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

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

Pythagoras’ theorem is named after the ancient Greek mathematician Pythagoras. It applies only to right-angled triangles and allows us to find an unknown side length when the other two sides are given. This topic is central to the Cambridge KS3 mathematics course and appears in many geometry, measurement and problem-solving questions.

勾股定理以古希腊数学家毕达哥拉斯命名。它只适用于直角三角形，当我们已知另外两条边时，可以用它来求未知的边长。这个主题是剑桥 KS3 数学课程的核心内容，出现在许多几何、测量和解决问题的问题中。

1. What is a Right-Angled Triangle? | 什么是直角三角形？

A right-angled triangle is a triangle in which one of the interior angles is exactly 90° (a right angle). The side opposite this right angle is always the longest side.

直角三角形是指其中一个内角恰好为 90°（直角）的三角形。直角所对的边总是最长的一条边。

In diagrams, the right angle is often marked with a small square. The other two angles are acute, which means they are less than 90° and add up to 90°.

在图中，直角通常用一个小的正方形符号标出。另外两个角是锐角，也就是说它们都小于 90°，并且它们的和为 90°。

2. The Hypotenuse: The Longest Side | 斜边：最长的一条边

The hypotenuse is the side opposite the right angle. It is always longer than either of the other two sides, which are called the legs or shorter sides.

斜边是直角所对的边。它总是比另外两条边（称为直角边或短边）中的任何一条都长。

Before using Pythagoras’ theorem, you must identify the hypotenuse correctly. Labelling sides as a, b and c with c as the hypotenuse helps avoid mistakes.

在使用勾股定理之前，你必须正确识别斜边。把边标记为 a、b 和 c，其中 c 为斜边，有助于避免错误。

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

Pythagoras’ theorem states that in a 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，在直角三角形中，斜边的平方等于另外两条边的平方之和。

This relationship can be written as: c² = a² + b², where c is the hypotenuse and a and b are the two shorter sides.

这个关系可以写成：c² = a² + b²，其中 c 是斜边，a 和 b 是两条较短的边。

c² = a² + b²

This formula only works for right-angled triangles. If the triangle does not have a right angle, the theorem cannot be used directly.

这个公式只适用于直角三角形。如果三角形没有直角，就不能直接使用该定理。

4. Key Formula and Its Meaning | 关键公式及其含义

The expression c² means c × c. Similarly, a² = a × a and b² = b × b. Squaring a length gives the area of a square with that side length.

表达式 c² 表示 c × c。同样地，a² = a × a，b² = b × b。将长度平方会得到以该长度为边长的正方形的面积。

Pythagoras’ theorem therefore has a geometric meaning: the area of the square on the hypotenuse equals the total area of the squares on the other two sides.

因此，勾股定理有一个几何意义：以斜边为边长的正方形面积等于以另外两条边为边长的正方形面积之和。

The equation can be rearranged to find any side. To find the hypotenuse, use c = √(a² + b²). To find a shorter side, use a = √(c² – b²).

这个方程可以变形来求任意一条边。求斜边时，用 c = √(a² + b²)。求一条短边时，用 a = √(c² – b²)。

c = √(a² + b²)

a = √(c² – b²)

5. Finding the Hypotenuse | 求斜边

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短边分别为 6 cm 和 8 cm。求斜边的长度。

Step 1: Write the formula c² = a² + b². Step 2: Substitute a = 6 and b = 8, so c² = 6² + 8² = 36 + 64 = 100.

步骤 1：写出公式 c² = a² + b²。步骤 2：代入 a = 6 和 b = 8，得到 c² = 6² + 8² = 36 + 64 = 100。

Step 3: Take the square root of both sides: c = √100 = 10 cm. Always include the units in your final answer.

步骤 3：两边取平方根：c = √100 = 10 cm。在你的最终答案中一定要写上单位。

This 6-8-10 triangle is one of the common Pythagorean triples: sets of whole numbers that satisfy the theorem.

这个 6-8-10 三角形是常见的勾股数之一：满足该定理的整数三元组。

6. Finding a Shorter Side | 求一条短边

Example: A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the remaining side.

例题：一个直角三角形的斜边为 13 cm，一条短边为 5 cm。求另一条边。

Use the rearranged formula a² = c² – b². Substitute c = 13 and b = 5: a² = 13² – 5² = 169 – 25 = 144.

使用变形后的公式 a² = c² – b²。代入 c = 13 和 b = 5：a² = 13² – 5² = 169 – 25 = 144。

Then a = √144 = 12 cm. This gives the 5-12-13 Pythagorean triple, another very useful set of numbers.

然后 a = √144 = 12 cm。这就得到了 5-12-13 勾股数，这是另一组非常有用的数。

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

在变形时，要用斜边的平方减去已知短边的平方，绝不能反过来。

7. Checking for Right Angles | 检验直角

Pythagoras’ theorem can also be used to test whether a triangle is right-angled. If the three sides satisfy a² + b² = c², the triangle is right-angled.

勾股定理还可以用来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，那么这个三角形就是直角三角形。

Example: A triangle has sides 9 cm, 12 cm and 15 cm. Check: 9² + 12² = 81 + 144 = 225, and 15² = 225. Since both sides are equal, the triangle is right-angled.

例题：一个三角形的三条边分别为 9 cm、12 cm 和 15 cm。检验：9² + 12² = 81 + 144 = 225，而 15² = 225。因为两边相等，所以这个三角形是直角三角形。

If the squared values are not equal, the triangle is not right-angled. Remember that c must be the longest side when testing.

如果平方值不相等，那么这个三角形就不是直角三角形。检验时要记住，c 必须是最长的边。

8. Real-Life Applications | 实际应用

Pythagoras’ theorem is used in many real-world situations, such as finding the diagonal of a rectangle, the shortest distance between two points, or the length of a ladder needed to reach a wall.

勾股定理在现实生活中有许多应用，例如求矩形的对角线、两点之间的最短距离，或求够到墙壁所需的梯子长度。

Example: A ladder is placed 3 m away from a vertical wall. The top of the ladder reaches 4 m up the wall. Find the length of the ladder.

例题：一把梯子放在离竖直墙壁 3 m 处。梯子顶部到达墙上 4 m 高的位置。求梯子的长度。

The ground and wall form a right angle, so the ladder is the hypotenuse: length² = 3² + 4² = 9 + 16 = 25, so length = √25 = 5 m.

地面和墙壁构成一个直角，所以梯子是斜边：长度² = 3² + 4² = 9 + 16 = 25，因此长度 = √25 = 5 m。

Other applications include finding distances on coordinate grids and calculating the length of a diagonal support in construction.

其他应用包括在坐标网格上求距离，以及计算建筑中对角支撑的长度。

9. Common Mistakes to Avoid | 常见错误

One common mistake is adding the two given sides without squaring them first. Always square each side before adding.

一个常见错误是先把两条已知边相加而没有先平方。一定要先平方每条边，然后再相加。

Another mistake is using the theorem on triangles that are not right-angled. Check that a right angle is present or that you are finding the diagonal of a rectangle or square.

另一个错误是对非直角三角形使用该定理。要检查是否存在直角，或者你是否在求矩形或正方形的对角线。

Students also sometimes forget to take the square root at the end. If you find c², remember to find c by taking the square root.

学生有时还会忘记最后取平方根。如果你求出的是 c²，要记得通过取平方根来求出 c。

Finally, do not forget to include units in practical problems. A length without a unit is incomplete in applied questions.

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