Pythagoras' Theorem | 勾股定理

📚 Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem is one of the most important results in IGCSE Mathematics. It connects the three sides of a right-angled triangle and allows you to find a missing length when the other two sides are known. The theorem is named after the ancient Greek mathematician Pythagoras, and it appears in many geometry, mensuration, and real-life problems.

勾股定理是 IGCSE 数学中最重要的结论之一。它将直角三角形的三条边联系起来，使你在已知另外两条边时能够求出一条缺失的边长。该定理以古希腊数学家毕达哥拉斯的名字命名，并出现在许多几何、测量和实际生活问题中。

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

Pythagoras’ theorem only works for right-angled triangles. A right-angled triangle has one interior angle of exactly 90°.

勾股定理只适用于直角三角形。直角三角形有一个内角恰好为 90°。

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

直角所对的边称为斜边。斜边始终是直角三角形中最长的边。

The other two sides are usually labelled a and b, and they form the right angle. These two sides are often called the shorter sides or the legs of the triangle.

另外两条边通常标记为 a 和 b，它们构成直角。这两条边通常被称为较短直角边或三角形的直角边。

a² + b² = c²

In this formula, c represents the hypotenuse, while a and b represent the two shorter sides.

在这个公式中，c 代表斜边，而 a 和 b 代表两条较短直角边。

2. The Formula and Its Rearrangements | 公式及其变形

The standard form of Pythagoras’ theorem is a² + b² = c², where c is the hypotenuse. To find the hypotenuse, you add the squares of the two shorter sides and then take the square root.

勾股定理的标准形式是 a² + b² = c²，其中 c 是斜边。要求斜边，你需要先求出两条较短边的平方和，然后再开平方。

To find a shorter side, you must rearrange the formula. For example, if you know c and b, you can find a by using a² = c² – b².

要求较短直角边，你必须对公式进行变形。例如，如果你已知 c 和 b，就可以使用 a² = c² – b² 来求出 a。

c = √(a² + b²)

a = √(c² – b²)

b = √(c² – a²)

Always check which side of the triangle is missing before choosing the correct form of the formula.

在选择正确的公式形式之前，始终要先检查三角形中缺失的是哪一条边。

3. Identifying the Hypotenuse | 识别斜边

The hypotenuse is always the side opposite the 90° angle. In a diagram, it is the slanted side that does not touch the right angle.

斜边始终是 90° 角所对的边。在图中，它是那条不接触直角的倾斜边。

Many students make mistakes by labelling the wrong side as c. Remember: the hypotenuse is the longest side, so it must have the greatest length in the triangle.

许多学生因为将错误的边标记为 c 而犯错。请记住：斜边是最长的边，所以它一定是三角形中长度最大的边。

If the triangle has side lengths of 3 cm, 4 cm, and 5 cm, the hypotenuse is 5 cm because 5 is the largest number and it is opposite the right angle.

如果三角形三条边的长度分别为 3 cm、4 cm 和 5 cm，那么斜边是 5 cm，因为 5 是最大的数，并且它位于直角的对面。

4. Finding the Hypotenuse | 求斜边

To find the hypotenuse, use the formula c = √(a² + b²). Start by squaring both shorter sides, add the results, and then take the square root.

要求斜边，请使用公式 c = √(a² + b²)。首先将两条较短边分别平方，将结果相加，然后再开平方。

Example: A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.

例题：一个直角三角形的两条较短边分别为 6 cm 和 8 cm。求斜边的长度。

c² = 6² + 8² = 36 + 64 = 100

c = √100 = 10 cm

The hypotenuse is 10 cm. Always write the correct unit in your final answer.

斜边长度为 10 cm。一定要在最终答案中写出正确的单位。

5. Finding a Shorter Side | 求较短直角边

When the hypotenuse and one shorter side are known, you must use subtraction. For example, if c = 13 cm and b = 5 cm, find a.

当已知斜边和一条较短直角边时，你必须使用减法。例如，如果 c = 13 cm 且 b = 5 cm，求 a。

a² = c² – b² = 13² – 5² = 169 – 25 = 144

a = √144 = 12 cm

The missing shorter side is 12 cm. Notice that the shorter side is always less than the hypotenuse, which is a useful check for your answer.

缺失的较短直角边为 12 cm。注意较短直角边始终小于斜边，这是检查答案是否有用的方法。

If your calculated shorter side turns out to be longer than the hypotenuse, you have probably added instead of subtracted.

如果你计算出的较短直角边竟然比斜边还长，那么你很可能做成了加法而不是减法。

6. Checking if a Triangle is Right-Angled | 判断三角形是否为直角三角形

Pythagoras’ theorem can also be used backwards to test whether a triangle is right-angled. If the three side lengths satisfy a² + b² = c², then the triangle has a right angle.

勾股定理还可以反向使用，以检验一个三角形是否为直角三角形。如果三条边满足 a² + b² = c²，那么这个三角形有一个直角。

Example: A triangle has sides of 9 cm, 12 cm, and 15 cm. Test whether it is right-angled.

例题：一个三角形的三条边分别为 9 cm、12 cm 和 15 cm。判断它是否为直角三角形。

First, identify the longest side as the possible hypotenuse: c = 15 cm. Then check:

首先，将最长边确定为可能的斜边：c = 15 cm。然后检验：

9² + 12² = 81 + 144 = 225

15² = 225

Since both values are equal, the triangle is right-angled.

由于两个值相等，因此这个三角形是直角三角形。

7. Real-Life Applications | 实际应用

Pythagoras’ theorem is used in navigation, construction, physics, and even computer graphics. It can calculate the shortest distance between two points when movements are at right angles.

勾股定理被用于导航、建筑、物理学甚至计算机图形学。当两个运动方向成直角时，它可以计算两点之间的最短距离。

A ladder leaning against a wall usually forms a right-angled triangle with the ground. The ladder is the hypotenuse, while the wall and the ground are the two shorter sides.

斜靠在墙上的梯子通常与地面构成一个直角三角形。梯子是斜边，而墙和地面是两条较短直角边。

The theorem also explains the distance between two points on a coordinate grid because the horizontal and vertical distances form two perpendicular sides of a right-angled triangle.

该定理也解释了坐标网格中两点之间的距离，因为水平距离和垂直距离构成了直角三角形的两条垂直边。

8. Pythagorean Triples | 勾股数

A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². The most common triples are (3, 4, 5), (5, 12, 13), and (7, 24, 25).

勾股数是一组满足 a² + b² = c² 的三个正整数。最常见的勾股数有 (3, 4, 5)、(5, 12, 13) 和 (7, 24, 25)。

Multiples of these triples also satisfy the theorem. For example, (6, 8, 10) is a multiple of (3, 4, 5), so it is also a Pythagorean triple.

这些勾股数的倍数也满足该定理。例如，(6, 8, 10) 是 (3, 4, 5) 的倍数，因此它也是一组勾股数。

Recognising common Pythagorean triples can save time in exams because you can write down the missing side without recalculation.

识别常见的勾股数可以在考试中节省时间，因为你可以直接写出缺失的边长而无需重新计算。

9. Common Mistakes | 常见错误

One common mistake is adding the squares when finding a shorter side. When finding a shorter side, you must subtract the known square from the hypotenuse squared.

一个常见错误是在求较短直角边时将平方相加。求较短直角边时，必须用斜边的平方减去已知直角边的平方。

Another common mistake is forgetting to take the square root at the end. The formula gives c², not c, so the final step must be a square root.

另一个常见错误是最后忘记开平方。公式给出的是 c²，而不是 c，因此最后一步必须开平方。

Some students also label the longest side as a or b and a shorter side as c. Always label the longest side as c before applying the theorem.

有些学生还会把最长的边标记为 a 或 b，而把较短边标记为 c。在应用定理之前，一定要把最长的边标记为 c。

10. Exam Tips for IGCSE | IGCSE 考试技巧

In IGCSE exams, always show your substitution into the formula before calculating. This allows method marks even if you make a small arithmetic error.

在 IGCSE 考试中，始终先写出代入公式的过程再进行计算。这样即使你出现了小的算术错误，也能获得方法分。

When a question uses a diagram, write the lengths on the diagram and label the hypotenuse clearly. This helps you choose the correct form of the formula.

当题目给出图形时，请在图形上标出长度，并清楚地标出斜边。这有助于你选择正确的公式形式。

Check that your answer is reasonable. The hypotenuse must be longer than either shorter side, and a shorter side must be less than the hypotenuse.

检查你的答案是否合理。斜边必须比任何一条较短直角边都长，而较短直角边必须小于斜边。

11. Worked Example: Distance on a Coordinate Plane | 例题详解：坐标平面上的距离

Problem: Find the distance between the points A(2, 3) and B(5, 7) using Pythagoras’ theorem.

问题：使用勾股定理求点 A(2, 3) 和点 B(5, 7) 之间的距离。

The horizontal difference is 5 – 2 = 3, and the vertical difference is 7 – 3 = 4. These two differences are the shorter sides of a right-angled triangle.

水平差为 5 – 2 = 3，垂直差为 7 – 3 = 4。这两个差值就是直角三角形的两条较短直角边。

d² = 3² + 4² = 9 + 16 = 25

d = √25 = 5

The distance between the points is 5 units. This is the basis of the distance formula used in coordinate geometry.

两点之间的距离为 5 个单位。这是坐标几何中距离公式的基础。

12. Summary | 小结

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.

勾股定理指出：在任何直角三角形中，斜边的平方等于另外两条边的平方之和。

Use c = √(a² + b²) to find the hypotenuse, and use a = √(c² – b²) to find a shorter side. Always identify the hypotenuse first and check your units.

使用 c = √(a² + b²) 求斜边，使用 a = √(c² – b²) 求较短直角边。始终先识别斜边并检查单位。

Practise with both numerical and real-life problems, and memorise the common Pythagorean triples to improve your speed and accuracy in the IGCSE exam.

请通过数值问题和实际生活问题进行练习，并记住常见的勾股数，以提高你在 IGCSE 考试中的速度和准确性。

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Pythagoras’ Theorem | 勾股定理