Pythagoras’ Theorem: Finding Missing Sides | 勾股定理：求缺失边长

📚 Pythagoras’ Theorem: Finding Missing Sides | 勾股定理：求缺失边长

Pythagoras’ theorem is one of the most important results in Key Stage 3 mathematics. It connects the three sides of a right-angled triangle and helps us solve problems involving lengths, distances, and real-world measurements. In this article, we will explain the theorem step by step, practise using the formula a² + b² = c², and apply it to a variety of Cambridge-style questions.

勾股定理是 KS3 数学中最重要的结论之一。它把直角三角形的三条边联系起来，帮助我们解决涉及长度、距离和实际测量的问题。本文将逐步解释这个定理，练习使用公式 a² + b² = c²，并将其应用到各种剑桥风格的题目中。

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1. What is Pythagoras’ Theorem? | 什么是勾股定理？

Pythagoras’ theorem applies only to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

勾股定理只适用于直角三角形。它指出：斜边的平方等于另外两条边的平方和。

a² + b² = c²

Here, c represents the hypotenuse, which is the longest side opposite the right angle. The letters a and b stand for the two shorter sides, often called the legs.

这里，c 代表斜边，也就是直角对面的最长边。字母 a 和 b 代表两条较短的边，通常称为直角边。

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2. Identifying the Hypotenuse | 识别斜边

The hypotenuse is always the side opposite the right angle. It is also the longest side of a right-angled triangle.

斜边总是直角对面的边。它也是直角三角形中最长的一条边。

For example, in a triangle with sides 6 cm, 8 cm, and 10 cm, the 10 cm side must be the hypotenuse because it is opposite the largest angle (90°) and is longer than 6 cm and 8 cm.

例如，在一个边长为 6 cm、8 cm 和 10 cm 的三角形中，10 cm 的边一定就是斜边，因为它对着最大的角（90°），并且比 6 cm 和 8 cm 都长。

Labelling the hypotenuse as c is a good habit before using the formula.

在使用公式之前，养成把斜边标记为 c 的好习惯。

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3. The Formula in Action: Finding the Hypotenuse | 公式应用：求斜边

Suppose a right-angled triangle has shorter sides of 3 cm and 4 cm. To find the hypotenuse c, we substitute into a² + b² = c².

假设一个直角三角形的两条直角边分别为 3 cm 和 4 cm。为了求斜边 c，我们代入 a² + b² = c²。

Step 1: square each shorter side: 3² = 9 and 4² = 16.

第一步：将每条直角边平方：3² = 9，4² = 16。

Step 2: add the squares: 9 + 16 = 25.

第二步：把平方相加：9 + 16 = 25。

Step 3: take the positive square root: √25 = 5.

第三步：取正的平方根：√25 = 5。

c = √(3² + 4²) = √25 = 5 cm

The hypotenuse is 5 cm. This is the classic 3-4-5 triangle.

斜边为 5 cm。这就是经典的 3-4-5 三角形。

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4. Finding a Shorter Side | 求一条直角边

Sometimes you know the hypotenuse and one shorter side. To find the other shorter side, rearrange the formula: a² = c² – b².

有时你已知斜边和一条直角边。为了求另一条直角边，需要调整公式：a² = c² – b²。

Example: a ladder of length 10 m leans against a wall. The foot of the ladder is 6 m from the wall. Find the height reached on the wall.

例子：一架长 10 m 的梯子斜靠在墙上。梯子脚离墙 6 m。求梯子在墙上达到的高度。

Here c = 10 m, b = 6 m, so the height a is found by a² = 10² – 6² = 100 – 36 = 64, giving a = √64 = 8 m.

这里 c = 10 m，b = 6 m，所以高度 a 由 a² = 10² – 6² = 100 – 36 = 64 求得，因此 a = √64 = 8 m。

a = √(c² – b²) = √(10² – 6²) = 8 m

Always subtract the square of the known shorter side from the square of the hypotenuse, never the other way round.

一定要用斜边的平方减去已知直角边的平方，千万不要反过来减。

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5. Pythagorean Triples | 勾股数

Some sets of whole numbers fit the theorem exactly. These are called Pythagorean triples.

有些整数组成的集合刚好满足勾股定理。这些数称为勾股数。

Common Pythagorean triples:

常见勾股数：

a	b	c
3	4	5
5	12	13
6	8	10
8	15	17

Recognising these triples saves time in exams. For example, if you see sides 5, 12, and 13, you can immediately say the triangle is right-angled.

在考试中识别这些勾股数可以节省时间。例如，如果你看到边长为 5、12 和 13，就可以立刻判断这个三角形是直角三角形。

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6. Using the Theorem in Real-Life Problems | 实际问题中的应用

Many Cambridge KS3 questions present Pythagoras in context: a ladder, a sail, a diagonal of a rectangle, or a journey north then east.

许多剑桥 KS3 题目会在实际情境中考查勾股定理：梯子、船帆、矩形的对角线，或者先向北再向东的行程。

Example: A boat sails 9 km north and then 12 km east. How far is it from its starting point?

例子：一艘船向北航行 9 km，再向东航行 12 km。它离出发点有多远？

The two legs are 9 km and

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