📚 Pythagoras' Theorem: Right-Angled Triangles and Applications | 毕达哥拉斯定理：直角三角形及其应用
In KS3 Cambridge mathematics, Pythagoras' theorem is one of the most useful results for solving problems involving right-angled triangles. It connects the three sides of a right-angled triangle through a simple squares relationship, and it appears in geometry, algebra, coordinate work and real-world measurement.
在剑桥 KS3 数学中，毕达哥拉斯定理是解决直角三角形相关问题最有用的结论之一。它通过一个简单的平方关系把直角三角形三条边联系起来，并广泛出现在几何、代数、坐标以及实际测量中。
1. What is Pythagoras' Theorem? | 什么是毕达哥拉斯定理？
Pythagoras' theorem applies only to right-angled triangles. It states that the square of the longest side is equal to the sum of the squares of the other two sides. This relationship was known to ancient Babylonian and Egyptian mathematicians but is named after the Greek mathematician Pythagoras.
毕达哥拉斯定理只适用于直角三角形。它指出最长边的平方等于另外两条边的平方之和。这个关系早在古巴比伦和古埃及数学家那里就已被发现，但以古希腊数学家毕达哥拉斯命名。
The side opposite the right angle is always the longest side, and the theorem tells us exactly how much longer its square is than the other two squares.
直角所对的边总是最长的边，而该定理准确地告诉我们它的平方比另外两个平方大多少。
2. Naming the Sides | 三角形各边的命名
In a right-angled triangle, the side opposite the right angle is called the hypotenuse. It is always the longest side because the right angle is the largest angle in the triangle.
在直角三角形中，直角所对的边叫做斜边。斜边总是最长的边，因为直角是三角形中最大的角。
The other two sides are often called the legs or shorter sides. They form the right angle and are usually labelled a and b.
另外两条边通常叫做直角边或较短的边。它们构成直角，通常标记为 a 和 b。
3. The Formula | 定理公式
If the two shorter sides are labelled a and b, and the hypotenuse is labelled c, then the theorem is written as:
如果两条直角边标记为 a 和 b，斜边标记为 c，那么定理可写成：
a² + b² = c²
This means you square the two shorter sides, add the results, and the total equals the square of the hypotenuse. To find a missing side, you often need to take a square root.
这意味着先将两条直角边平方，再相加，所得总和等于斜边的平方。要求出缺失的边，通常需要开平方。
4. Finding the Hypotenuse | 求斜边
To find the hypotenuse, use the rearranged form c = √a² + b². For example, if a = 6 cm and b = 8 cm, then:
求斜边时，使用变形公式 c = √a² + b²。例如，如果 a = 6 cm，b = 8 cm，那么：
c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
So the hypotenuse is 10 cm. Notice that squaring removes the units until the final square root brings them back.
所以斜边是 10 cm。注意平方会使单位暂时消失，最后开平方后单位又恢复。
5. Finding a Shorter Side | 求直角边
To find a missing shorter side, rearrange the formula as a = √c² − b². If the hypotenuse is 13 m and one shorter side is 5 m, then:
求缺失的直角边时，将公式变形为 a = √c² − b²。如果斜边为 13 m，一条直角边为 5 m，则：
a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12 m
Make sure you subtract the smaller square from the larger square; otherwise you may get a negative value under the square root.
务必用较大的平方减去较小的平方；否则平方根下可能出现负数。
6. The Converse of the Theorem | 定理的逆定理
The converse of Pythagoras' e triangle is right-angled. This is useful for checking whether a triangle has a right angle without measuring angles.
毕达哥拉斯定理的逆定理指出，如果一个三角形的三条边满足 a² + b² = c²，那么这个三角形是直角三角形。这在不测量角度的情况下判断三角形是否有直角时非常有用。
es 9 cm, 12 cm and 15 cm is right-angled because 9² + 12² = 81 + 144 = 225, and 15² = 225.
例如，一个边长为 9 cm、12 cm 和 15 cm 的三角形是直角三角形，因为 9² + 12² = 81 + 144 = 225，而 15² = 225。
7. Pythagorean Triples | 毕达哥拉斯三元组
a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8, 15, 17).
毕达哥拉斯三元组是指满足 a² + b² = c² 的三个正整数。常见例子包括 (3, 4, 5)、(5, 12, 13)、(7, 24, 25) 和 (8, 15, 17)。
