Pythagoras' Theorem: Finding Side Lengths in Right-Angled Triangles | 勾股定理：求直角三角形边长

📚 Pythagoras’ Theorem: Finding Side Lengths in Right-Angled Triangles | 勾股定理：求直角三角形边长

Welcome to your KS3 Cambridge Mathematics guide on Pythagoras’ theorem. This article will help you understand the relationship between the sides of a right-angled triangle, learn how to find missing lengths, and apply the theorem to real-world problems. We will use clear English and Chinese explanations paired together, so you can learn the concept thoroughly and boost your confidence for checkpoint assessments.

欢迎来到 KS3 剑桥数学勾股定理学习指南。本文将帮助你理解直角三角形三边之间的关系，学会如何求未知边长，并将定理应用于实际问题。我们将采用中英双语配对讲解，让你扎实掌握这一概念，增强在 Checkpoint 测评中的信心。

Pythagoras’ theorem is a key topic in the Cambridge Lower Secondary Mathematics curriculum, especially in Stage 9. It links algebra and geometry, and it forms the foundation for later work in trigonometry and coordinate geometry.

勾股定理是剑桥初中数学课程（尤其是 Stage 9）的核心主题之一。它连接了代数与几何，并为后续的三角函数、坐标几何等内容打下基础。

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

Pythagoras’ theorem is named after the ancient Greek mathematician Pythagoras, who lived around 570–495 BCE. The theorem describes a special relationship that holds true for every right-angled triangle. A right-angled triangle is a triangle that has one angle exactly equal to 90°.

勾股定理以古希腊数学家毕达哥拉斯（约公元前570–495年）命名。该定理描述了所有直角三角形都满足的一种特殊边关系。直角三角形是指其中一个角恰好等于90°的三角形。

The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This statement can be written as a simple equation that allows you to find a missing side if you know the other two.

该定理表明，在直角三角形中，斜边（直角所对的边）长度的平方等于另外两条边长度的平方和。这个关系可以用一个简洁的方程表示，只要知道其中两条边的长度，就可以求出未知边的长度。

The side opposite the right angle is called the hypotenuse. It is always the longest side in a right-angled triangle. The other two sides are often called the legs or shorter sides.

直角所对的边称为斜边，它始终是直角三角形中最长的边。另外两条边通常称为直角边或短边。

2. The Key Formula a² + b² = c² | 核心公式 a² + b² = c²

The formula that represents Pythagoras’ theorem is a² + b² = c², where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. It does not matter which leg you label as a and which as b, as long as c is the hypotenuse.

表示勾股定理的公式为 a² + b² = c²，其中 c 代表斜边的长度，a 和 b 代表两条直角边的长度。至于哪条直角边标为 a、哪条标为 b 并不重要，只要确保 c 是斜边即可。

a² + b² = c²

This equation means that if you take the length of one leg, square it, take the length of the other leg, square it, and then add those two values together, you obtain exactly the square of the hypotenuse length. Squaring a number means multiplying the number by itself. For instance, 3² = 3 × 3 = 9.

这个等式表示，将一条直角边的长度平方，再将另一条直角边的长度平方，然后将两个平方值相加，所得之和恰好等于斜边长度的平方。平方就是一个数乘以它本身，例如 3² = 3 × 3 = 9。

Because the hypotenuse is always the longest side, its square is always larger than the square of either leg alone. This property helps you check whether a problem has been set up correctly.

由于斜边总是最长边，它的平方一定大于任何一条直角边的平方。这一性质有助于你检查题目设置是否正确。

3. Identifying the Hypotenuse | 识别斜边

Before using the theorem, you must be able to identify the hypotenuse correctly. Look for the side that is opposite the right angle. You can also look for the longest side, as the hypotenuse is always the longest side in a right-angled triangle.

在使用定理前，必须能正确识别斜边。找找直角所对的那条边。你也可以寻找最长的边，因为在直角三角形中斜边总是最长边。

Sometimes the right angle is marked with a small square, which is a standard geometric symbol for a 90° angle. If you see this square, the side directly across from it is the hypotenuse.

有时直角会用一个小正方形标记，这是90°角的标准几何符号。如果看到这个正方形标记，那么正对着它的那条边就是斜边。

Label the hypotenuse as c in your diagram. Then label the other two sides as a and b. This labelling will help you substitute the correct values into the formula a² + b² = c².

在图上将斜边标为 c，然后将另外两条边标为 a 和 b。做好标注有助于你将正确的数值代入公式 a² + b² = c²。

4. Finding the Hypotenuse – Worked Example | 求斜边 – 示例讲解

Suppose a right-angled triangle has legs of lengths 6 cm and 8 cm. Find the length of the hypotenuse. Step 1: Write down the formula: a² + b² = c².

假设一个直角三角形的两条直角边长分别为6厘米和8厘米，求斜边的长度。第1步：写出公式 a² + b² = c²。

Step 2: Substitute a = 6 and b = 8. So 6² + 8² = c². Calculate the squares: 36 + 64 = c². Add them: 100 = c².

第2步：代入 a = 6，b = 8，得到 6² + 8² = c²。计算平方：36 + 64 = c²。相加得 100 = c²。

Step 3: To find c, take the square root of both sides. The square root of c² is c, and the square root of 100 is 10. Therefore, c = 10 cm. Note that length is positive, so we ignore the negative root.

第3步：为求 c，对等式两边同时开平方根。c² 的平方根是 c，100 的平方根是 10。因此 c = 10 厘米。注意边长是正数，所以忽略负根。

c = √(6² + 8²) = √(36 + 64) = √100 = 10

If the squares do not give a perfect square number, leave the answer as a square root in simplified form or rounded to a suitable number of decimal places as instructed.

如果平方和不是完全平方数，那么答案可保留为简化后的二次根式，或按题目要求四舍五入到合适的小数位。

5. Finding a Shorter Side – Worked Example | 求直角边 – 示例讲解

The formula a² + b² = c² can be rearranged to find a missing shorter side. If you are given the hypotenuse and one leg, you can find the other leg by subtracting the square of the known leg from the square of the hypotenuse.

公式 a² + b² = c² 可以变形，用于求未知的直角边。如果已知斜边和一条直角边，可以通过斜边的平方减去已知直角边的平方，再开方求得另一条直角边。

Suppose the hypotenuse is 13 m and one leg is 5 m. Let the unknown leg be a. Write the equation: a² + 5² = 13². Then a² + 25 = 169. Subtract 25 from both sides: a² = 144. Take the square root: a = √144 = 12 m.

假设斜边为13米，一条直角边为5米。设未知直角边为 a。列出方程：a² + 5² = 13²，即 a² + 25 = 169。两边同时减去25：a² = 144。开平方根得 a = √144 = 12 米。

The rearrangement can be written directly as a = √(c² − b²). Always subtract the known leg square from the hypotenuse square, never the other way around, because the hypotenuse square is the largest.

上述变形可直接写作 a = √(c² − b²)。务必用斜边平方减去已知直角边平方，不能反过来，因为斜边平方是最大的。

Double-check your answer: 5² + 12² = 25 + 144 = 169, and 13² = 169. The equation holds true.

检验答案：5² + 12² = 25 + 144 = 169，13² = 169，等式成立。

6. Checking for a Right-Angled Triangle | 验证直角三角形

Pythagoras’ theorem also works in reverse. If the three sides of a triangle satisfy the equation a² + b² = c², with c being the longest side, then the triangle must be right-angled, and the angle opposite side c is 90°.

勾股定理的逆定理同样成立。如果一个三角形的三边长满足 a² + b² = c²（其中 c 为最长边），则该三角形一定是直角三角形，且 c 边所对的角为90°。

For example, consider a triangle with sides 9 cm, 12 cm, and 15 cm. Check whether 9² + 12² equals 15². 9² = 81, 12² = 144, sum = 225. 15² = 225. Since the two values are equal, the triangle is right-angled.

例如，考虑一个边长分别为9厘米、12厘米、15厘米的三角形。检验 9² + 12² 是否等于 15²。9² = 81，12² = 144，和为225。15² = 225。两边相等，因此该三角形是直角三角形。

This method is very useful for checking whether a corner is perfectly square in construction or in coordinate geometry problems.

这种方法在建筑中检验角是否为直角，或在坐标几何问题中非常有用。

7. Pythagorean Triples | 勾股数组

A Pythagorean triple consists of three positive integers (a, b, c) that satisfy a² + b² = c². The angles in such a triangle are determined by the ratios of the sides, but the triple itself is just a set of whole numbers.

勾股数组是由三个正整数 (a, b, c) 组成的数组，满足 a² + b² = c²。这种三角形中的角由边长之比决定，而数组本身只是一组整数。

The simplest and most famous triple is (3, 4, 5). Any set of multiples of this triple, such as (6, 8, 10) or (9, 12, 15), is also a Pythagorean triple. Knowing common triples can save time in examinations.

最简单且最著名的勾股数组是 (3, 4, 5)。这一组数的任何倍数，如 (6, 8, 10) 或 (9, 12, 15)，也都是勾股数组。熟记常用数组能节省考试时间。

Here are a few common Pythagorean triples you might encounter in KS3 Cambridge Mathematics:

下面是你在 KS3 剑桥数学中可能遇到的几组常见勾股数组：

a (leg 1 / 直角边1)	b (leg 2 / 直角边2)	c (hypotenuse / 斜边)
3	4	5
5	12	13
7	24	25
8	15	17

Remember, you can create new triples by multiplying each number in an existing triple by the same positive integer.

记住，你可以将现有数组中的每一个数乘以同一个正整数，从而得到新的勾股数组。

8. Real-Life Applications | 实际应用

Pythagoras’ theorem is not just a classroom exercise; it has many practical uses. For example, if you need to find the diagonal length of a rectangular screen or a door, you can use the theorem. The diagonal acts as the hypotenuse of the right-angled triangle formed by the rectangle’s length and width.

勾股定理不仅仅是课堂练习，它有许多实际用途。例如，要计算矩形屏幕或房门的对角线长度，就可以运用该定理。对角线相当于由矩形的长和宽构成的直角三角形的斜边。

Another common application is in navigation. A plane flying north and then east traces a right-angled path. The direct distance from the starting point is the hypotenuse. Similarly, a ladder leaning against a vertical wall forms a right-angled triangle with the ground.

另一个常见应用是在导航方面。飞机先向北飞再向东飞，其路径构成一个直角三角形，起点到终点的直线距离即为斜边。同样，一架梯子斜靠在竖直墙壁上也与地面构成直角三角形。

To solve such problems, sketch the situation, identify the right angle, label the sides, and decide whether you need to find the hypotenuse or a shorter side. Then substitute the known lengths and solve.

解答此类问题时，先画出示意图，找出直角，标出各边，明确需要求斜边还是直角边，然后代入已知长度并求解。

9. Common Errors and How to Avoid Them | 常见错误与应对方法

One common mistake is misidentifying the hypotenuse. Always check for the side opposite the right angle and confirm it is the longest. Adding the squares of the hypotenuse and a leg to find the other leg is incorrect; you must subtract.

一个常见错误是错误识别斜边。务必确认直角所对的边，并检查它是否最长。用斜边平方加上直角边平方去求另一个直角边是错误的；必须用减法。

Another error is forgetting to take the square root at the end. After finding c² = 100, students sometimes write c = 100 instead of c = √100 = 10. Take care to complete the final step.

另一个错误是忘记在最后一步开平方根。求出 c² = 100 后，有些学生会误写为 c = 100，而正确的结果应是 c = √100 = 10。注意把最后一步做完。

Also, watch out for mixing units. Ensure all given lengths are in the same unit before squaring. If one side is in centimetres and another in metres, convert them to the same unit first.

另外，要注意统一单位。在平方之前必须确保所有已知长度使用相同的单位。如果一条边用厘米而另一条边用米，首先要将它们转换为相同单位。

When rounding an irrational square root answer, do not round intermediate steps. Keep the exact square root in the calculator and round only the final answer to the required number of decimal places.

当无理数开平方运算需要四舍五入时，中间步骤不要取近似值。将计算器中的精确根值保留，最后再按所要求的小数位数对最终答案进行四舍五入。

10. Practice Questions and Summary | 练习题与总结

Let us consolidate your understanding with a few quick questions. Try to answer them using the methods described above, and check your solutions against the steps we discussed.

让我们通过几道快速练习题来巩固理解。尝试用上述方法作答，并与我们讨论过的步骤对照检查。

A right-angled triangle has legs of length 9 cm and 12 cm. Find the hypotenuse.
一个直角三角形两条直角边长分别为9厘米和12厘米，求斜边。

The hypotenuse of a right-angled triangle is 25 m, and one leg is 24 m. Calculate the length of the other leg.
一个直角三角形的斜边为25米，一条直角边为24米。求另一条直角边的长度。

Decide whether a triangle with sides 10 cm, 24 cm, and 26 cm is right-angled.
判断边长分别为10厘米、24厘米和26厘米的三角形是否为直角三角形。

A ladder of length 5 m leans against a wall, with its foot 1.5 m from the wall. How high up the wall does the ladder reach?
一架长5米的梯子斜靠在墙上，梯脚离墙1.5米。梯子顶端能达到墙多高的位置？

In summary, Pythagoras’ theorem a² + b² = c² applies only to right-angled triangles. The hypotenuse is the side opposite the right angle and is always the longest. To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract the square of the known leg from the square of the hypotenuse and then take the square root.

总结一下，勾股定理 a² + b² = c² 仅适用于直角三角形。斜边是直角所对的边，也是最长边。求斜边时，将两条直角边的平方相加再开方。求直角边时，用斜边的平方减去已知直角边的平方再开方。

Keep practising with different sets of numbers and word problems. The more you apply the theorem, the more natural it will feel to recognise right-angled situations and set up the equation correctly.

多练习不同类型的数字和应用题。运用定理的次数越多，你就越能自然地识别出直角三角形的情景并正确列出方程。

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Pythagoras’ Theorem: Finding Side Lengths in Right-Angled Triangles | 勾股定理：求直角三角形边长