Pythagoras' Theorem in Right-Angled Triangles | 直角三角形中的勾股定理

📚 Pythagoras’ Theorem in Right-Angled Triangles | 直角三角形中的勾股定理

Pythagoras’ theorem is one of the most widely used results in geometry. It links the three sides of a right-angled triangle and allows you to calculate an unknown length when the other two sides are known. In this article, you will learn how to identify the hypotenuse, apply the formula a² + b² = c², solve problems in both pure mathematics and everyday life, and avoid common mistakes.

勾股定理是几何中应用最广泛的结论之一。它将直角三角形三条边联系在一起，使你能够在已知另外两条边的情况下求出一条未知边的长度。本文将教你如何识别斜边、应用公式 a² + b² = c²、解决纯数学和现实生活中的问题，并避免常见错误。

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

Pythagoras’ theorem states that in any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

勾股定理指出，在任何一个直角三角形中，斜边长度的平方等于另外两条直角边长度的平方之和。

It only works for right-angled triangles, so you must always check that one angle is exactly 90° before applying the rule.

这个定理只适用于直角三角形，因此在应用之前必须确认三角形中有一个角恰好为 90°。

The theorem is named after the ancient Greek mathematician Pythagoras, although the relationship was known to earlier Babylonian and Chinese mathematicians as well.

这一定理以古希腊数学家毕达哥拉斯命名，不过巴比伦和中国数学家更早以前就已经知道这一关系。

2. The Formula a² + b² = c² | 公式 a² + b² = c²

For any right-angled triangle, the standard form of the theorem is written as follows.

对于任意直角三角形，该定理的标准形式写作如下。

a² + b² = c²

Here, c represents the length of the hypotenuse, while a and b represent the lengths of the two shorter sides, often called the legs.

这里，c 表示斜边的长度，a 和 b 表示两条较短边的长度，它们通常被称为直角边。

You can rearrange the formula to calculate any side of the triangle. To find the hypotenuse c, use:

你可以重新整理公式来求三角形的任意一条边。若要求斜边 c，使用：

c = √(a² + b²)

To find one of the shorter sides, rearrange it as:

要求其中一条直角边，则重新整理为：

a = √(c² – b²)

或

b = √(c² – a²)

The two shorter sides a and b are interchangeable. What matters is that c must always be the longest side, opposite the right angle.

两条直角边 a 和 b 可以互换位置。重要的是 c 必须始终是最长边，且对着直角。

3. Identifying the Hypotenuse | 识别斜边

Before substituting numbers into the formula, you need to identify the hypotenuse correctly. The hypotenuse is always the longest side, and it is always opposite the right angle.

在把数值代入公式之前，你需要正确识别斜边。斜边始终是最长边，并且始终位于直角的对面。

Do not assume that a side is the hypotenuse just because it is drawn at the top or looks the longest. Always check which side is opposite the 90° angle.

不要因为某条边画在上方或看起来最长就认定它是斜边。务必检查哪条边位于 90° 角的对面。

For example, in triangle ABC, if angle C is 90°, then side AB is the hypotenuse.

例如，在三角形 ABC 中，如果角 C 是 90°，那么边 AB 就是斜边。

Misidentifying the hypotenuse is one of the most common errors in Pythagoras’ theorem questions.

错误识别斜边是勾股定理题目中最常见的错误之一。

4. Finding the Hypotenuse | 求斜边

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

例题：一个直角三角形两条直角边的长度分别为 6 cm 和 8 cm。求斜边的长度。

Step 1: Write down the formula for the hypotenuse.

步骤一：写出求斜边的公式。

c = √(a² + b²)

Step 2: Substitute the two known lengths into the formula.

步骤二：将两个已知长度代入公式。

c = √(6² + 8²)

Step 3: Calculate the squares and add them together.

步骤三：计算平方并相加。

c = √(36 + 64) = √100

Step 4: Take the square root of the total.

步骤四：求出总数的平方根。

c = 10 cm

So the hypotenuse is 10 cm long.

因此斜边长度为 10 cm。

5. Finding a Shorter Side | 求一条直角边

Worked example: A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the length of the other shorter side.

例题：一个直角三角形的斜边为 13 cm，一条直角边为 5 cm。求另一条直角边的长度。

Step 1: Use the rearranged formula for a shorter side.

步骤一：使用重新整理后求直角边的公式。

a = √(c² – b²)

Step 2: Substitute the hypotenuse and the known shorter side.

步骤二：代入斜边和已知直角边的长度。

a = √(13² – 5²)

Step 3: Calculate the squares and subtract.

步骤三：计算平方并相减。

a = √(169 – 25) = √144

Step 4: Take the square root.

步骤四：开平方根。

a = 12 cm

So the missing shorter side is 12 cm long.

因此缺失的直角边长度为 12 cm。

Always check that your answer is shorter than the hypotenuse. If you obtain a longer value, you have probably made a substitution error.

始终要检查答案是否小于斜边。如果你得到的结果比斜边更长，很可能是代入时出错了。

6. Checking for a Right Angle | 判断是否为直角三角形

Pythagoras’ theorem also works in reverse. If the three sides of a triangle satisfy a² + b² = c², where c is the longest side, then the triangle must be right-angled.

勾股定理反过来也成立。如果一个三角形的三条边满足 a² + b² = c²，其中 c 是最长边，那么这个三角形必定是直角三角形。

Worked example: Determine whether a triangle with sides 9 cm, 12 cm and 15 cm is right-angled.

例题：判断边长分别为 9 cm、12 cm 和 15 cm 的三角形是否为直角三角形。

Step 1: Identify the longest side. Here, 15 cm is the longest side.

步骤一：找出最长边。这里最长边是 15 cm。

Step 2: Check whether the square of the longest side equals the sum of the squares of the other two sides.

步骤二：检验最长边的平方是否等于另外两条边的平方之和。

9² + 12² = 81 + 144 = 225

15² = 225

Since 9² + 12² = 15², the triangle is right-angled.

因为 9² + 12² = 15²，所以这个三角形是直角三角形。

This reverse check is useful in construction, navigation and geometry proofs.

这种反向检验在建筑、导航和几何证明中非常有用。

7. Pythagorean Triples | 勾股数（毕氏三元组）

A Pythagorean triple is a set of three positive integers that

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Pythagoras’ Theorem in Right-Angled Triangles | 直角三角形中的勾股定理