📚 Pythagoras’ Theorem and Its Applications | 勾股定理及其应用
Pythagoras’ Theorem is a fundamental relationship in Euclidean geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is one of the most frequently tested topics in IGCSE Mathematics, appearing in both Paper 1 and Paper 2, and it also forms the basis for coordinate geometry, trigonometry and 3D mensuration problems.
勾股定理（毕达哥拉斯定理）是欧氏几何中最基本的关系之一。它指出：在一个直角三角形中，斜边长度的平方等于另外两条直角边长度的平方之和。该定理是IGCSE数学中最高频的考点之一，在Paper 1和Paper 2中都会出现，同时也是坐标几何、三角学和三维立体计算的基础。
1. Statement of the Theorem | 定理表述
For a right-angled triangle, let the two shorter sides (the legs) be a and b, and let the longest side (the hypotenuse) be c. The theorem gives the algebraic relationship:
a² + b² = c²
对直角三角形，设两条较短的直角边为 a 和 b，最长边（斜边）为 c，则代数关系为：
a² + b² = c²
This formula only works for right-angled triangles. The side c must always be the side opposite the 90° angle. If you substitute the wrong side, your answer will be incorrect, so always identify the hypotenuse first.
这个公式只适用于直角三角形。c 必须始终是 90° 角所对的边。如果代入了错误的边，结果必然出错，因此解题时应先找出斜边。
2. Identifying the Hypotenuse | 识别斜边
The hypotenuse is the side opposite the right angle, and it is always the longest side in a right-angled triangle. The two sides that form the right angle are called the legs. When solving a problem, draw the triangle and label the sides clearly; this simple step prevents most careless errors.
斜边是直角所对的边，永远是直角三角形中最长的边。构成直角的两条边称为直角边。解题时画出三角形并清楚标注各边，这一简单步骤可以避免大部分粗心错误。
In a triangle with a right angle at vertex C, the hypotenuse is the segment AB. If the triangle is drawn in a rotated position, the right-angle symbol (a small square) is the best indicator of which side is the hypotenuse.
若三角形在顶点 C 处为直角，则斜边是线段 AB。即使三角形被旋转了方向，直角符号（小正方形）才是判断哪条边是斜边的最佳标记。
3. Finding the Hypotenuse | 求斜边
When both legs are known, rearrange the theorem to make c the subject:
c = √(a² + b²)
已知两条直角边时，将定理变形，使 c 成为主项：
c = √(a² + b²)
Worked example: a = 6, b = 8, so c = √(6² + 8²) = √(36 + 64) = √100 = 10. Because 100 is a perfect square, the answer is an integer. Always square the two legs first, then add, and finally take the square root.
例题：a = 6，b = 8，则 c = √(6² + 8²) = √(36 + 64) = √100 = 10。因为 100 是完全平方数，所以答案是整数。计算顺序是先分别平方两条直角边，再相加，最后开平方根。
4. Finding a Shorter Side | 求直角边
When the hypotenuse and one leg are known, subtract the square of the known leg from the square of the hypotenuse:
a = √(c² − b²) or b = √(c² − a²)
当已知斜边和一条直角边时，用斜边的平方减去已知直角边的平方，再开方：
a = √(c² − b²) 或 b = √(c² − a²)
Worked example: c = 13, b = 5, so a = √(13² − 5²) = √(169 − 25) = √144 = 12. The order of operations matters: always subtract the smaller square from the larger one, because c is the largest side.
例题：c = 13，b = 5，则 a = √(13² − 5²) = √(169 − 25) = √144 = 12。运算顺序很关键：务必用较大的平方减去较小的平方，因为 c 是最长边。
5. Pythagorean Triples | 勾股数
A Pythagorean triple is a set of three positive integers a, b, c that satisfy a² + b² = c². Recognising common triples saves time in non-calculator papers. Frequently used triples include:
勾股数是满足 a² + b² = c² 的三个正整数。在非计算器试卷中，熟记常见勾股数可以节省大量时间。常用勾股数包括：
- (3, 4, 5)
- (5, 12, 13)
- (7, 24, 25)
- (8, 15, 17)
- (9, 40, 41)
Any multiple of a Pythagorean triple is also a triple. For example, (6, 8, 10) = 2 × (3, 4, 5), and (10, 24, 26) = 2 × (5, 12, 13). If you see two sides such as 6 and 10, you can immediately suspect the third side is 8.
勾股数的任意倍数仍然是勾股数。例如，(6, 8, 10) = 2 × (3, 4, 5)，(10, 24, 26) = 2 × (5, 12, 13)。如果看到两条边分别是 6 和 10，可以立即猜出第三边可能是 8。
6. Worded Problems | 文字应用题
A typical exam problem involves a ladder leaning against a vertical wall. If the ladder has length L and its foot is d metres from the wall, the height h reached on the wall is given by:
h = √(L² − d²)
典型的考题是梯子斜靠在竖直墙上的问题。若梯子长度为 L，梯脚离墙 d 米，则梯顶到达的高度 h 为：
h = √(L² − d²)
Always draw a diagram and mark the right angle. The wall and the ground meet at 90°, so the ladder is the hypotenuse. Do not confuse the horizontal and vertical distances; the biggest side of the triangle must be treated as the hypotenuse.
解题时必须画图并标出直角。墙与地面成 90°，因此梯子是斜边。切勿混淆水平距离与竖直距离，三角形中最长的边必须作为斜边处理。
Another common application is the diagonal of a rectangle. A rectangle with length w and height h has a diagonal equal to √(w² + h²). This appears in questions about television screens, tennis courts and areas of land.
另一个常见应用是矩形的对角线。长为 w、高为 h 的矩形，其对角线长度为 √(w² + h²)。这类问题常出现在电视屏幕、网球场和土地面积的题目中。
7. Distance between Two Points | 两点间距离
In coordinate geometry, the distance d between two points (x₁, y₁) and (x₂, y₂) is found using the formula:
d = √((x₂ − x₁)² + (y₂ − y₁)²)
在坐标几何中，两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离 d 用以下公式计算：
d = √((x₂ − x₁)² + (y₂ − y₁)²)
This is Pythagoras’ Theorem in disguise: the horizontal difference x₂ − x₁ and the vertical difference y₂ − y₁ form the two legs of a right-angled triangle, and the distance is the hypotenuse. Worked example: the distance between (1, 2) and (4, 6) is √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5.
这本质上是勾股定理的另一种形式：水平差 x₂ − x₁ 与竖直差 y₂ − y₁ 构成直角三角形的两条直角边，距离就是斜边。例题：点 (1, 2) 与 (4, 6) 的距离为 √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5。
8. Three-Dimensional Applications | 三维应用
In a cuboid with length l, width w and height h, the space diagonal that joins opposite corners has length:
d = √(l² + w² + h²)
在长、宽、高分别为 l、w、h 的长方体中，连接相对顶点的空间对角线长度为：
d = √(l² + w² + h²)
Worked example: a box with l = 3, w = 4, h = 12 has diagonal d = √(9 + 16 + 144) = √169 = 13. This is a hidden triple (3, 4, 12, 13). For pyramids and cones, look for a right-angled triangle in a vertical cross-section; the slant height and the vertical height are the two legs or leg and hypotenuse.
例题：l = 3，w = 4，h = 12 的箱子，其体对角线 d = √(9 + 16 + 144) = √169 = 13。这是一个隐藏的勾股数组 (3, 4, 12, 13)。对于棱锥和圆锥，要在竖直截面中寻找直角三角形；斜高与竖直高分别是直角边和斜边。
9. Surds and Approximations | 无理数与近似值
Many answers involve irrational numbers. For example, a right-angled triangle with legs 1 and 1 has hypotenuse √2 ≈ 1.414. Unless the question specifies a degree of accuracy, leave the answer in exact surd form such as √5 or 2√3.
很多答案涉及无理数。例如，两条直角边均为 1 的直角三角形，斜边为 √2 ≈ 1.414。除非题目明确指定精度，否则应保留精确的根号形式，如 √5 或 2√3。
If a question asks for an approximation, use the calculator and round to the required accuracy, for example 3 significant figures or 1 decimal place. Do not round intermediate values; keep the exact value until the final step to avoid accumulating errors.
如果题目要求近似值，则用计算器计算并按指定精度四舍五入，例如保留三位有效数字或一位小数。不要在中间步骤四舍五入；应保留精确值到最后一步，以避免误差累积。
10. Converse of Pythagoras | 逆定理
The converse states: if a triangle has side lengths a, b
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