Pythagoras' Theorem: Page 157 Exercise 1 Explained | 勾股定理：第157页练习1深度解析

📚 Pythagoras’ Theorem: Page 157 Exercise 1 Explained | 勾股定理：第157页练习1深度解析

Welcome to our focused revision guide built around the type of questions found in Page 157, Exercise 1 of the Cambridge KS3 Mathematics textbook. Whether you are preparing for a topic test or simply cementing your understanding, this article will walk you through Pythagoras’ theorem from first principles to advanced applications in 3D. Every explanation is paired in English and Chinese so you can grasp the reasoning in both languages with confidence.

欢迎来到我们围绕剑桥KS3数学教材第157页练习1题型打造的专项复习指南。无论你是在准备单元测验还是巩固理解，本文都将带你从勾股定理的基本原理一直深入到三维应用。每一段解释都以中英双语成对呈现，让你用两种语言自信地掌握推理过程。

1. Introduction to Pythagoras’ Theorem | 勾股定理简介

Pythagoras’ theorem describes a special relationship between the three sides of a right‑angled triangle. It is named after the ancient Greek mathematician Pythagoras and is one of the most powerful tools used throughout secondary school mathematics and beyond.

勾股定理描述了直角三角形三条边之间的一种特殊关系。它得名于古希腊数学家毕达哥拉斯，是整个中学数学乃至后续学习中最强大的工具之一。

In any right‑angled triangle, the hypotenuse is the side opposite the right angle – it is always the longest side. The other two sides are called the legs, and they meet to form the right angle.

在任何直角三角形中，斜边是与直角相对的边——它永远是最长的边。另外两条边称为直角边，它们相交构成直角。

2. Understanding the Right‑Angled Triangle | 理解直角三角形

Before using the theorem, you must be able to identify the hypotenuse. Look for the square corner symbol that marks the right angle. The side that does not touch this corner is the hypotenuse.

在应用定理之前，你必须能够识别斜边。寻找标记直角的方形角符号。没有接触这个角的那条边就是斜边。

Label the hypotenuse as c, and label the two shorter sides as a and b. It does not matter which leg is called a and which is called b – both play the same role in the formula.

将斜边标记为 c，两条较短的直角边标记为 a 和 b。哪条直角边叫 a 哪条叫 b 并不重要——两者在公式中作用相同。

a² + b² = c²

3. The Theorem Formula | 定理公式

The standard form of Pythagoras’ theorem is written as a² + b² = c², where c is the length of the hypotenuse, and a and b are the lengths of the two legs. The squares are literal: you square each side length, add the squares of the legs, and you get the square of the hypotenuse.

勾股定理的标准形式写作 a² + b² = c²，其中 c 是斜边的长度，a 和 b 是两条直角边的长度。平方指的是字面意义上的平方：将每条边长平方，将两直角边的平方相加，就得到斜边的平方。

This equality always holds for right‑angled triangles. When sides are whole numbers that satisfy the equation, such as 3, 4, 5, we call them Pythagorean triples.

这个等式对直角三角形永远成立。当边长是满足该等式的整数时，例如3、4、5，我们称之为勾股数组。

4. Finding the Hypotenuse | 求斜边

To find the hypotenuse, substitute the leg lengths into a² + b² = c², calculate a² + b², and then take the square root of the result. For example, if a = 6 cm and b = 8 cm, then a² + b² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.

要求斜边，将直角边长代入 a² + b² = c²，计算 a² + b²，然后对结果开平方根。例如，如果 a = 6 cm, b = 8 cm，则 a² + b² = 6² + 8² = 36 + 64 = 100，因此 c = √100 = 10 cm。

Always remember to take the square root as the final step – a common mistake is to leave the answer as c² rather than c.

务必记住最后一步是开平方根——一个常见错误是把答案写成 c² 而非 c。

5. Finding a Shorter Side | 求直角边

When you know the hypotenuse and one leg, you can find the missing shorter side by rearranging the formula. If a is missing and c and b are known, use a² = c² − b².

当已知斜边和一条直角边时，你可以通过重新整理公式来求缺失的直角边。如果 a 未知而 c 和 b 已知，使用 a² = c² − b²。

Then take the square root to find a. For example, with c = 13 m and b = 5 m, a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 m.

然后开平方根求得 a。例如，c = 13 m，b = 5 m，a² = 13² − 5² = 169 − 25 = 144，因此 a = √144 = 12 m。

Remember that subtraction must happen before taking the root, and the hypotenuse squared is always the largest term.

记住必须是先减后开方，且斜边的平方永远是最大的那一项。

6. Checking for Right‑Angled Triangles | 验证直角三角形

The converse of Pythagoras’ theorem helps you test whether a triangle is right‑angled. If the three side lengths satisfy a² + b² = c² (with c being the longest side), the triangle is definitely right‑angled.

勾股定理的逆定理可以帮助你检验一个三角形是否为直角三角形。如果三条边长满足 a² + b² = c²（c 为最长边），那么这个三角形一定是直角三角形。

This check is widely used in construction and design. For example, a triangle with sides 5 cm, 12 cm, 13 cm gives 5² + 12² = 25 + 144 = 169 = 13², confirming a perfect right angle.

这种验证方法广泛应用于建筑和设计中。例如，边长分别为5 cm、12 cm、13 cm的三角形满足 5² + 12² = 25 + 144 = 169 = 13²，从而证实一个完美的直角。

7. Square Roots and Exact Answers | 平方根与精确值

Not every square root gives a neat integer answer. When c² is not a perfect square, leave your answer in surd form or round to a required degree of accuracy. For instance, if a = 1 cm and b = 2 cm, c = √(1² + 2²) = √5 cm. Leaving it as √5 is more precise than writing 2.236.

并非每一个平方根都会给出整齐的整数答案。当 c² 不是完全平方数时，将答案保留为根号形式，或按要求精确到指定位数。例如，如果 a = 1 cm, b = 2 cm，则 c = √(1² + 2²) = √5 cm。保留为 √5 比写作 2.236 更精确。

In KS3 Cambridge assessments, you are often asked to give answers to 1 decimal place or 2 decimal places. Make sure you use the appropriate rounding after calculating.

在剑桥KS3的测试中，经常要求将答案保留至1位或2位小数。确保在计算后使用正确的四舍五入。

8. Applying Pythagoras in Word Problems | 应用题中的勾股定理

Many real‑life problems can be solved with Pythagoras’ theorem. Draw a clear diagram, turn the situation into a right‑angled triangle, and label the known sides carefully.

许多现实问题都可以用勾股定理解决。画一张清晰的示意图，把情境转化为一个直角三角形，并仔细标注已知边长。

A classic example is a ladder leaning against a wall. The ladder forms the hypotenuse, the wall is one leg, and the ground distance is the other leg. If a 5 m ladder reaches a point 4 m up the wall, the foot of the ladder is √(5² − 4²) = 3 m away from the wall.

一个经典例子是靠墙的梯子。梯子构成斜边，墙壁是一条直角边，地面距离是另一条直角边。如果一架5米长的梯子靠在墙上达到4米高处，那么梯脚离墙的距离为 √(5² − 4²) = 3 m。

Always check that your answer makes sense – the hypotenuse must be longer than either leg.

一定要检查答案是否合理——斜边必须比任意一条直角边长。

9. Pythagoras in 3D: Cuboids | 三维中的勾股定理：长方体

Pythagoras’ theorem extends naturally into three dimensions. Inside a cuboid, you can first use the theorem to find the diagonal of the base, then combine that with the height to find the space diagonal.

勾股定理自然地延伸至三维空间。在长方体中，你可以先运用定理求出底面的对角线，然后再将其与高度结合以求得空间对角线。

For a cuboid with length l, width w and height h, the space diagonal d is given by d = √(l² + w² + h²). This is a direct application of Pythagoras twice – once in the base and once with the height.

对于一个长 l 、宽 w 、高 h 的长方体，空间对角线 d 可通过 d = √(l² + w² + h²) 求得。这是勾股定理的直接两次应用——先在底面应用一次，再结合高度应用一次。

Example: l = 3 cm, w = 4 cm, h = 12 cm → d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 cm.

示例： l = 3 cm, w = 4 cm, h = 12 cm → d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 cm。

10. Pythagoras in 3D: Pyramids and Other Shapes | 三维中的其他形状

In a right square‑based pyramid, you often need to find the slant height or the length of a sloping edge. Draw the right‑angled triangle inside the shape – the vertical height is one leg, half the base diagonal is another leg, and the sloping edge is the hypotenuse.

在正四棱锥中，你经常需要求斜高或侧棱的长度。在形状内部画出直角三角形——垂直高度是一条直角边，底面半对角线是另一条直角边，侧棱为斜边。

Break the problem into two steps: first find the base diagonal using Pythagoras on the base square, then halve it and use Pythagoras again with the vertical height.

将问题分成两步：先在底面正方形上用勾股定理求出底面对角线，然后取其一半，再结合垂直高度再次应用勾股定理。

This layered thinking appears frequently in Cambridge problem‑solving questions. Sketching the right‑angled triangle is the key to success.

这种分层思路在剑桥的解决问题类题目中频繁出现。画出相关的直角三角形是成功的关键。

11. Common Mistakes to Avoid | 常见错误

One typical error is adding before subtracting when finding a shorter side. Always isolate the missing squared term first. Another mistake is forgetting to take the square root at the end.

一个典型错误是在求直角边时先加后减。请务必先移项求出缺失边的平方项。另一个错误是最后忘记开平方根。

Students sometimes label the hypotenuse incorrectly, especially when the diagram is rotated. Remember: the hypotenuse is always opposite the right angle, regardless of the triangle’s orientation.

同学们有时会标错斜边，特别是当图形旋转后。记住：斜边永远在直角的对边，与三角形的朝向无关。

Rounding too early can also cause inaccuracies. Always keep exact values (surd form or calculator memory) until the very last step, then round as required.

过早四舍五入也会导致不准确。始终将精确值（根式或计算器存储值）保留到最后一步，然后按要求取近似值。

12. Practice Questions from Page 157 | 第157页的练习

Below are typical exercises similar to those found on Page 157, Exercise 1. Try them after studying the sections above.

下面是类似于第157页练习1的典型习题。学完上面各节后尝试完成。

A right‑angled triangle has legs of length 9 cm and 12 cm. Find the hypotenuse.

一个直角三角形的直角边长分别为9 cm和12 cm。求斜边的长度。
The hypotenuse of a right‑angled triangle is 17 m, and one leg is 8 m. Find the missing leg.

一个直角三角形的斜边为17 m，一条直角边为8 m。求缺失的直角边长。
A rectangle measures 6 cm by 8 cm. Calculate the length of its diagonal.

一个矩形长6 cm、宽8 cm。计算其对角线的长度。
A ship sails 20 km east and then 21 km north. How far is it from its starting point?

一艘船向东航行20 km，然后向北航行21 km。它离出发点有多远？
Find the space diagonal of a box with length 2 m, width 3 m, and height 6 m.

求长2 m、宽3 m、高6 m的长方体的空间对角线。

Work through each problem step by step, drawing a diagram and writing the equation a² + b² = c² (or its rearranged form) before substituting the values. Check that your answer is sensible – for instance, the hypotenuse must be greater than either leg.

逐步完成每道题，画出图形，写出方程 a² + b² = c²（或其变形形式），再代入数值。检查答案是否合理——比如斜边必须比任意一条直角边长。

This set of questions reinforces everything covered in the guide and prepares you perfectly for any topic assessment on Pythagoras’ theorem.

这套题目强化了本指南涵盖的所有内容，能让你完美应对勾股定理的任何主题测验。

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Pythagoras’ Theorem: Page 157 Exercise 1 Explained | 勾股定理：第157页练习1深度解析