📚 Pythagoras' Theorem in Right-Angled Triangles | 直角三角形中的勾股定理
Pythagoras' theorem is one of the most important results in KS3 mathematics. It connects the three sides of a right-angled triangle and allows you to find a missing side when the other two sides are known. This article explains the theorem, how to use it in different situations, and common pitfalls to avoid.
勾股定理是 KS3 数学中最重要的定理之一。它建立了直角三角形三条边之间的关系，当已知其中两条边时，可以用来求出缺失的第三条边。本文讲解这一定理、在不同情形下的使用方法以及需要避免的常见错误。
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°. The side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are often called the legs or shorter sides.
直角三角形是指其中一个内角恰好为 90° 的三角形。直角所对的边称为斜边，它始终是最长的边。另外两条边通常称为直角边或较短的边。
When you are given a diagram, look for the small square symbol placed at one angle. This square indicates the right angle and helps you identify the correct sides before applying the theorem.
在给出的图形中，请寻找位于某个角处的小正方形符号。这个正方形表示直角，可帮助你在应用定理之前正确识别边。
2. The Theorem and Its Formula | 定理及其公式
Pythagoras' theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the hypotenuse is labelled c and the other two sides are a and b, the formula is:
勾股定理指出，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。如果斜边标记为 c，另外两条边标记为 a 和 b，则公式为：
c² = a² + b²
This formula only works for right-angled triangles. It is used to find the length of one side when the lengths of the other two sides are known.
这个公式只适用于直角三角形。当已知另外两条边的长度时，可以用它来求第三条边的长度。
3. Identifying the Hypotenuse | 识别斜边
Before substituting values into the formula, always check which side is the hypotenuse. The hypotenuse is opposite the right angle and is longer than either of the other two sides. Labelling the hypotenuse as c prevents mistakes in the order of subtraction.
在将数值代入公式之前，请始终确认哪条边是斜边。斜边位于直角的对面，比另外两条边中的任何一条都长。将斜边标记为 c 可以避免减法顺序上的错误。
If the diagram is rotated or drawn to scale, remember that the hypotenuse is not always at the bottom or sloping in a familiar direction. Use the right-angle symbol to locate it reliably.
如果图形经过旋转或按比例绘制，请记住斜边并不总是在底部或沿着熟悉的方向倾斜。使用直角符号可以可靠地找到它。
4. Finding the Hypotenuse: Worked Example | 求斜边：例题
4 cm. To find the hypotenuse c, use the formula c² = a² + b². Substitute a = 3 and b = 4:
一个直角三角形的两条直角边长度分别为 3 cm 和 4 cm。要求斜边 c，使用公式 c² = a² + b²。将 a = 3 和 b = 4 代入：
c² = 3² + 4² = 9 + 16 = 25
Then take the square root of both sides: c = √25 = 5 cm. The hypotenuse is 5 cm. This is the well-known 3-4-5 triangle.
然后对等式两边取平方根：c = √25 = 5 cm。斜边为 5 cm。这就是著名的 3-4-5 直角三角形。
5. Finding a Shorter Side: Worked Example | 求直角边：例题
shorter side is 5 cm. Let the missing side be a. Then c² = a² + b² becomes 13² = a² + 5².
当你需要求一条直角边时，首先重新整理公式。假设斜边为 13 cm，一条直角边为 5 cm。设缺失的边为 a。那么 c² = a² + b² 变为 13² = a² + 5²。
169 = a² + 25
a = √144 = 12 cm. Always check that the shortest side result is less than the hypotenuse.
两边同时减去 25：a² = 169 − 25 = 144。因此 a = √144 = 12 cm。请始终检查求得的直角边是否小于斜边。
6. Testing for a Right-Angled Triangle | 判断是否为直角三角形
The converse of Pythagoras' c is the longest side), then the triangle is right-angled. This is useful when no angle is marked on a diagram.
勾股定理的逆定理指出，如果三角形的三条边满足 a² + b² = c²（其中 c 为最长边），那么这个三角形就是直角三角形。当图中没有标出直角时，这个性质非常有用。
² + 8² = 36 + 64 = 100 and 10² = 100, so the sides match and the triangle is right-angled.
例如，边长分别为 6 cm、8 cm 和 10 cm 的三角形有 6² + 8² = 36 + 64 = 100，且 10² = 100，因此边长匹配，该三角形是直角三角形。
7. Real-Life Applications | 实际应用
Pythagoras' theorem is widely used in real-life problems, such as finding the length of a ladder leaning against a wall or calculating the distance between two points on a map. You often need to sketch a right-angled triangle from the given information.
勾股定理在现实问题中应用广泛，例如求斜靠在墙上的梯子的长度，或计算地图上两点之间的距离。你通常需要根据已知信息画出直角三角形的草图。
t reached on the wall is found by using 5² = h² + 3², giving h² = 25 − 9 = 16, so h = 4 m.
例如，一把长 5 m 的梯子底部离墙 3 m。梯子在墙上达到的垂直高度可通过 5² = h² + 3² 求出，得到 h² = 25 − 9 = 16，因此 h = 4 m。
8. Common Mistakes to Avoid | 常见错误
A very common error is adding instead of subtracting when finding a shorter side. Always rearrange the formula so the missing shorter side is squared on its own before taking the square root.
一个很常见的错误是在求直角边时把加法做成了减法。请始终先重新整理公式，使缺失的直角边单独平方后再取平方根。
Another mistake is forgetting to write the units, especially when converting between centimetres, metres and millimetres. Keep all side lengths in the same unit before using the formula.
另一个错误是忘记写明单位，尤其是在厘米、米和毫米之间换算时。在使用公式之前，请将所有边长保持为相同的单位。
9. Practice Questions and Answers | 练习题与答案
Try these practice questions to check your understanding. The answers are given below, but make sure you show full working.
尝试以下练习题，检查你的理解。答案在下方给出，但请务必写出完整的计算过程。
- Q1: A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse. Ans: 13 cm
中文：一个直角三角形两条直角边分别为 5 cm 和 12 cm，求斜边。答案：13 cm。 - Q2: A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the other shorter side. Ans: 15 cm
中文：一个直角三角形斜边为 17 cm，一条直角边为 8 cm，求另一条直角边。答案：15 cm。 - Q3: Do the sides 9 cm, 12 cm and 15 cm form a right-angled triangle? Ans: Yes, because 9² + 12² = 225 = 15².
中文：边长 9 cm、12 cm、15 cm 是否构成直角三角形？答案：是，因为 9² + 12² = 225 = 15²。
10. Summary and Key Points | 总结与要点
To use Pythagoras' er side: a² = c² − b². Always take the positive square root and include the correct units.
要正确使用勾股定理，首先确定直角和斜边。求斜边用 c² = a² + b²，求直角边则重新整理为 a² = c² − b²。始终取正的平方根，并写出正确的单位。
Practise with different orientations and word problems so that you can recognise right-angled triangles in many contexts. This will build confidence for Cambridge KS3 assessments and later study of trigonometry.
通过不同方向和文字题进行练习，以便能够在多种情境中识别直角三角形。这将为
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