📚 Pythagoras' Theorem: Finding Missing Sides and Checking Right Angles | 勾股定理：求未知边与验证直角
Pythagoras' theorem is a central topic in the Cambridge KS3 mathematics curriculum and a foundation for later work in IGCSE and A-Level. It links algebra with geometry and gives us a reliable way to calculate unknown lengths in right-angled triangles, and to test whether a triangle contains a right angle. This guide explains the rule clearly, shows step-by-step methods for both directions of the problem, and helps you avoid the most common mistakes in tests.
勾股定理是剑桥KS3数学课程的核心主题，也是后续IGCSE与A-Level学习的基础。它把代数与几何联系起来，为我们提供了一种可靠的方法，既能计算直角三角形中的未知边长，也能判断一个三角形是否含有直角。本指南清晰地解释这一定理，展示两个方向问题的分步方法，并帮助你避免考试中最常见的错误。
1. The Right-Angled Triangle and Its Sides | 直角三角形及其边
A right-angled triangle is a triangle with one interior angle exactly equal to 90°. The longest side is always opposite this right angle and is called the hypotenuse.
直角三角形是一个内角恰好等于90°的三角形。最长边总是位于直角的对面，称为斜边。
The two sides that form the right angle are known as the shorter sides or legs. In most questions they are labelled a and b, while the hypotenuse is labelled c. When you draw a diagram, mark the right angle first and then place c opposite it.
构成直角的两条边称为较短边或直角边。在大多数题目中，它们被标记为 a 和 b，斜边标记为 c。画图时，先标出直角，然后把 c 放在直角的对面。
Always identify the hypotenuse before substituting values into the formula. A common mistake is to place a shorter side in the position of c, which leads to an incorrect result.
在代入公式之前，务必先确定斜边。一个常见错误是把较短边放到 c 的位置，这会导致结果错误。
Useful labels to remember: the right angle marker is the small square in the corner; the hypotenuse touches neither side of that square.
需要记住的标记：直角符号是角落里的小正方形；斜边不与这个小正方形的任何一条边相接。
2. The Rule: a² + b² = c² | 定理：a² + b² = c²
Pythagoras' theorem states that in any right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
勾股定理指出：在任意直角三角形中，两条较短边的平方和等于斜边的平方。
a² + b² = c²
Here a and b are the lengths of the shorter sides, and c is the length of the hypotenuse. The theorem only works when the triangle contains a right angle. If a triangle is not right-angled, this relationship does not hold.
这里的 a 和 b 是两条较短边的长度，c 是斜边的长度。该定理只在三角形含有直角时成立。如果三角形不是直角三角形，这一关系就不成立。
An equivalent rearrangement is often useful when you need a shorter side:
当需要求较短边时，等价的重排形式往往很有用：
a² = c² − b² and b² = c² − a²
Notice the subtraction. The hypotenuse term c² is always on its own, and you subtract the known shorter side squared from it.
注意这里是减法。斜边项 c² 始终单独在一边，用它减去已知较短边的平方。
3. Key Vocabulary and Glossary | 关键术语表
Learning the vocabulary helps you read questions accurately. Below is a quick reference table of terms that appear in KS3 papers.
掌握术语有助于你准确理解题目。下面是KS3试卷中常见术语的速查表。
| English | 中文 | Meaning |
| Hypotenuse | 斜边 | Longest side, opposite the right angle |
| Leg / shorter side | 直角边 | Either side forming the right angle |
| Square | 平方 | A number multiplied by itself |
| Square root | 平方根 | The inverse of squaring, written √ |
| Converse | 逆定理 | The reverse statement, used to test for 90° |
| Pythagorean triple | 勾股数 | Whole-number set such as 3, 4, 5 |
Keep this table in your notes. Examiners frequently use the word 'converse', and misreading it turns a checking question into a length question.
把这张表记在笔记里。考官经常使用"逆定理"这个词，一旦看错，就会把验证题做成求长度的题。
4. Method 1: Finding the Hypotenuse | 方法一：求斜边
When the two shorter sides are known and you need the hypotenuse, add the squares and take the square root.
当已知两条较短边、需要求斜边时，把两个平方相加，再取平方根。
Step-by-step method:
分步方法：
- Write down the theorem: a² + b² = c².
- Substitute the two known shorter sides.
- Square each value and add them.
- Take the square root of the total to find c.
- Round only at the end, and state the unit.
- 写下定理：a² + b² = c²。
- 代入已知的两条较短边。
- 分别平方后相加。
- 对总和开平方求出 c。
- 只在最后一步取近似值，并写上单位。
orked example: a = 6 cm, b = 8 cm. Then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
例题：a = 6 cm，b = 8 cm。则 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 cm。
Always check that your answer is longer than both given sides. If it is not, you have made an error.
务必检查答案是否比两条已知边都长。如果不是，说明你出错了。
5. Method 2: Finding a Shorter Side | 方法二：求一条较短边
When the hypotenuse and one shorter side are known, rearrange the formula by subtracting rather than adding.
当已知斜边和一条较短边时，要重排公式，用减法而不是加法。
b² = c² − a² → b = √c² − a²
This is the step where most marks are lost. Students who add instead of subtract usually end up with an answer larger than the hypotenuse, which is impossible.
这一步是失分最多的地方。把减号写成加号的学生，往往会得到一个比斜边还大的答案，而这是不可能的。
ed example: c = 13 cm, a = 5 cm. Then b² = 13² − 5² = 169 − 25 = 144, so b = √144 = 12 cm.
例题：c = 13 cm，a = 5 cm。则 b² = 13² − 5² = 169 − 25 = 144，所以 b = √144 = 12 cm。
Note the pattern here: 5, 12, 13 is another Pythagorean triple, just like 6, 8, 10 and 3, 4, 5. Recognising these saves time in exams.
注意这里的规律：5、12、13 是另一组勾股数，与 6、8、10 以及 3、4、5 一样。识别它们能在考试中节省时间。
6. Method 3: Checking Whether a Triangle Has a Right Angle | 方法三：判断三角形是否有直角
The converse of Pythagoras' theorem lets you test any triangle when all three sides are known. Identify the longest side and treat it as c.
勾股定理的逆定理让你在已知三边时判断任意三角形。先找出最长边，把它当作 c。
Then compare the two quantities:
然后比较这两个量：
a² + b² versus c²
- If a² + b² = c², the triangle is right-angled.
- If a² + b² > c², the triangle is acute-angled (all angles less than 90°).
- If a² + b² < c², the triangle is obtuse-angled (one angle greater than 90°).
- 如果 a² + b² = c²，则三角形是直角三角形。
- 如果 a² + b² > c²，则三角形是锐角三角形（所有角都小于90°）。
- 如果 a² + b² < c²，则三角形是钝角三角形（有一个角大于90°）。
cm, 25 cm. Here 7² + 24² = 49 + 576 = 625 and 25² = 625, so the triangle is right-angled.
例题：三边为 7 cm、24 cm、25 cm。这里 7² + 24² = 49 + 576 = 625，而 25² = 625，所以该三角形是直角三角形。
e 52 < 64, the angle opposite the side of length 8 cm is obtuse, so the triangle is obtuse-angled.
例题：三边为 4 cm、6 cm、8 cm。这里 4² + 6² = 16 + 36 = 52，而 8² = 64。由于 52 < 64，长度为 8 cm 的边所对的角是钝角，所以该三角形是钝角三角形。
7. Common Pythagorean Triples | 常用勾股数
Pythagorean triples are sets of three whole numbers that satisfy the theorem exactly. Learning them speeds up calculation and helps you spot errors.
勾股数是恰好满足该定理的三个整数。记住它们能加快计算速度，并帮助你发现错误。
| a | b | c | Check |
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
Any multiple of a triple is also a triple. For example, doubling 3, 4, 5 gives 6, 8, 10, and tripling gives 9, 12, 15.
任何勾股数的整数倍仍然是勾股数。例如，3、4、5 乘以2得到 6、8、10，乘以3得到 9、12、15。
In an exam, if you calculate a hypotenuse and the answer is not one of these familiar forms, re-check your squares before doubting the method.
在考试中，如果你算出的斜边不是这些熟悉的形式，先重新检查平方计算，再怀疑方法本身。
8. Distance and Diagonal Problems | 距离与对角线问题
The theorem applies whenever a right angle is present in a diagram, even if no triangle is drawn. A rectangle's diagonal, a ladder against a wall, and the distance between two points on a grid are all standard examples.
只要图形中出现直角，该定理就适用，即使图上没有画出三角形。矩形的对角线、靠在墙上的梯子、以及网格上两点之间的距离，都是标准例子。
nd width W, the diagonal d satisfies d² = L² + W². So a 5 m by 12 m rectangle has diagonal √25 + 144 = √169 = 13 m.
对于长为 L、宽为 W 的矩形，对角线 d 满足 d² = L² + W²。所以 5 m × 12 m 的矩形对角线为 √25 + 144 = √169 = 13 m。
tal gap is Δx = x₂ − x₁ and the vertical gap is Δy = y₂ − y₁, giving the distance formula:
对于坐标 (x₁, y₁) 和 (x₂, y₂)，水平差为 Δx = x₂ − x₁，竖直差为 Δy = y₂ − y₁，于是得到距离公式：
d = √Δx² + Δy²
This is Pythagoras' theorem in disguise, and it becomes the basis of the circle equation later in IGCSE.
这其实就是伪装的勾股定理，它之后会成为IGCSE中圆的方程的基础。
9. Real-Life Contexts: Ladders, Ramps and Screens | 实际情境：梯子、坡道与屏幕
Word problems often hide the triangle in a story. The safest approach is always the same: sketch the situation, mark the right angle, and label the hypotenuse.
应用题常常把三角形藏在故事情节里。最稳妥的处理方式总是一样的：画出情境草图，标出直角，并标出斜边。
- A 4 m ladder leaning against a wall with its foot 1.5 m from the base reaches a height of √4² − 1.5² = √13.75 ≈ 3.71 m.
- A ramp rises 1.2 m over a horizontal run of 5 m; its sloping length is √5² + 1.2² = √26.44 ≈ 5.14 m.
- 9 ratio has a width near 121 cm and a height near 68 cm, since 121² + 68² ≈ 55² in inches.
- 一把 4 m 的梯子靠在墙上，梯脚离墙脚 1.5 m，则它能到达的高度为 √4² − 1.5² = √13.75 ≈ 3.71 m。
- 一条坡道在水平方向前进 5 m 时升高 1.2 m，其斜面长度为 √5² + 1.2² = √26.44 ≈ 5.14 m。
- 一台 55 英寸、比例为 16 : 9 的电视屏幕，宽度约 121 cm，高度约 68 cm，因为以英寸计时 121² + 68² ≈ 55²。
In each case, ask yourself first: which length is the hypotenuse? Only then decide whether to add or subtract.
在每一题中，先问自己：哪一条是斜边？然后才决定该相加还是相减。
10. Non-Calculator Skills and Exact Answers | 非计算器技巧与精确答案
y hand. Know the square numbers up to 15² = 225 and the roots of perfect squares fluently.
KS3试卷常设非计算器部分，所以你必须手算平方与平方根。要熟练掌握到 15² = 225 的平方数，以及完全平方数的根。
When the answer is not a whole number, leave it as a simplified surd rather than a rounded decimal unless the question says otherwise. For example, √50 simplifies to 5√2.
当答案不是整数时，除非题目另有要求，否则应保留为化简后的根式，而不是取近似小数。例如 √50 可化简为 5√2。
- √8 = 2√2 √18 = 3√2 √32 = 4√2 √72 = 6√2
- √12 = 2√3 √27 = 3√3 √48 = 4√3 √75 = 5√3
- √20 = 2√5 √45 = 3√5 √80 = 4√5 √125 = 5√5
To simplify a surd, look for the largest square factor. So √50 = √25 × 2 = 5√2, and √200 = √100 × 2 = 10√2.
化简根式时，找出最大的平方因子。于是 √50 = √25 × 2 = 5√2，√200 = √100 × 2 = 10√2。
11. Common Mistakes and Exam Techniques | 常见错误与应试技巧
Most lost marks in this topic come from a small number of repeated slips. Check your working against this list before you move on.
本主题的失分大多来自少数几个反复出现的失误。继续答题前，用下面的清单检查你的解题过程。
- Adding the two shorter sides before squaring, instead of squaring first.
- Using the hypotenuse as a shorter side in a subtraction question.
- Forgetting to take the square root at the end.
- Rounding too early and carrying the error through.
- Omitting units, or mixing metres and centimetres in one calculation.
- Assuming a triangle is right-angled without checking the converse.
- 先相加两条较短边再平方，而不是先平方。
- 在减法题中把斜边当作较短边代入。
- 最后忘记开平方。
- 过早取近似值，使误差一路累积。
- 漏写单位，或在同一次计算中混用米和厘米。
- 未用逆定理验证就默认三角形是直角三角形。
A reliable layout earns method marks even when the final number is wrong: write the formula, substitute, calculate the squares, then state the root with its unit.
规范的书写能在最终答案错误时依然拿到方法分：写出公式、代入、计算平方、最后写出根与单位。
12. Quick Revision Summary | 快速复习小结
Use this summary as a five-minute checklist the night before a test.
考前一夜，可用这份小结做五分钟自查。
- Right-angled triangle: one angle is 90°; the hypotenuse is the longest side, opposite the right angle.
- Finding the hypotenuse: c = √a² + b².
- Finding a shorter side: subtract, b = √c² − a².
- Checking a right angle: compare a² + b² with c² using the longest side as c.
- Memorise the triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.
- Keep exact surd answers unless the question asks for a decimal.
- 直角三角形：有一个角为90°；斜边是最长边，位于直角对面。
- 求斜边：c = √a² + b²。
- 求较短边：用减法，b = √c² − a²。
- 验证直角：以最长边作 c，比较 a² + b² 与 c²。
- 记住勾股数 3-4-5、5-12-13、8-15-17、7-24-25 及其倍数。
- 除非题目要求小数，否则保留精确的根式答案。
Pythagoras' theorem rewards careful reading more than clever calculation. Identify the hypotenuse first, decide whether to add or subtract, and always sanity-check that your answer is physically possible.
勾股定理更奖励仔细读题，而不是聪明的计算。先确定斜边，判断该加还是该减，并始终检查答案在物理上是否可能。
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