Pythagoras' Theorem: Finding Missing Sides and Real-Life Applications | 勾股定理：求缺失边长与实际应用

📚 Pythagoras’ Theorem: Finding Missing Sides and Real-Life Applications | 勾股定理：求缺失边长与实际应用

Pythagoras’ theorem is one of the most important results in KS3 Cambridge mathematics. It connects the three sides of a right-angled triangle and allows us to calculate an unknown side length when the other two sides are known. This topic appears frequently in checkpoint tests, end-of-stage exams and real-world problem solving. In this article, we will cover the statement of the theorem, how to identify the hypotenuse, how to find both the hypotenuse and a shorter side, common mistakes, and exam-style applications.

勾股定理是剑桥 KS3 数学中最重要的结论之一。它把直角三角形三条边的长度联系起来，使我们能够在已知另外两条边时求出未知边长。这个考点经常出现在 Checkpoint 测试、阶段末考试和实际问题中。在本文中，我们将学习定理的表述、如何识别斜边、如何求斜边和直角边、常见错误以及考试风格的应用。

1. Introduction to Right-Angled Triangles | 直角三角形入门

A right-angled triangle is any triangle that contains one 90° angle. The side directly opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are called the legs or shorter sides, and they form the right angle.

直角三角形是包含一个 90° 角的三角形。直角正对的边叫作斜边，它总是最长的边。另外两条边叫作直角边或较短边，它们构成直角。

In Cambridge KS3 problems, the right angle is usually marked with a small square. Recognising this symbol helps you identify the triangle type immediately. Without a right angle, Pythagoras’ theorem cannot be used directly.

在剑桥 KS3 题目中，直角通常用一个小的正方形符号标出。识别这个符号能帮助你立即判断三角形的类型。如果没有直角，就不能直接使用勾股定理。

2. Statement of Pythagoras’ Theorem | 勾股定理的表述

Pythagoras’ theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If the hypotenuse is labelled c and the other two sides are labelled a and b, then the relationship is written as:

勾股定理指出，在任何一个直角三角形中，斜边的平方等于另外两条边的平方之和。如果把斜边记为 c，另外两条边记为 a 和 b，那么这个关系可以写成：

c² = a² + b²

Here, a and b are interchangeable because addition is commutative, but c must always be the hypotenuse. This formula is the foundation for all missing-side calculations. You should memorise it and be able to rearrange it.

这里 a 和 b 可以互换，因为加法满足交换律，但 c 必须始终是斜边。这个公式是所有缺失边长计算的基础。你应当记住它，并能够对它进行变形。

3. Identifying the Hypotenuse | 识别斜边

The hypotenuse is the side opposite the right angle. It does not touch the right angle. In a diagram, it is usually the slanted side when the right angle is at the bottom left or bottom right of the triangle. A common exam skill is being able to label the hypotenuse correctly before substituting values.

斜边是直角对面的边。它不与直角接触。在图中，当直角位于三角形左下角或右下角时，斜边通常是倾斜的那条边。考试中的常见技能是在代入数值之前能够正确地标出斜边。

For example, if a triangle has sides of 6 cm, 8 cm and 10 cm, the side of 10 cm must be the hypotenuse because it is the longest side. You can check this by confirming that it is opposite the right angle.

例如，如果一个三角形的三条边分别为 6 cm、8 cm 和 10 cm，那么 10 cm 这条边一定是斜边，因为它是最长的边。你可以通过确认它是否正对直角来检验。

4. Finding the Hypotenuse | 求斜边

When you know the two shorter sides and need to find the hypotenuse, use the formula c² = a² + b² directly. First substitute the known values, then add the squares, and finally take the square root of the result. Do not forget the square root step.

当你已知两条直角边而需要求斜边时，直接使用公式 c² = a² + b²。首先代入已知值，然后求平方和，最后对结果开平方。不要忘记开平方这一步。

Example: A right-angled triangle has legs of 5 cm and 12 cm. Find the hypotenuse.

例题：一个直角三角形的两条直角边分别为 5 cm 和 12 cm，求斜边。

c² = 5² + 12² = 25 + 144 = 169 → c = √169 = 13 cm

Always include the units in your final answer. In Cambridge checkpoint questions, a missing unit can lose a mark even if the number is correct.

在最终答案中一定要写单位。在剑桥 Checkpoint 题目中，即使数值正确，缺少单位也可能被扣分。

5. Finding a Shorter Side | 求直角边

If you know the hypotenuse and one of the shorter sides, you need to rearrange the theorem. The formula becomes a² = c² – b² or b² = c² – a². Subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.

如果你已知斜边和其中一条直角边，就需要对定理进行变形。公式变为 a² = c² – b² 或 b² = c² – a²。用斜边的平方减去已知直角边的平方，然后开平方。

Example: A right-angled triangle has a hypotenuse of 10 cm and one shorter side of 6 cm. Find the other shorter side.

例题：一个直角三角形的斜边为 10 cm，一条直角边为 6 cm，求另一条直角边。

a² = 10² – 6² = 100 – 36 = 64 → a = √64 = 8 cm

Notice that the unknown side is not always labelled a. In an exam, label the sides yourself before substituting. This helps avoid mixing up the hypotenuse and the shorter sides.

注意未知边并不总是标记为 a。在考试中，先自己标出各边，再代入数值。这有助于避免混淆斜边和直角边。

6. Pythagorean Triples | 勾股数

A Pythagorean triple is a set of three positive integers that satisfy the equation a² + b² = c². The most common triple is (3, 4, 5), because 3² + 4² = 9 + 16 = 25 = 5². Other examples include (5, 12, 13), (7, 24, 25) and (8, 15, 17).

勾股数是一组满足方程 a² + b² = c² 的正整数。最常见的勾股数是 (3, 4, 5)，因为 3² + 4² = 9 + 16 = 25 = 5²。其他例子包括 (5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。

Recognising these triples can save time in exams. If you see a right-angled triangle with legs 6 cm and 8 cm, you can quickly state the hypotenuse is 10 cm because 6-8-10 is a multiple of 3-4-5.

识别这些勾股数可以在考试中节省时间。如果你看到一个直角三角形两条直角边为 6 cm 和 8 cm，你可以快速说出斜边是 10 cm，因为 6-8-10 是 3-4-5 的倍数。

7. Real-Life Applications | 实际应用

Pythagoras’ theorem is used in many real-life situations, such as finding the diagonal of a rectangle, the distance between two points on a map, or the length of a ladder needed to reach a wall. In these problems, you often have to draw a right-angled triangle first.

勾股定理在许多实际情境中都有应用，例如求矩形的对角线、地图上两点之间的距离，或者够到墙壁所需的梯子长度。在这些问题中，你通常需要先画出一个直角三角形。

Example: A ladder is placed 3 m from a wall and reaches 4 m up the wall. How long is the ladder?

例题：一架梯子放在离墙 3 m 的地方，靠在墙上达到 4 m 的高度。梯子有多长？

ladder² = 3² + 4² = 9 + 16 = 25 → ladder = √25 = 5 m

Always check that the answer makes sense in context. A ladder length should be longer than either the horizontal or vertical distance but shorter than their sum.

始终检查答案在情境中是否合理。梯子的长度应当大于水平距离或垂直距离，但小于两者之和。

8. Proof of the Theorem | 定理的证明

There are many proofs of Pythagoras’ theorem. One simple visual proof uses squares built on each side of a right-angled triangle. The area of the square on the hypotenuse is exactly equal to the combined areas of the squares on the other two sides.

勾股定理有多种证明方法。一个简单的直观证明使用直角三角形各边上的正方形。斜边上的正方形面积恰好等于另外两条边上的正方形面积之和。

For a 3-4-5 triangle, the area of the square on the hypotenuse is 25 square units, while the squares on the shorter sides have areas 9 and 16 square units. Since 9 + 16 = 25, the theorem holds.

对于 3-4-5 三角形，斜边上的正方形面积为 25 平方单位，而两条直角边上的正方形面积分别为 9 和 16 平方单位。由于 9 + 16 = 25，定理成立。

9. Converse of Pythagoras’ Theorem | 勾股定理的逆定理

The converse of Pythagoras’ theorem states that if a triangle has sides a, b and c such that a² + b² = c², then the triangle is right-angled, with the right angle opposite the side c. This is useful for testing whether a triangle is right-angled.

勾股定理的逆定理指出，如果一个三角形的三条边 a、b 和 c 满足 a² + b² = c²，那么这个三角形就是直角三角形，且直角正对边 c。这可以用来检验一个三角形是否为直角三角形。

Example: Is a triangle with sides 9 cm, 12 cm and 15 cm right-angled?

例题：边长分别为 9 cm、12 cm 和 15 cm 的三角形是直角三角形吗？

9² + 12² = 81 + 144 = 225 = 15², so yes, it is right-angled.

This is a frequent question in Cambridge exams. Remember to square the two shorter sides first and then compare the result with the square of the longest side.

这是剑桥考试中常见的题型。记住先求两条较短边的平方和，再与最长边的平方进行比较。

10. Common Mistakes | 常见错误

One common mistake is using the theorem on a triangle that is not right-angled. Always check for the right-angle symbol. Another mistake is adding or subtracting the wrong sides. When finding a shorter side, you must subtract, not add.

一个常见错误是在非直角三角形上使用定理。一定要检查是否有直角符号。另一个错误是加错或减错边。求直角边时，必须用减法，而不是加法。

Students also forget to take the square root at the end. If c² = 169, then c = 13, not 169. Finally, remember to give the correct unit, such as cm, m or mm, in the final answer.

学生也常常忘记最后开平方。如果 c² = 169，那么 c = 13，而不是 169。最后，记住在最终答案中给出正确的单位，如 cm、m 或 mm。

11. Exam-Style Practice | 考试风格练习

Try these exam-style questions to check your understanding. First, find the hypotenuse of a right-angled triangle with legs 7 cm and 24 cm. Second, find the missing shorter side if the hypotenuse is 17 cm and one shorter side is 8 cm. Third, determine whether a triangle with sides 5 m, 10 m and 12 m is right-angled.

尝试以下考试风格的问题来检查你的理解。第一，求两条直角边分别为 7 cm 和 24 cm 的直角三角形的斜边。第二，如果斜边为 17 cm，一条直角边为 8 cm，求另一条直角边。第三，判断边长分别为 5 m、10 m 和 12 m 的三角形是否为直角三角形。

Question 1: c² = 7² + 24² = 49 + 576 = 625 → c = 25 cm
问题 1：c² = 7² + 24² = 49 + 576 = 625 → c = 25 cm
Question 2: a² = 17² – 8² = 289 – 64 = 225 → a = 15 cm
问题 2：a² = 17² – 8² = 289 – 64 = 225 → a = 15 cm
Question 3: 5² + 10² = 25 + 100 = 125, but 12² = 144, so it is not right-angled.
问题 3：5² + 10² = 25 + 100 = 125，而 12² = 144，所以它不是直角三角形。

Practising these types of questions will help you gain confidence before your Cambridge checkpoint test.

练习这些类型的题目将帮助你在剑桥 Checkpoint 测试前建立信心。

12. Summary | 小结

Pythagoras’ theorem only works on right-angled triangles. The formula c² = a² + b² connects the hypotenuse c with the shorter sides a and b. To find the hypotenuse, add the squares and take the square root. To find a shorter side, subtract the squares and take the square root.

勾股定理只适用于直角三角形。公式 c² = a² + b² 将斜边 c 与直角边 a 和 b 联系起来。要求斜边，先求平方和再开平方。要求直角边，先求平方差再开平方。

Remember to check for the right-angle symbol, label sides carefully, use correct units, and watch out for Pythagorean triples. With regular practice, you can apply the theorem accurately in both pure and real-life problems.

记住检查直角符号、仔细标注各边、使用正确的单位，并留意勾股数。通过经常练习，你可以在纯数学问题和实际应用中准确使用这一定理。

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Pythagoras’ Theorem: Finding Missing Sides and Real-Life Applications | 勾股定理：求缺失边长与实际应用