📚 Mastering Pythagoras' Theorem | 掌握毕达哥拉斯定理（勾股定理）
In KS3 Cambridge maths, Pythagoras' theorem is one of the most powerful tools for solving problems involving right-angled triangles. It links the three sides of a right-angled triangle in a simple equation and appears in many topics, including shapes, coordinates, and real-life applications.
在 KS3 剑桥数学中，毕达哥拉斯定理（勾股定理）是解决直角三角形问题最有力的工具之一。它用一个简单的方程把直角三角形三条边联系起来，并出现在许多主题中，包括图形、坐标和实际生活应用。
This revision guide walks through the core ideas step by step, from identifying the hypotenuse to applying the theorem in exam-style questions. By the end, you should be able to set up the formula confidently and avoid the most common pitfalls.
本复习指南将带你一步步梳理核心概念，从识别斜边到在考试题型中应用该定理。学完后，你应该能够自信地列出公式，并避免最常见的错误。
1. The Right-Angled Triangle | 直角三角形
A right-angled triangle is a triangle with one angle exactly 90 degrees. The side opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are often called the legs or shorter sides.
直角三角形是一个角恰好为 90 度的三角形。直角对面的边总是最长边，称为斜边。另外两条边通常称为直角边或较短边。
always represents the hypotenuse, while a and b represent the two shorter sides in any order. Getting the labels right is the first step in every Pythagoras problem.
在公式 a² + b² = c² 中，字母 c 始终表示斜边，而 a 和 b 按任意顺序表示两条直角边。在每个勾股定理问题中，正确标出字母是第一步。
Always sketch the triangle and mark the right angle first. Then label the side directly opposite the right angle as c, because that is the hypotenuse.
始终先画出三角形并标出直角。然后把直角正对的边标为 c，因为这条边就是斜边。
2. Statement of the Theorem | 定理陈述
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. This relationship is written as:
毕达哥拉斯定理指出，在任意直角三角形中，斜边的平方等于另外两条边的平方之和。这个关系可以写成：
a² + b² = c²
Here a and b are the two shorter sides, and c is the hypotenuse. The theorem only works for right-angled triangles, so you must always check that a right angle is present before using it.
这里 a 和 b 是两条直角边，c 是斜边。该定理只适用于直角三角形，因此在使用之前必须确认三角形中有一个直角。
heorem can be rearranged to find any side of the triangle. To find the hypotenuse, use c = √a² + b². To find a shorter side, use a = √c² − b² or b = √c² − a².
该定理可以变形来求三角形的任意一边。求斜边时用 c = √a² + b²。求直角边时用 a = √c² − b² 或 b = √c² − a²。
3. Finding the Hypotenuse | 求斜边
horter sides of 3 cm and 4 cm. To find the hypotenuse, substitute a = 3 and b = 4 into c = √a² + b².
假设一个直角三角形的两条直角边分别是 3 厘米和 4 厘米。要求斜边，就把 a = 3 和 b = 4 代入 c = √a² + b²。
c = √3² + 4² = √9 + 16 = √25 = 5 cm
The hypotenuse is therefore 5 cm. Notice that you must square each side first, then add, and only take the square root at the end. Many students forget the final square root step.
因此斜边是 5 厘米。注意你必须先分别计算平方，再相加，最后才开平方根。许多学生容易忘记最后的开平方步骤。
In general, when the two shorter sides are a and b, the hypotenuse is always longer than either of them on its own, but shorter than the sum of the two shorter sides. This gives a useful check for your answer.
一般来说，当两条直角边为 a 和 b 时，斜边总是比任意一条直角边长，但比两条直角边之和短。这可以作为一个有用的答案检验方法。
4. Finding a Shorter Side | 求直角边
If you know the hypotenuse and one shorter side, you can find the other shorter side by rearranging the theorem. For example, if the hypotenuse is 13 cm and one shorter side is 5 cm, find the missing side.
如果你知道斜边和一条直角边，就可以通过变形公式求出另一条直角边。例如，如果斜边是 13 厘米，一条直角边是 5 厘米，求缺失的边。
b = √c² − a² = √13² − 5² = √169 − 25 = √144 = 12 cm
The missing shorter side is 12 cm. Here we subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
缺失的直角边是 12 厘米。这里我们用斜边的平方减去已知直角边的平方，然后开平方根。
Be careful to set the equation up correctly: the hypotenuse is always the largest number, so you must subtract the smaller square from the larger square. If you get a negative number under the square root, you have labelled the sides incorrectly.
注意要正确建立方程：斜边总是最大的数，所以你必须用较大的平方减去较小的平方。如果平方根下面得到负数，说明你把边的标号弄错了。
5. Checking for a Right Angle | 判断直角三角形
Pythagoras' b² = c², then the triangle must be right-angled, with the longest side as the hypotenuse.
毕达哥拉斯定理也可以反过来使用。如果一个三角形的三条边满足 a² + b² = c²，那么这个三角形一定是直角三角形，最长边为斜边。
For example, a triangle with sides 6 cm, 8 cm and 10 cm gives:
例如，一个边长为 6 厘米、8 厘米和 10 厘米的三角形满足：
6² + 8² = 36 + 64 = 100 = 10²
Since the squares match, the triangle is right-angled. This is a quick way to test whether a triangle with given side lengths contains a right angle, without drawing it accurately.
由于平方相等，这个三角形是直角三角形。这是一种快速检验给定边长三角形是否含有直角的方法，无需精确作图。
If the two values are not equal, then the triangle is not right-angled. It may be acute or obtuse, but that is beyond the KS3 Pythagoras requirement.
如果两边平方和不等于最长边的平方，那么这个三角形就不是直角三角形。它可能是锐角三角形或钝角三角形，但这超出了 KS3 勾股定理的要求范围。
6. Pythagorean Triples | 勾股数
on a² + b² = c². The most famous example is (3, 4, 5), because 3² + 4² = 9 + 16 = 25 = 5².
勾股数是一组满足方程 a² + b² = c² 的正整数。最著名的例子是 (3, 4, 5)，因为 3² + 4² = 9 + 16 = 25 = 5²。
Other common triples include (5, 12, 13), (8, 15, 17) and (7, 24, 25). Multiples of a triple also work: for example, (6, 8, 10) is just the (3, 4, 5) triple multiplied by 2.
其他常见的勾股数包括 (5, 12, 13)、(8, 15, 17) 和 (7, 24, 25)。勾股数的倍数也成立：例如 (6, 8, 10) 就是 (3, 4, 5) 乘以 2 得到的。
Recognising these triples can save time in exams. If you spot that two sides are 6 and 8, you immediately know the hypotenuse is 10 without needing to calculate the square roots.
在考试中识别这些勾股数可以节省时间。如果你发现两条边是 6 和 8，就能立刻知道斜边是 10，而不需要计算平方根。
7. Coordinates and Distance | 坐标与距离
Pythagoras' theorem is also used to find the distance between two points on a coordinate grid. The horizontal and vertical distances between the points act as the two shorter sides of a right-angled triangle.
毕达哥拉斯定理还用于求坐标网格上两点之间的距离。两点之间的水平距离和垂直距离相当于直角三角形两条直角边。
If point A is (x₁, y₁) and point B is (x₂, y₂), then the distance d between them is given by:
如果点 A 是 (x₁, y₁)，点 B 是 (x₂, y₂)，那么它们之间的距离 d 可以用以下公式表示：
d = √((x₂ − x₁)² + (y₂ − y₁)²)
For example, the distance from (2, 3) to (6, 6) is √((6 − 2)² + (6 − 3)²) = √4² + 3² = √16 + 9 = √25 = 5 units.
例如，点 (2, 3) 到点 (6, 6) 的距离是 √((6 − 2)² + (6 − 3)²) = √4² + 3² = √16 + 9 = √25 = 5 个单位。
This method is especially useful when the two points are not on the same horizontal or vertical line, because you can form a right-angled triangle and apply the theorem directly.
当两点不在同一条水平线或垂直线上时，这个方法特别有用，因为你可以构造一个直角三角形并直接应用该定理。
8. Real-Life Applications | 实际应用
Pythagoras' theorem appears in many everyday situations. For example, a ladder leaning against a wall forms a right-angled triangle with the ground and the wall. If you know the height reached by the ladder and its distance from the wall, you can find the length of the ladder.
毕达哥拉斯定理出现在许多日常生活中。例如，靠在墙上的梯子与地面和墙构成一个直角三角形。如果你知道梯子达到的高度和它离墙的距离，就可以求出梯子的长度。
r a television screen. A rectangle with width w and height h has a diagonal d given by d = √w² + h². This is because the diagonal splits the rectangle into two right-angled triangles.
另一个常见应用是求矩形或电视屏幕的对角线。宽为 w、高为 h 的矩形，其对角线 d 可由 d = √w² + h² 得到。这是因为对角线把矩形分成两个直角三角形。
In navigation and map reading, Pythagoras' theorem can estimate the shortest distance between two points when you know the east-west and north-south separations. It turns real-world measurements into a straightforward calculation.
在导航和地图阅读中，当你知道两点的东西向和南北向间隔时，毕达哥拉斯定理可以估算两点间的最短距离。它把实际测量转化为简单的计算。
9. Common Mistakes to Avoid | 常见错误
b² = 25, the hypotenuse is 5, not 25. Always apply the square root as the final operation.
最常见的错误之一是最后忘记开平方根。如果你得到 a² + b² = 25，斜边是 5，而不是 25。始终把开平方根作为最后一步运算。
tely: (3 + 4)² is not the same as 3² + 4². The correct calculation is 9 + 16 = 25, not 49.
另一个错误是在平方之前先相加边长。该定理要求分别平方：(3 + 4)² 不等于 3² + 4²。正确计算是 9 + 16 = 25，而不是 49。
Students sometimes label the longest side as a or b and then try to find c using addition. Remember that c is always the hypotenuse, so if you know the hypotenuse, you must subtract to find a shorter side.
学生有时把最长边标为 a 或 b，然后试图用加法求 c。记住 c 一定是斜边，所以如果你知道斜边，求直角边时必须用减法。
Finally, avoid rounding intermediate steps too early. Keep the full value in your calculator until the end, then round to the required degree of accuracy. This prevents rounding errors from building up.
最后，避免过早对中间步骤取近似值。在计算器中保留完整数值直到最后，再按要求精确到指定小数位。这可以防止舍入误差累积。
10. Exam-Style Practice Tips | 考试练习技巧
ula a² + b² = c² before substituting any numbers, so the examiner can follow your working.
在考试题中，先画出三角形并仔细标出边 a、b 和 c。在代入任何数字之前写下公式 a² + b² = c²，这样阅卷老师就能看懂你的解题过程。
Show every step of your working, including the squared values and the square root. Even if your final answer is wrong, you may earn marks for correct method and partial substitution.
展示每一步计算过程，包括平方值和平方根。即使最终答案错误，你也可能因为正确的方法和部分代入而获得分数。
Always state the units clearly, such as cm, m, or units. If the question asks for the answer to a given number of decimal places or significant figures, make sure you round only at the final answer.
始终清楚地写出单位，例如 cm、m 或 units。如果题目要求保留指定的小数位数或有效数字，确保只在最终答案处进行四舍五入。
Finally, check your answer against the context. The hypotenuse must be the longest side, and a shorter side must be less than the hypotenuse. If your answer breaks these rules, go back and check your labelling and calculation.
最后，结合实际情境检查答案。斜边一定是最长边，直角边一定小于斜边。如果你的答案违反这些规则，回去检查边的标号和计算过程。
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