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' .pdf or revising for a Cambridge Checkpoint test, you need to be confident in finding missing side lengths and checking whether a triangle is right-angled.

勾股定理是 KS3 数学中最重要的结论之一。它建立了直角三角形三条边之间的关系，并被用于许多实际问题中。无论你是在完成 p208_2.pdf 这样的课本练习，还是在为剑桥 Checkpoint 测试复习，你都需要熟练掌握求未知边长和判断三角形是否为直角三角形的方法。

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

If a triangle has a right angle (90°), the longest side is called the hypotenuse. Pythagoras' theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

如果一个三角形有一个直角（90°），那么最长的边叫做斜边。勾股定理指出，斜边的平方等于另外两条直角边的平方和。

c² = a² + b²

Here c is the hypotenuse, while a and b are the two shorter sides, often called the legs. This formula only works for right-angled triangles.

这里 c 是斜边，a 和 b 是两条较短的直角边。这个公式只适用于直角三角形。

2. Key Vocabulary and Notation | 关键术语与符号

Before solving problems, make sure you know these words and symbols: right angle, hypotenuse, leg, square, square root, and Pythagoras' theorem itself.

在解题之前，请务必掌握以下术语和符号：直角、斜边、直角边、平方、平方根以及勾股定理本身。

• Hypotenuse / 斜边: the side opposite the right angle, always the longest side. / 正对直角的边，始终是最长边。
• Legs / 直角边: the two sides that form the right angle. / 构成直角的两条边。
• Square / 平方: multiplying a number by itself, e.g. 5² = 25. / 一个数乘以它本身，例如 5² = 25。
• Square root / 平方根: the opposite of squaring, e.g. √25 = 5. / 平方的逆运算，例如 √25 = 5。

Being accurate with notation matters in exams: always label the hypotenuse as c and the shorter sides as a and b.

在考试中准确的符号使用很重要：始终将斜边标为 c，将较短边标为 a 和 b。

3. Identifying the Hypotenuse | 识别斜边

Always locate the right angle first. The hypotenuse is the side directly across from the right angle, and it is always the longest side of the triangle.

一定要先找到直角。斜边是正对直角的那条边，它始终是三角形中最长的边。

For example, in a triangle with sides 6 cm, 8 cm, and 10 cm, the hypotenuse is 10 cm because it is opposite the right angle and it is the largest value.

例如，在一个边长分别为 6 cm、8 cm 和 10 cm 的三角形中，斜边是 10 cm，因为它正对直角且数值最大。

A common error is to label the longest side as a or b when it should be c. This makes the calculation incorrect, so always check the position of the right angle before substituting.

一个常见错误是把最长边标成 a 或 b，而不是 c。这会导致计算错误，因此在代入之前一定要检查直角的位置。

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4. Using the Theorem to Find the Hypotenuse | 利用定理求斜边

To find the hypotenuse, square both shorter sides, add the results, and then take the square root of the total.

要求斜边，先将两条直角边分别平方，再把结果相加，然后取总和的平方根。

Example 1: A right-angled triangle has legs of 3 cm and 4 cm. Find the hypotenuse.

例题 1：一个直角三角形的两条直角边分别为 3 cm 和 4 cm，求斜边。

c² = 3² + 4²
c² = 9 + 16 = 25
c = √25 = 5 cm

s 5 cm. Notice that you must take the square root at the end: c² = 25 gives c = 5, not 25.

所以斜边为 5 cm。注意最后必须开平方根：c² = 25 得到 c = 5，而不是 25。

Example 2: Find the hypotenuse when the shorter sides are 7 cm and 24 cm.

例题 2：当两条直角边分别为 7 cm 和 24 cm 时，求斜边。

c² = 7² + 24²
c² = 49 + 576 = 625
c = √625 = 25 cm

The hypotenuse is 25 cm. This is a useful Pythagorean triple to remember: 7, 24, 25.

斜边为 25 cm。这是一组值得记住的勾股数：7、24、25。

5. Finding a Shorter Side | 求直角边

If you know the hypotenuse and one shorter side, rearrange the theorem to find the missing leg. Subtract the known square from the square of the hypotenuse, then take the square root.

如果你知道斜边和一条直角边，可以重新整理公式来求另一条直角边。用斜边的平方减去已知直角边的平方，然后取平方根。

a² = c² - b²
b² = c² - a²

Example: The hypotenuse is 13 cm, and one leg is 5 cm. Find the other leg.

例题：斜边为 13 cm，一条直角边为 5 cm，求另一条直角边。

a² = 13² - 5²
a² = 169 - 25 = 144
a = √144 = 12 cm

The missing shorter side is 12 cm. Always subtract when finding a leg, because the hypotenuse is the largest side and must be the sum of the two squares.

缺失的直角边长为 12 cm。求直角边时一定要用减法，因为斜边是最大边，它等于两个直角边平方之和。

6. Checking for a Right-Angled Triangle | 检验直角三角形

You can use the converse of Pythagoras' theorem to test whether a triangle is right-angled. Square all three sides and check whether the largest square equals the sum of the other two squares.

你可以使用勾股定理的逆定理来检验一个三角形是否为直角三角形。将三条边都平方，检查最大的平方是否等于另外两个平方之和。

Example 1: Is a triangle with sides 5 cm, 6 cm, and 7 cm right-angled?

例题 1：三边长分别为 5 cm、6 cm 和 7 cm 的三角形是直角三角形吗？

7² = 49
5² + 6² = 25 + 36 = 61

49 is not equal to 61, so the triangle is not right-angled.

49 不等于 61，所以这个三角形不是直角三角形。

Example

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