications and Exam Tips | 综合应用与考试技巧
Many Checkpoint questions combine Pythagoras and trigonometry with word problems such as ladders, ramps, bearings and coordinates. Decide first whether the problem involves only side lengths or also an acute angle.
许多 Checkpoint 题目会把毕达哥拉斯定理和三角函数与梯子、坡道、方位角和坐标等实际应用题结合起来。先判断题目只涉及边长，还是也涉及锐角。
Use Pythagoras when only side lengths are involved and trigonometry when an acute angle is given or required. Draw a clear diagram, label all given sides and angles, and write down the formula before substituting.
只在涉及边长时使用毕达哥拉斯定理，在给出或要求锐角时使用三角函数。画出清晰的示意图，标出所有已知边和角，并在代入前写出公式。
| Do | 应该做 | Avoid | 避免 |
|---|---|
| Draw and label diagrams | Mixing O and A |
| Check for a right angle | Using Pythagoras in a non-right triangle |
| Show every step of working | Forgetting units or rounding too early |
Check the reasonableness of every answer. A hypotenuse should always be longer than each of the other two sides, and a sine or cosine value should never exceed 1 for a right-angled triangle with real side lengths.
检查每个答案的合理性。斜边应始终比另外两条边都长，对于边长真实的直角三角形，正弦或余弦值绝不会超过 1。
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