📚 Pythagoras' Theorem: Finding Missing Sides | 勾股定理：求未知边
In many Cambridge KS3 mathematics exercises, such as the problem set on page 158, question 2, you are given a right-angled triangle with two known sides and asked to find the third side. This skill depends on a simple but powerful rule called Pythagoras' theorem.
在许多剑桥 KS3 数学练习中，例如第 158 页的第 2 题，你会看到一个直角三角形，已知两条边，要求第三条边。这个技能依赖一个简单却强大的规则，即勾股定理。
1. What is Pythagoras' theorem? | 什么是勾股定理？
Pythagoras' theorem connects the three sides of any right-angled triangle. It states that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
勾股定理将任意直角三角形的三条边联系起来。它指出：斜边上的正方形面积等于另外两条直角边上的正方形面积之和。
are a and b, and the hypotenuse is c, then a² + b² = c². Here c must be the longest side.
用符号表示，如果两条较短的直角边为 a 和 b，斜边为 c，那么 a² + b² = c²。这里的 c 必须是最长的边。
a² + b² = c²
2. Key vocabulary and notation | 关键术语与记号
The side opposite the right angle is called the hypotenuse. It is always the longest side. The other two sides are often called the legs or shorter sides.
直角对面的边叫做斜边。它总是最长的边。另外两条边通常称为直角边或较短的边。
- Right angle = 90° | 直角 = 90°
- Hypotenuse = side c | 斜边 = 边 c
- Legs = sides a and b | 直角边 = 边 a 和 b
Always label the hypotenuse as c before substituting numbers. This small habit prevents many errors.
在代入数字之前，始终把斜边标为 c。这个小习惯能避免许多错误。
3. Identifying the hypotenuse | 识别斜边
To use the theorem correctly, first locate the right angle. The side directly across from it is the hypotenuse. Do not assume it is the slanted side.
要正确使用定理，首先找到直角。直角对面的边就是斜边。不要以为斜边一定是斜着的那条边。
In a triangle where the vertical side is 6, the horizontal side is 8, and the sloping side is 10, the sloping side is the hypotenuse because it is opposite the right angle.
在一个三角形中，竖直边为 6，水平边为 8，斜边为 10，那么斜边就是 10，因为它在直角的对面。
4. Formula and rearrangement | 公式与变形
To find the hypotenuse, use c = √a² + b². To find a shorter side, rearrange the formula to a = √c² − b² or b = √c² − a².
要求斜边，使用 c = √a² + b²。要求一条直角边，将公式变形为 a = √c² − b² 或 b = √c² − a²。
he square root symbol √ returns the positive length. Length is never negative in geometry.
平方根符号 √ 只取正的长度。在几何中，长度永远不会是负数。
5. Worked example: finding the hypotenuse | 例题：求斜边
A right-angled triangle has shorter sides of 5 cm and 12 cm. Find the hypotenuse.
一个直角三角形的两条直角边分别为 5 厘米和 12 厘米。求斜边。
Write the formula: c² = 5² + 12².
写出公式：c² = 5² + 12²。
Work out the squares: c² = 25 + 144 = 169.
计算平方：c² = 25 + 144 = 169。
Take the square root: c = √169 = 13 cm.
取平方根：c = √169 = 13 厘米。
6. Worked example: finding a shorter side | 例题：求直角边
A right-angled triangle has hypotenuse 10 m and one shorter side 6 m. Find the other shorter side.
一个直角三角形的斜边为 10 米，一条直角边为 6 米。求另一条直角边。
Use a² = c² − b², so a² = 10² − 6² = 100 − 36 = 64.
使用 a² = c² − b²，因此 a² = 10² − 6² = 100 − 36 = 64。
Then a = √64 = 8 m.
然后 a = √64 = 8 米。
7. Checking with a known triple | 用已知勾股数检查
Some right-angled triangles use standard whole-number triples, such as 3-4-5, 5-12-13, and 8-15-17. Recognising these can help you check answers quickly.
一些直角三角形使用标准的整数勾股数，例如 3-4-5、5-12-13 和 8-15-17。识别这些数可以帮助你快速检查答案。
If you obtain 6-8-10, that is the 3-4-5 triple multiplied by 2: 6² + 8² = 10².
如果你得到 6-8-10，那是 3-4-5 勾股数乘以 2：6² + 8² = 10²。
8. Common mistakes | 常见错误
One common mistake is adding the hypotenuse and a leg instead of subtracting when finding a shorter side. Always rearrange carefully.
一个常见错误是在求直角边时，把斜边和直角边相加而不是相减。务必仔细变形公式。
Another mistake is rounding too early. Keep the exact square until the final step, then round if the question asks for it.
另一个错误是过早四舍五入。保留精确的平方直到最后一步，如果题目要求再四舍五入。
9. Exam-style practice questions | 考试型练习题
Try these questions to test your understanding.
尝试以下问题来检验你的理解。
- Find the hypotenuse when the legs are 9 cm and 12 cm. | 直角边为 9 厘米和 12 厘米时，求斜边。
- Find the missing shorter side when the hypotenuse is 25 cm and one leg is 7 cm. | 斜边为 25 厘米，一条直角边为 7 厘米时，求另一条直角边。
- A ladder of length 5.2 m leans against a wall. The foot is 2.0 m from the wall. How high up the wall does the ladder reach? | 一个长 5.2 米的梯子斜靠在墙上，梯脚距墙 2.0 米。梯子能达到多高？
10. Real-life application: ladder problem | 实际应用：梯子问题
In the ladder problem, the wall and ground form a right angle. The ladder is the hypotenuse, the ground distance is one leg, and the height is the other leg.
在梯子问题中，墙和地面形成直角。梯子是斜边，地面距离是一条直角边，高度是另一条直角边。
Using Pythagoras: height = √5.2² − 2.0² = √27.04 − 4 = √23.04 ≈ 4.8 m.
使用勾股定理：高度 = √5.2² − 2.0² = √27.04 − 4 = √23.04 ≈ 4.8 米。
11. Summary | 小结
To solve a missing-side problem, identify the hypotenuse, write the correct form of Pythagoras' theorem, substitute known sides, and use square roots carefully.
解决求边问题时，先识别斜边，写出勾股定理的正确形式，代入已知边，并小心使用平方根。
Always check that the hypotenuse is the longest answer. If a shorter side turns out longer than the hypotenuse, you have made a sign error.
始终检查斜边是否是最长的边。如果直角边结果比斜边还长，说明符号出错了。
12. How to use this in the classroom | 如何在课堂上使用
quares on each side of a right triangle. This visual proof helps explain why a² + b² = c².
老师常让学生画出直角三角形每条边上的正方形。这个直观证明有助于解释为什么 a² + b² = c²。
Practise with mixed questions, including finding both the hypotenuse and a shorter side, so the rearrangement becomes automatic.
混合练习求斜边和直角边的题目，让公式变形变得自动化。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图