📚 Cambridge Year 10 Maths: Formulas & Theorems Quick Reference Handbook | 剑桥 Year 10 数学：公式定理速查手册
This quick reference handbook compiles all the essential formulas, theorems, and rules needed for the Cambridge IGCSE Year 10 Mathematics course. It covers number, algebra, geometry, trigonometry, coordinate geometry, probability, statistics, vectors, transformations, and functions. Each entry is presented clearly in English and Chinese, with key formulas highlighted for rapid revision.
本速查手册汇集了剑桥 IGCSE Year 10 数学课程所需的所有核心公式、定理和规则，涵盖数论、代数、几何、三角学、坐标几何、概率、统计、向量与变换以及函数。每个条目都提供清晰的中英双语解释，并突出显示关键公式，便于快速复习。
1. Number | 数论
Percentage increase and decrease: Multiply the original amount by (1 ± percentage / 100).
New = Original × (1 ± r / 100)
百分数增减：原量乘以 (1 ± 百分率 / 100)。
P × r × t, where P is principal, r is annual rate (as a decimal), and t is time in years.
I = P × r × t
单利：I = P × r × t，其中 P 为本金，r 为年利率（小数形式），t 为年数。
Compound interest: Amount after n compounding periods.
A = P (1 + r/100)ⁿ
复利：n 个复利周期后的总金额。A = P (1 + r/100)ⁿ。
Reverse percentages: Original = Final amount ÷ (1 ± r / 100).
Original = Final ÷ (1 ± r/100)
反向百分数：原量 = 最终量 ÷ (1 ± r/100)。
e a quantity Q into parts with ratio a : b, the parts are (a/(a+b)) × Q and (b/(a+b)) × Q.
Part = (ratio part / sum of ratios) × Total
按比例分配：将总量 Q 按 a : b 分配，各部分为 (a/(a+b)) × Q 和 (b/(a+b)) × Q。
Speed, density, and pressure relationships.
Speed = Distance / Time, Density = Mass / Volume, Pressure = Force / Area
速度、密度与压力关系：速度 = 距离 / 时间，密度 = 质量 / 体积，压力 = 力 / 面积。
2. Algebra | 代数
Laws of indices: For any base a, the key rules are as follows.
aᵐ × aⁿ = am+n, (aᵐ)ⁿ = amn, aᵐ ÷ aⁿ = am−n, a⁻ⁿ = 1 / aⁿ, a1/n = ⁿ√a
指数定律：aᵐ × aⁿ = am+n，(aᵐ)ⁿ = amn，aᵐ ÷ aⁿ = am−n，a⁻ⁿ = 1 / aⁿ，a1/n = ⁿ√a。
Expanding brackets: Multiply each term in the first bracket by each term in the second.
(a + b)(c + d) = ac + ad + bc + bd
展开括号：将第一个括号中的每一项与第二个括号中的每一项相乘。(a + b)(c + d) = ac + ad + bc + bd。
Factorising: Look for a common factor, or use the form for quadratic trinomials.
x² + bx + c = (x + p)(x + q), where p + q = b, pq = c
因式分解：先提取公因子，二次三项式可用十字相乘法寻找 p 和 q 使 p+q=b, pq=c。
Difference of two squares:
a² − b² = (a − b)(a + b)
平方差公式：a² − b² = (a − b)(a + b)。
Quadratic formula: Solves ax² + bx + c = 0.
x = [−b ± √b² − 4ac] / (2a)
二次公式：求解 ax² + bx + c = 0 的根。x = [−b ± √b² − 4ac] / (2a)。
Arithmetic sequence: nth term and sum of first n terms.
uₙ = a + (n − 1)d, Sₙ = n/2 (2a + (n − 1)d)
等差数列：第 n 项 uₙ = a + (n − 1)d，前 n 项和 Sₙ = n/2 (2a + (n − 1)d)。
Geometric sequence (Extended): nth term and sum of first n terms.
uₙ = arn−1, Sₙ = a(1 − rⁿ) / (1 − r) (for |r| < 1, sum to infinity also valid: S∞ = a/(1−r))
等比数列（扩展）：第 n 项 uₙ = arn−1，前 n 项和 Sₙ = a(1 − rⁿ) / (1 − r)。
Rearranging formulae: Isolate the required variable using inverse operations.
E.g., from y = mx + c, x = (y − c) / m
变换公式：运用逆运算将所求变量独立出来，如 y = mx + c ⇒ x = (y − c) / m。
3. Geometry | 几何
Angles on a straight line sum to 180°, angles around a point sum to 360°.
Straight line: total 180°; Around a point: total 360°
平角 180°，周角 360°。
When parallel lines are cut by a transversal: alternate angles are equal, corresponding angles are equal, co‑interior angles sum to 180°.
Alternate ∠s =, Corresponding ∠s =, Co‑interior ∠s sum to 180°
平行线被截线所截图：内错角相等，同位角相等，同旁内角互补（和为180°）。
Sum of interior angles of an n‑sided polygon: (n − 2) × 180°; sum of exterior angles is always 360°.
Interior sum = (n − 2) × 180°, Exterior sum = 360°
n 边形内角和 = (n − 2) × 180°，外角和恒为 360°。
Pythagoras' theorem: In a right‑angled triangle with legs a, b and hypotenuse c.
a² + b² = c²
毕达哥拉斯定理：直角三角形中，a² + b² = c²（c 为斜边）。
Similar shapes: Corresponding lengths are proportional. If the linear scale factor is k, then the area scale factor is k² and the volume scale factor is k³.
Length ratio = k, Area ratio = k², Volume ratio = k³
相似形：对应边成比例。若长度比为 k，则面积比为 k²，体积比为 k³。
Congruence conditions for triangles: SSS, SAS, ASA, AAS, RHS (right‑angle, hypotenuse, side).
SSS, SAS, ASA, AAS, RHS
三角形全等的判定条件：边边边、边角边、角边角、角角边、直角斜边边。
segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius; tangents from an external point are equal in length.
∠ at centre = 2 × ∠ at circumference; ∠ in semicircle = 90°; tangents from a point are equal
圆定理：圆心角 = 2 × 圆周角；直径所对圆周角 = 90°；同弧所对圆周角相等；圆内接四边形对角互补；切线与半径垂直；圆外一点引出的两条切线长度相等。
4. Mensuration | 测量
Area of a triangle: half the base times the perpendicular height, or using two sides and the included angle.
A = ½ × b × h, or A = ½ ab sin C
三角形面积：底乘高的一半，或 A = ½ ab sin C（已知两边及其夹角）。
Area of a trapezium: half the sum of parallel sides multiplied by the distance between them.
A = ½ (a + b) h
梯形面积：½ (上底 + 下底) × 高。
Circumference and area of a circle.
C = 2πr = πd, A = πr²
圆的周长 C = 2πr = πd，面积 A = πr²。
Arc length and sector area of a circle (θ in degrees).
Arc length = (θ/360) × 2πr, Sector area = (θ/360) × πr²
弧长与扇形面积（角度制）：弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
Cylinder: volume and surface area (r = radius, h = height).
Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πr(r + h)
圆柱：体积 = πr²h，侧面积 = 2πrh，全面积 = 2πr(r + h)。
Cone: volume and surface area (l = slant height).
Volume = ⅓ πr²h
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