📚 Pythagoras’ Theorem: Finding Missing Sides | 毕达哥拉斯定理：求未知边长
Pythagoras’ theorem is one of the most useful results in KS3 geometry. It connects the three sides of a right-angled triangle and allows you to find an unknown side when the other two are known. This article reviews the key methods and common pitfalls in exercises like the one on page 179, question 2.
毕达哥拉斯定理是 KS3 几何中最有用的结论之一。它把直角三角形的三条边联系起来，让你在已知另外两条边时求出未知边。本文复习类似第 179 页第 2 题练习中的关键方法和常见错误。
1. Naming the Sides of a Right-Angled Triangle | 直角三角形的边的命名
In any right-angled triangle, the longest side is called the hypotenuse. It is always opposite the right angle. The other two sides are often labelled a and b, and they meet at the right angle.
在任何直角三角形中，最长的一条边叫作斜边。它总是对着直角。另外两条边通常标记为 a 和 b，它们在直角处相交。
A common KS3 question gives you a diagram with sides labelled. Your first job is to identify which side is the hypotenuse. Never assume the side at the top or bottom is automatically the hypotenuse.
KS3 常见题目会给出标好字母的图形。你的第一步是判断哪条边是斜边。不要假设顶部或底部的边一定就是斜边。
2. The Theorem Statement | 定理表述
Pythagoras’ theorem states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. We usually write this as c² = a² + b², where c is the hypotenuse.
毕达哥拉斯定理指出，在直角三角形中，斜边的平方等于另外两条边的平方之和。我们通常写成 c² = a² + b²，其中 c 是斜边。
c² = a² + b²
This formula only works for right-angled triangles. If the triangle does not have a right angle, you cannot use it directly.
这个公式只适用于直角三角形。如果三角形没有直角，就不能直接使用它。
3. Rearranging the Formula | 公式变形的两种形式
You can rearrange the formula to find different sides. To find the hypotenuse, use c = √(a² + b²). To find one of the shorter sides, use a = √(c² − b²) or b = √(c² − a²).
你可以把公式变形来求不同的边。求斜边时用 c = √(a² + b²)。求一条直角边时用 a = √(c² − b²) 或 b = √(c² − a²)。
c = √(a² + b²) | a = √(c² − b²)
The minus sign is essential when finding a shorter side. Many students forget and add the squares instead of subtracting.
求直角边时减号非常重要。许多学生忘记减去平方，而是把平方相加。
4. Finding the Hypotenuse | 求斜边
To find the hypotenuse, square the two shorter sides, add them, then take the square root. For example, if a = 6 cm and b = 8 cm, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
求斜边时，先把两条直角边平方，相加，再开平方。例如，如果 a = 6 厘米，b = 8 厘米，那么 c² = 6² + 8² = 36 + 64 = 100，所以 c = √100 = 10 厘米。
c² = 6² + 8² = 36 + 64 = 100 → c = √100 = 10 cm
Write the full working line by line. Even if you can see the answer, the method is what earns marks in Cambridge KS3 assessments.
逐行写出完整过程。即使你能直接看出答案，方法仍然是剑桥 KS3 评估中得分的关键。
5. Finding a Shorter Side | 求直角边
When finding a shorter side, subtract the square of the known shorter side from the square of the hypotenuse. Example: if c = 13 cm and b = 5 cm, then a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 cm.
求直角边时，用斜边的平方减去已知直角边的平方。例如：如果 c = 13 厘米，b = 5 厘米，那么 a² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 厘米。
a² = 13² − 5² = 169 − 25 = 144 → a = √144 = 12 cm
Always check you are subtracting the side you know, not the side you are trying to find. If the answer looks longer than the hypotenuse, something is wrong.
一定要检查你减去的是已知边，而不是你正在求的边。如果答案看起来比斜边还长，那就说明出错了。
6. Checking Your Answer | 检验答案
After you find an answer, substitute all three sides back into the original equation. For a 6-8-10 triangle, check 6² + 8² = 10², which gives 36 + 64 = 100. This confirms the result.
求出答案后，把三条边代回原方程检验。对于 6-8-10 三角形，检查 6² + 8² = 10²，得到 36 + 64 = 100。这可以确认结果正确。
You can also use estimation: if the hypotenuse is 10 cm, each shorter side must be less than 10 cm. A shorter side larger than the hypotenuse is a red flag.
你还可以用估算检验：如果斜边是 10 厘米，那么每条直角边都必须小于 10 厘米。直角边比斜边长就是一个危险信号。
7. Applying Pythagoras to Word Problems | 应用毕达哥拉斯定理解应用题
Word problems often describe a ladder leaning against a wall, a ship sailing east and north, or a rectangle with a diagonal. Draw the right-angled triangle first, label the known lengths, and mark the unknown.
应用题经常描述梯子斜靠在墙上、船先向东再向北航行，或者长方形的一条对角线。先画出直角三角形，标出已知长度，再标出未知量。
Then decide whether the unknown is the hypotenuse or a shorter side. Choose the correct form of the formula before you calculate.
然后判断未知量是斜边还是直角边。计算之前先选择正确的公式形式。
For example, a ladder 5 m long rests against a wall, with its foot 3 m from the wall. The height reached is √(5² − 3²) = √(25 − 9) = √16 = 4 m.
例如，一架 5 米长的梯子斜靠在墙上，梯脚离墙 3 米。它到达的高度为 √(5² − 3²) = √(25 − 9) = √16 = 4 米。
8. Pythagorean Triples | 勾股数
Some whole-number triples appear very often. The 3-4-5 triangle is the most famous, because 3² + 4² = 9 + 16 = 25 = 5². Multiples such as 6-8-10 and 9-12-15 also work.
一些整数三元组经常出现。3-4-5 三角形最著名，因为 3² + 4² = 9 + 16 = 25 = 5²。它的倍数如 6-8-10 和 9-12-15 也成立。
| Triangle 三角形 | Check 检验 | Result 结果 |
|---|---|---|
| 3-4-5 | 3² + 4² = 5² | 9 + 16 = 25 |
| 6-8-10 | 6² + 8² = 10² | 36 + 64 = 100 |
| 5-12-13 | 5² + 12² = 13² | 25 + 144 = 169 |
Recognising these triples can save time in mental maths, but you should still show the method in written work.
识别这些勾股数可以节省心算时间，但在书面作业中你仍然应该展示计算过程。
9. Common Mistakes to Avoid | 常见错误
A common mistake is adding squares when you should subtract. For finding a shorter side, write a² = c² − b², not a² = c² + b². Another mistake is forgetting to take the square root at the end.
常见错误是在该减的时候把平方相加。求直角边时，要写 a² = c² − b²，而不是 a² = c² + b²。另一个错误是最后忘记开平方。
Also, never round too early. Keep the exact square root until the final step, then round to the required number of decimal places. Premature rounding can change the final answer.
还有，不要过早四舍五入。把精确的平方根保留到最后一步，再根据要求的小数位数取近似值。过早取近似值可能改变最终答案。
10. Practice Questions and Answers | 练习题与答案
Try this quick practice: a right-angled triangle has a hypotenuse of 15 cm and one shorter side of 9 cm. Find the missing side. Answer: a² = 15² − 9² = 225 − 81 = 144, so a = √144 = 12 cm.
试做这道快速练习：一个直角三角形的斜边为 15 厘米，一条直角边为 9 厘米。求另一条直角边。答案：a² = 15² − 9² = 225 − 81 = 144，所以 a = √144 = 12 厘米。
Another practice: if a = 7 cm and b = 24 cm, find c. Answer: c² = 7² + 24² = 49 + 576 = 625, so c = √625 = 25 cm.
再练一道：如果 a = 7 厘米，b = 24 厘米，求 c。答案：c² = 7² + 24² = 49 + 576 = 625，所以 c = √625 = 25 厘米。
11. Connecting to the Coordinate Plane | 与坐标平面的联系
Pythagoras’ theorem also helps you find the distance between two points on a coordinate grid. The horizontal and vertical differences are the shorter sides, and the straight-line distance is the hypotenuse.
毕达哥拉斯定理还能帮助你在坐标网格上求两点之间的距离。水平差和垂直差是两条直角边，直线距离就是斜边。
For points (1, 2) and (4, 6), the horizontal difference is 3 and the vertical difference is 4. Distance = √(3² + 4²) = √25 = 5 units.
对于点 (1, 2) 和 (4, 6)，水平差为 3，垂直差为 4。距离 = √(3² + 4²) = √25 = 5 个单位。
12. Summary | 小结
In summary, identify the hypotenuse
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