Pythagoras’ Theorem and Its Applications | 勾股定理及其应用

📚 Pythagoras’ Theorem and Its Applications | 勾股定理及其应用

Pythagoras’ theorem is one of the most famous results in geometry and a key topic in KS3 Cambridge Mathematics. On page 245 of the Cambridge Checkpoint Maths Practice Book, students explore how to find missing lengths in right‑angled triangles and how to apply the theorem to real‑world problems. This article recaps the essential theory, worked examples, common pitfalls, and extended applications to build a solid revision foundation.

勾股定理是几何学中最著名的结论之一，也是剑桥 KS3 数学的重要课题。在剑桥 Checkpoint 数学练习册第 245 页，同学们学习了如何在直角三角形中求未知边长，以及如何将定理应用到实际问题中。本文梳理了核心理论、典型例题、常见错误和拓展应用，帮助大家打下扎实的复习基础。

1. The Theorem Statement | 定理陈述

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 sides are labelled a, b and c, with c as the hypotenuse, the relationship is written as:

在任何一个直角三角形中，斜边的平方等于两条直角边的平方和。如果把三条边分别记作 a、b 和 c，其中 c 为斜边，则关系式写作：

c² = a² + b²

This formula only works for right‑angled triangles. It connects the three side lengths directly, which means that if you know any two sides, you can always calculate the third. In KS3 problems, the values are often whole numbers, but the theorem works for decimals and fractions as well.

这个公式仅适用于直角三角形。它把三条边长直接联系起来，意味着只要知道任意两条边，就一定能算出第三条边。在 KS3 的题目中，数值常常是整数，但定理对小数和分数同样适用。

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2. Identifying the Hypotenuse | 识别斜边

The hypotenuse is the longest side of a right‑angled triangle and is always opposite the right angle (90°). It is never adjacent to the right angle. In diagrams, look for the side that does not touch the square corner symbol ┐. Labelling the hypotenuse with c is a standard convention, but any letter can be used.

斜边是直角三角形中最长的一条边，并且总是正对着直角（90°）。它永远不会与直角相邻。在图形中，要找到那条不接触直角符号 ┐ 的边。通常用 c 表示斜边，但也可以使用其它字母。

If the triangle is drawn in different orientations, do not assume the sloping side is always the hypotenuse. Turn the diagram in your mind so that the right angle sits at the bottom left, then the side directly opposite, usually the longest, is the hypotenuse.

如果三角形以不同方向绘制，不要想当然地认为倾斜的边就一定是斜边。可以在脑海中旋转图形，让直角位于左下角，那么正对着直角的那条边——通常是最长的——就是斜边。

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3. Finding the Hypotenuse | 求斜边长

When the two shorter sides (often called legs) are known, find the hypotenuse by squaring both, adding them, and then taking the square root. For example, if a = 3 and b = 4, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5.

当已知两条较短的直角边时，求斜边的方法是：将它们分别平方，求和，再开平方。例如，a = 3、b = 4，则 c² = 3² + 4² = 9 + 16 = 25，所以 c = √25 = 5。

Remember that the square root operation gives the positive root because length is always positive. Many KS3 questions use exact square roots, so learners should be comfortable with perfect squares up to 15² = 225. If the sum is not a perfect square, leave the answer in surd form or round as directed.

记住，开平方运算取正根，因为长度总是正值。许多 KS3 题目得到的是完全平方数，因此同学们应当熟记 15² = 225 以内的平方数。如果和不是完全平方数，可保留根式形式或按要求四舍五入。

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4. Finding a Shorter Side | 求直角边

When the hypotenuse and one leg are known, rearrange the theorem to find the missing leg. From c² = a² + b², we obtain a² = c² − b². Always subtract the square of the known leg from the square of the hypotenuse, then take the square root. The result will be smaller than the hypotenuse.

当已知斜边和一条直角边时，需要变形公式来求未知的直角边。由 c² = a² + b² 可得 a² = c² − b²。务必用斜边的平方减去已知直角边的平方，再开平方。得到的结果一定比斜边短。

For instance, a ladder of length 5 m leans against a wall with its foot 2 m from the wall. The height reached is √(5² − 2²) = √(25 − 4) = √21 ≈ 4.58 m. This subtraction order is critical: many students mistakenly add instead of subtract.

例如，一架长 5 m 的梯子斜靠在墙上，梯脚距墙 2 m，到达的高度为 √(5² − 2²) = √(25 − 4) = √21 ≈ 4.58 m。这个减法顺序非常关键：很多同学会错误地用加法代替减法。

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5. Pythagorean Triples | 勾股数

A Pythagorean triple consists of three positive integers (a, b, c) that satisfy c² = a² + b². The most common triples are (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17) and their multiples. Recognising these can save time in exams and help check answers quickly.

勾股数（勾股三元组）是指满足 c² = a² + b² 的三个正整数 (a, b, c)。最常见的勾股数组有 (3, 4, 5)、(5, 12, 13)、(7, 24, 25)、(8, 15, 17) 以及它们的倍数。熟记这些数组能在考试中节省时间，并快速检查答案。

Triple (a, b, c)	Check c² = a² + b²
3, 4, 5	25 = 9 + 16
5, 12, 13	169 = 25 + 144
6, 8, 10 (×2)	100 = 36 + 64
9, 12, 15 (×3)	225 = 81 + 144

If you multiply each number in a triple by the same scale factor, you get another Pythagorean triple. This is a useful shortcut: for example, a triangle with sides 6, 8 and 10 must be right‑angled because it is a multiple of (3, 4, 5).

如果把勾股数组中的每个数乘以相同的倍数，会得到一组新的勾股数。这是一条很有用的捷径：比如，边长为 6、8、10 的三角形一定是直角三角形，因为它是 (3, 4, 5) 的倍数。

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6. Converse of Pythagoras | 勾股定理的逆定理

The converse states: if a triangle has sides of lengths a, b and c such that c² = a² + b², then the triangle is right‑angled with the right angle opposite side c. This is extremely useful for testing whether a triangle is right‑angled without a protractor.

逆定理指出：如果一个三角形的三边长 a、b、c 满足 c² = a² + b²，那么这个三角形是直角三角形，且直角正对边 c。这为不用量角器检验直角三角形提供了极为有用的方法。

For example, a triangle with sides 9 cm, 12 cm and 15 cm satisfies 15² = 9² + 12² (225 = 81 + 144), so the triangle contains a right angle opposite the 15 cm side. If the equality does not hold, the triangle is either acute or obtuse.

例如，一个三角形三边长分别为 9 cm、12 cm 和 15 cm，满足 15² = 9² + 12²（225 = 81 + 144），所以该三角形含有一个直角，且直角正对 15 cm 的边。如果等式不成立，那么这个三角形是锐角三角形或钝角三角形。

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7. Real‑Life Word Problems | 实际应用题

Pythagoras’ theorem appears in many practical contexts: a ladder leaning against a wall, the diagonal of a rectangle, the shortest distance across a park, or the length of a ramp. Always begin by drawing a labelled right‑angled triangle and identifying which side needs to be found.

勾股定理出现在许多实际情境中：梯子斜靠墙壁、矩形的对角线、穿过公园的最短路径，或者坡道的长度。解题时首先要画出带标注的直角三角形，并明确需要求哪条边。

In a typical question: ‘A rectangular football pitch is 100 m by 64 m. Calculate the length of the diagonal.’ The diagonal forms the hypotenuse of two right‑angled triangles. The answer is √(100² + 64²) = √(10000 + 4096) = √14096 ≈ 118.7 m. Stating the final answer with appropriate units and correct rounding is essential.

在典型题目中：“一个矩形足球场长 100 m、宽 64 m，计算对角线长。”对角线构成两个直角三角形的斜边。答案为 √(100² + 64²) = √(10000 + 4096) = √14096 ≈ 118.7 m。给出最终答案时必须带合适的单位并正确四舍五入。

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8. Pythagoras in 3D | 立体图形中的勾股定理

The theorem can be extended to find the space diagonal of a cuboid or the slant height of a cone. To find the space diagonal of a box with length l, width w and height h, first find the base diagonal d₁ = √(l² + w²), then use it with height to get the space diagonal d = √(d₁² + h²) = √(l² + w² + h²).

该定理可以拓展到求长方体的空间对角线或圆锥的斜高。要计算一个长、宽、高分别为 l、w、h 的长方体的空间对角线，首先求底面对角线 d₁ = √(l² + w²)，再用它与高度求得空间对角线 d = √(d₁² + h²) = √(l² + w² + h²)。

This two‑step method is common in KS3 extension work. For a box measuring 6 cm by 2 cm by 3 cm, the space diagonal is √(6² + 2² + 3²) = √(36 + 4 + 9) = √49 = 7 cm. Always sketch the shape and highlight the right‑angled triangles involved.

这种两步法是 KS3 拓展练习中的常见内容。对于一个尺寸为 6 cm × 2 cm × 3 cm 的长方体，空间对角线为 √(6² + 2² + 3²) = √(36 + 4 + 9) = √49 = 7 cm。解这类题时一定要画出立体示意图，标出涉及的直角三角形。

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9. Checking for Right Angles in Construction | 施工中直角的检验

Builders and carpenters use the 3‑4‑5 rule to ensure corners are exactly 90°. They measure 3 units along one edge, 4 units along the other, and check that the diagonal between the two marks is exactly 5 units. If it is, the angle is a perfect right angle.

建筑工人和木匠使用“3‑4‑5 法”确保墙角正好是 90°。他们沿一条边量出 3 个单位，沿另一条边量出 4 个单位，然后检查两个标记之间的对角线是否正好是 5 个单位。如果是，这个角就是标准直角。

This method relies on the converse of Pythagoras. In practice, multiples like 60 cm, 80 cm and 100 cm are often used for larger constructions. This application shows how the theorem connects classroom maths to real‑world precision.

这个方法正是基于勾股定理的逆定理。实际施工中常用 60 cm、80 cm 和 100 cm 这样的倍数来进行更大尺度的测量。这一应用展示了课堂数学如何与现实中的精度要求紧密相连。

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10. Common Misconceptions and Exam Tips | 常见误区与应考技巧

Mistake 1: Adding the squares of the hypotenuse and a leg when finding a shorter side. Always subtract: shorter side = √(hypotenuse² − known leg²). Mistake 2: Forgetting to take the square root at the end. The final length must be the square root of the sum or difference, not just the squared value.

误区一：在求直角边时把斜边和直角边的平方相加。一定要用减法：直角边 = √(斜边² − 已知直角边²)。误区二：最后忘记开平方。最终边长必须是和或差的平方根，不能只停留在平方值上。

Mistake 3: Misidentifying the hypotenuse when the right angle is not marked conventionally. Always confirm which side is opposite the 90° angle. Exam tip: show full working by writing the equation c² = a² + b², substitute values, and then solve. This earns method marks even if the final answer is slightly off.

误区三：当直角不是习惯位置时认错斜边。一定要确认哪条边正对 90° 角。应考技巧：写出完整的解题过程，先列出方程 c² = a² + b²，代入数值，再求解。即使最终答案稍有不准确，也能拿到过程分。

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11. Practice Questions Walkthrough | 典型例题演练

Question: A right‑angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the missing side. Solution: Let missing side = x. Then 13² = x² + 5² → 169 = x² + 25 → x² = 144 → x = √144 = 12 cm. Always check: does 13² = 12² + 5²? 169 = 144 + 25, correct.

题目：一个直角三角形斜边长 13 cm，一条直角边长 5 cm，求另一条直角边。解答：设未知边为 x，则 13² = x² + 5² → 169 = x² + 25 → x² = 144 → x = √144 = 12 cm。务必检验：13² = 12² + 5²？169 = 144 + 25，正确。

Question: A ship sails 8 km east and then 15 km north. How far is it from its starting point? Solution: The path forms a right‑angled triangle with legs 8 km and 15 km. The direct distance is the hypotenuse: √(8² + 15²) = √(64 + 225) = √289 = 17 km. The answer is a whole number because (8, 15, 17) is a Pythagorean triple.

题目：一艘船向东航行 8 km，再向北航行 15 km，它离出发点多远？解答：航线构成直角边为 8 km 和 15 km 的直角三角形。直线距离即为斜边：√(8² + 15²) = √(64 + 225) = √289 = 17 km。答案为整数，因为 (8, 15, 17) 是一组勾股数。

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12. Summary and Key Connections | 总结与知识链接

Pythagoras’ theorem is a tool for finding side lengths in right‑angled triangles, testing for right angles, and solving real‑life distance problems. It links directly to topics such as area, perimeters, coordinate geometry (distance between two points), and trigonometry in later years. Mastering the basics in KS3 lays the groundwork for GCSE and beyond.

勾股定理是求直角三角形边长、检验直角以及解决实际距离问题的利器。它直接关联到面积、周长、坐标几何（两点间距离）以及高年级的三角学等课题。在 KS3 打好基础，将为 GCSE 及更高层次的学习铺平道路。

Keep a list of common Pythagorean triples handy, practise drawing and labelling right‑angled triangles, and always decide whether the missing side is the hypotenuse or a leg before applying the formula. Consistent practice with word problems and 3D extensions will build both confidence and accuracy.

手边常备一份常见勾股数列表，多做绘制并标注直角三角形的练习，并在套用公式前先判断所求边是斜边还是直角边。通过不断练习应用题和立体拓展题，能够建立起信心，同时提高答题的准确性。

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