📚 Pythagoras’ Theorem | 勾股定理
Pythagoras’ theorem is a cornerstone of KS3 geometry. It describes a special relationship between the side lengths of any right-angled triangle. Once you master this rule, you can find missing sides, test angles, and solve practical problems involving distances.
勾股定理是 KS3 几何的核心内容。它描述了直角三角形三条边之间的特殊关系。掌握这个规律后，你就能求出未知边长、判断角度，并解决与距离有关的实际问题。
1. What is Pythagoras’ Theorem? | 什么是勾股定理？
In any right-angled triangle, the side opposite the right angle is called the hypotenuse. The theorem states that the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. In symbols, if the shorter sides are a and b, and the hypotenuse is c, then a² + b² = c².
在任何直角三角形中，直角的对边称为斜边。定理指出，斜边上的正方形面积等于另外两条直角边上的正方形面积之和。用符号表示，如果较短的两条边为 a 和 b，斜边为 c，那么 a² + b² = c²。
a² + b² = c²
2. The Formula and Notation | 公式与记号
Always label the longest side as c. The other two sides can be labelled a and b in either order because addition is commutative. Do not confuse c with the right angle itself; c is the length of the side opposite the right angle. The theorem only works for right-angled triangles, so look for the square symbol in the diagram.
始终把最长边标记为 c。另外两条边可以任意标记为 a 或 b，因为加法满足交换律。不要将 c 与直角本身混淆；c 是直角对边的长度。该定理只适用于直角三角形，因此在图中要寻找直角符号。
c² = a² + b² and c = √(a² + b²)
3. Finding the Hypotenuse | 求斜边
Suppose a right-angled triangle has a = 6 cm and b = 8 cm. Substitute into c² = a² + b². This gives c² = 6² + 8² = 36 + 64 = 100. Take the positive square root: c = √100 = 10 cm. Always give the positive root because length cannot be negative.
假设一个直角三角形的两条直角边为 a = 6 cm 和 b = 8 cm。代入 c² = a² + b²，得到 c² = 6² + 8² = 36 + 64 = 100。取正平方根：c = √100 = 10 cm。一定要取正根，因为长度不能为负数。
c = √(a² + b²)
4. Finding a Shorter Side | 求直角边
When you know the hypotenuse and one shorter side, rearrange the formula. Subtract the known square from the hypotenuse squared: a² = c² − b². Example: c = 13 cm, b = 5 cm, then a² = 13² − 5² = 169 − 25 = 144, so a = √144 = 12 cm. Check that the result is shorter than the hypotenuse.
当已知斜边和一条直角边时，需要重新整理公式。用斜边的平方减去已知直角边的平方：a² = c² − b²。例如：c = 13 cm，b = 5 cm，那么 a² = 13² − 5² = 169 − 25 = 144，所以 a = √144 = 12 cm。检查结果是否比斜边短。
a² = c² − b²
5. Checking for a Right Angle | 判断直角三角形
You can use the converse of Pythagoras’ theorem to test whether a triangle is right-angled. Arrange the side lengths in increasing order and let c be the largest. If a² + b² = c², the triangle is right-angled. Example: sides 9 cm, 12 cm, 15 cm give 9² + 12² = 81 + 144 = 225 = 15², so the triangle is right-angled.
你可以利用勾股定理的逆定理来判断一个三角形是否为直角三角形。将三条边按从小到大排列，令最大边为 c。如果 a² + b² = c²，那么这个三角形就是直角三角形。例如：边长为 9 cm、12 cm、15 cm 的三角形满足 9² + 12² = 81 + 144 = 225 = 15²，因此它是直角三角形。
6. Pythagorean Triples | 勾股数
A Pythagorean triple consists of three whole numbers that satisfy a² + b² = c². The most common triples are (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17). Multiples of these, such as (6, 8, 10), are also valid. Recognising triples can save time in mental calculations.
勾股数是指满足 a² + b² = c² 的三个整数。最常见的勾股数有 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。它们的倍数，例如 (6, 8, 10)，同样有效。识别勾股数可以节省心算时间。
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
7. Real-Life Applications | 实际应用
Pythagoras’ theorem appears in many practical situations. For example, a ladder leaning against a wall forms a right-angled triangle with the ground. If a 5 m ladder reaches 4 m up a wall, the distance of the foot from the wall is x, where x² + 4² = 5², so x = 3 m. Other applications include finding the diagonal of a television, the shortest path across a park, or the distance between two points on a map.
勾股定理出现在许多实际情境中。例如，靠在墙上的梯子与地面构成一个直角三角形。如果 5 m 长的梯子顶端距离地面 4 m，那么梯子底部到墙的距离为 x，满足 x² + 4² = 5²，所以 x = 3 m。其他应用包括求电视屏幕的对角线、穿过公园的最短路径，或地图上两点之间的距离。
8. Using Pythagoras in 3D | 三维中的勾股定理
In a cuboid, you can find the space diagonal by applying Pythagoras twice. First find the diagonal of the base, then combine it with the height. The combined formula is d² = l² + w² + h². Example: a box has length 3 cm, width 4 cm and height 12 cm. Then d² = 3² + 4² + 12² = 9 + 16 + 144 = 169, so d = √169 = 13 cm.
在长方体中，可以通过两次应用勾股定理来求空间对角线。先求出底面对角线，再与高度结合。综合公式为 d² = l² + w² + h²。例如：一个盒子的长为 3 cm、宽为 4 cm、高为 12 cm。那么 d² = 3² + 4² + 12² = 9 +
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