📚 Year 10 CIE Maths: Formula & Theorem Quick Reference Handbook | Year 10 CIE 数学：公式定理速查手册
This essential quick reference handbook compiles the key formulas and theorems that every Year 10 student following the CIE IGCSE Mathematics syllabus needs at their fingertips. It covers algebra, geometry, trigonometry, statistics, and more, presented with paired English and Chinese explanations to support bilingual learners and reinforce understanding. Use this guide for rapid revision, homework assistance, and exam preparation.
这本必备速查手册汇集了每位学习 CIE IGCSE 数学课程的十年级学生需要掌握的关键公式与定理。内容涵盖代数、几何、三角学、统计等，并采用中英双语对照讲解，助力双语学习者巩固理解。请利用本指南进行快速复习、辅助作业和备考。
1. Algebraic Identities and Factorisation | 代数恒等式与因式分解
The difference of two squares is a fundamental identity: a² − b² = (a − b)(a + b). It is used to factorise expressions like x² − 9 into (x − 3)(x + 3).
平方差公式是一个基本恒等式：a² − b² = (a − b)(a + b)。它可用于将形如 x² − 9 的式子分解为 (x − 3)(x + 3)。
Perfect square trinomials follow these patterns: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Recognising these helps in completing the square.
完全平方三项式遵循以下模式：a² + 2ab + b² = (a + b)² 和 a² − 2ab + b² = (a − b)²。识别这些有助于配方。
For factorising quadratic expressions ax² + bx + c, we look for two numbers that multiply to ac and add to b. Then we split the middle term and factor by grouping.
对于二次式 ax² + bx + c 的因式分解，我们先寻找两个数，使其乘积为 ac 且和为 b，然后拆分中间项并分组分解。
| Identity in English | 中文恒等式 |
|---|---|
| a² − b² = (a − b)(a + b) | 平方差 |
| (a + b)² = a² + 2ab + b² | 和的完全平方 |
| (a − b)² = a² − 2ab + b² | 差的完全平方 |
| a³ + b³ = (a + b)(a² − ab + b²) | 立方和 |
| a³ − b³ = (a − b)(a² + ab + b²) | 立方差 |
2. Equations and Inequalities | 方程与不等式
To solve a quadratic equation ax² + bx + c = 0, you can use the quadratic formula: x = (−b ± √b² − 4ac) / (2a). The discriminant Δ = b² − 4ac determines the nature of the roots.
要解二次方程 ax² + bx + c = 0，可以使用二次公式：x = (−b ± √b² − 4ac) / (2a)。判别式 Δ = b² − 4ac 决定根的性质。
tly one real root (a repeated root); and when Δ < 0, there are no real roots, but two complex roots.
当 Δ > 0 时，方程有两个相异实根；当 Δ = 0 时，有一个实根（重根）；当 Δ < 0 时，没有实根，而有两个复数根。
For linear simultaneous equations, substitution and elimination are the standard methods. Always check your solutions by substituting back into the original equations.
对于线性联立方程，代入法和消元法是标准方法。务必将解代回原方程进行检验。
When solving inequalities, remember that multiplying or dividing by a negative number reverses the inequality sign: if a < b and c < 0, then ac > bc.
解不等式时，要注意乘以或除以负数会反转不等号：若 a < b 且 c < 0，则 ac > bc。
Quadratic inequalities such as x² − 5x + 6 > 0 are solved by factorising, finding critical values, and testing intervals on a number line.
解二次不等式如 x² − 5x + 6 > 0，需先因式分解，求出临界值，然后在数轴上检验区间。
3. Ratio, Proportion and Variation | 比、比例与变分
If two quantities are in direct proportion, their ratio stays constant: y = kx, where k is the constant of proportionality. The graph is a straight line through the origin.
若两个量成正比例，其比值保持不变：y = kx，其中 k 是比例常数。图像是一条过原点的直线。
For inverse proportion, the product of the two quantities is constant: y = k/x or xy = k. The graph is a rectangular hyperbola.
对于反比例，两个量的乘积为常数：y = k/x 或 xy = k。图像是直角双曲线。
In variation problems, you first find k using given data, then use the resulting equation to find unknown values. Always state the relationship clearly, e.g., 'y varies directly as the square of x' gives y = kx².
在变分问题中，先用已知数据求出 k，再使用所得方程求未知量。要明确表述关系，例如“y 随 x 的平方正变”给出 y = kx²。
Ratios can be simplified like fractions, and dividing a quantity in a given ratio involves finding the total number of parts and sharing proportionally.
比可以像分数一样化简，按给定比例分配一个量需要先求出总份数，再按比例分摊。
4. Functions and Graphs | 函数与图像
A function f maps each input x to exactly one output f(x). The domain is the set of all possible inputs; the range is the set of all possible outputs.
函数 f 将每个输入 x 映射到唯一的输出 f(x)。定义域是所有可能输入的集合；值域是所有可能输出的集合。
Composite functions combine two functions: (f ∘ g)(x) = f(g(x)). Apply the inner function first, then the outer function.
复合函数将两个函数结合：(f ∘ g)(x) = f(g(x))。先应用内层函数，再应用外层函数。
: if f(a) = b, then f⁻¹(b) = a. To find it, write y = f(x), swap x and y, and solve for y.
反函数 f⁻¹ 逆转 f 的作用：若 f(a) = b，则 f⁻¹(b) = a。求反函数时，写出 y = f(x)，交换 x 和 y，然后解出 y。
Linear graphs follow y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1.
线性图像遵循 y = mx + c，其中 m 是斜率，c 是 y 轴截距。平行线斜率相等；垂直线斜率之积为 −1。
abolas. The sign of a determines whether the parabola opens up (a > 0) or down (a < 0). The vertex can be found by completing the square or using (−b/(2a), f(−b/(2a))).
二次图像 y = ax² + bx + c 是抛物线。a 的符号决定开口向上 (a > 0) 还是向下 (a < 0)。顶点可通过配方或使用 (−b/(2a), f(−b/(2a))) 求得。
5. Geometry: Angles and Polygons | 几何：角与多边形
When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side are supplementary (sum to 180°).
当一条截线与两条平行线相交时，同位角相等，内错角相等，同旁内角互补（和为 180°）。
The sum of interior angles of a triangle is always 180°. An exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和恒为 180°。三角形的一个外角等于与它不相邻的两个内角之和。
For any polygon with n sides, the sum of interior angles is (n − 2) × 180°. The sum of exterior angles, one at each vertex, is always 360°.
对于任意 n 边形，内角和为 (n − 2) × 180°。在每个顶点取一个外角，所有外角之和恒为 360°。
In a circle, the angle at the centre is twice the angle at the circumference subtended by the same arc. Angles in the same segment are equal.
在圆中，圆心角是同一弧所对圆周角的两倍。同弧上的圆周角相等。
6. Mensuration: Length, Area and Volume | 测量：长度、面积与体积
The area of a rectangle is length × width. For a triangle, use ½ × base × height. A parallelogram's area is base × perpendicular height.
矩形面积 = 长 × 宽。三角形面积用 ½ × 底 × 高。平行四边形的面积 = 底 × 垂直高。
The area of a trapezium is given by ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.
梯形面积公式为 ½ × (a + b) × h，其中 a 和 b 是平行边，h 是它们之间的垂直距离。
For a circle, circumference = 2πr and area = πr². The length of an arc is (θ/360) × 2πr, and the area of a sector is (θ/360) × πr².
对于圆，周长 = 2πr，面积 = πr²。弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
Volume of a prism = area of cross-section × length. A cylinder has volume πr²h and curved surface area 2πrh. A pyramid's volume is ⅓ × base area × height.
棱柱体积 = 底面积 × 长。圆柱体积为 πr²h，侧面积为 2πrh。棱锥体积 = ⅓ × 底面积 × 高。
A cone has volume ⅓πr²h and curved surface area πrl, where l is the slant height. For a sphere, volume = ⁴⁄₃πr³ and surface area = 4πr².
圆锥体积 = ⅓πr²h，侧面积 = πrl（l 为斜高）。球的体积 = ⁴⁄₃πr³，表面积 = 4πr²。
7. Trigonometry | 三角学
In a right-angled triangle, the basic trigonometric ratios are: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. The mnemonic SOH CAH TOA is helpful.
在直角三角形中，基本三角比是：sin = 对边/斜边、cos = 邻边/斜边、tan = 对边/邻边。助记口诀 SOH CAH TOA 很有用。
For any triangle, the sine rule states: a/sin A = b/sin B = c/sin C. The cosine rule: a² = b² + c² − 2bc cos A. These are used when the triangle is not right-angled.
对于任意三角形，正弦定理指出：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。这些适用于非直角三角形。
The area of any triangle can be found using ½ ab sin C, where a and b are two sides and C is the included angle.
任意三角形的面积可用 ½ ab sin C 求得，其中 a 和 b 为两条边，C 为其夹角。
ic values for 0°, 30°, 45°, 60°, and 90° should be memorised, e.g., sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
应当熟记 0°、30°、45°、60° 和 90° 的精确三角值，例如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
8. Vectors | 向量
A vector is a quantity with both magnitude and direction. It can be represented as a column vector (x y)ᵀ or as ai + bj. The magnitude of vector v = (x y)ᵀ is √x² + y².
向量是既有大小又有方向的量。它可以表示为列向量 (x y)ᵀ 或 ai + bj 形式。向量 v = (x y)ᵀ 的模为 √x² + y²。
Vectors are added by adding corresponding components. Scalar multiplication multiplies each component by the scalar. The negative of a vector reverses its direction.
向量相加是将对应分量相加。数乘向量的每个分量都乘以该标量。向量的负号表示方向相反。
The position vector of a point P relative to an origin O is written as OP. The vector from A to B is AB = OB − OA. In geometry, vectors can be used to prove that lines are parallel or that points are collinear.
点 P 相对于原点 O 的位置向量记作 OP。从 A 到 B 的向量为 AB = OB − OA。在几何中，向量可用来证明直线平行或点共线。
AB = k CD, then AB is parallel to CD. If AB = k BC, then points A, B, and C are collinear.
若 AB = k CD，则 AB 与 CD 平行。若 AB = k BC，则
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