📚 Year 9 CIE Maths: Formula & Theorem Quick Reference Handbook | Year 9 CIE 数学：公式定理速查手册
This quick-reference handbook brings together the essential formulas and theorems you will meet in the Year 9 CIE Mathematics course. Use it to check your recall, to spot patterns, and to build confidence before topic tests and end-of-year exams. Each entry gives a concise statement in English followed by its Chinese equivalent so that bilingual learners can grasp the meaning from both directions.
这本速查手册汇集了你在 Year 9 CIE 数学课程中会遇到的所有重要公式和定理。你可以用它来检验记忆、发现规律，并在单元测验和年终大考前建立信心。每个条目先给出简洁的英文陈述，再给出对应的中文版本，方便双语学习者从两个方向理解含义。
1. Algebra Fundamentals | 代数基础
a(b + c) = ab + ac
The distributive law lets you multiply a term across a bracket. It works the same way when you expand or factorise expressions.
分配律允许你将括号外的项乘入括号内。它在展开或因式分解表达式时同样适用。
(a + b)(a − b) = a² − b²
This is the difference of two squares. Recognising the form lets you factorise quickly without writing out all the middle terms.
这是平方差公式。识别出这种形式后，你可以直接因式分解，而不必写出中间所有项。
(a ± b)² = a² ± 2ab + b²
The perfect-square expansion appears whenever a binomial is multiplied by itself. Remember that the middle term is always twice the product of a and b.
完全平方展开式出现在二项式自乘时。记住中间项总是 a 与 b 乘积的两倍。
2. Linear Equations | 线性方程
If ax + b = c, then x = (c − b) / a
To solve a linear equation, reverse the operations: subtract or add to remove the constant term, then divide by the coefficient of x.
解一元一次方程时，逆序操作：先减去或加上常数项，再除以 x 的系数。
When an equation contains brackets, expand them first, then collect like terms. Check your solution by substituting it back into the original equation.
当方程含有括号时，先展开，再合并同类项。将解代回原方程可以检验答案。
3. Inequalities | 不等式
If a > b, then a + c > b + c for any c
Adding or subtracting the same number from both sides leaves the inequality sign unchanged.
在不等式两边同加或同减一个数，不等号的方向不变。
If a > b and c > 0, then ac > bc. If c < 0, then ac < bc.
Multiplying or dividing by a positive number keeps the sign the same. Multiplying or dividing by a negative number reverses the inequality sign.
乘以或除以正数时不等号方向不变；乘以或除以负数时必须反转不等号。
4. Sequences | 数列
Arithmetic sequence: aₙ = a₁ + (n − 1)d
In an arithmetic sequence the difference between consecutive terms is constant. The nth term is found by adding the common difference (d) repeatedly.
在等差数列中，相邻两项的差为常数。通项公式为 a₁ 加上 (n−1) 个公差 d。
Geometric sequence: aₙ = a₁ × rⁿ⁻¹
For a geometric sequence each term is multiplied by a constant ratio r to get the next term. The nth term uses the first term multiplied by r raised to the power (n−1).
在等比数列中，每一项乘以公比 r 得到下一项。第 n 项是首项乘以 r 的 (n−1) 次幂。
When given a simple linear or quadratic pattern, look for a rule linking the position number to the term. Often the nth term can be written as an expression in n.
当给出一列简单的线性或二次规律时，寻找项的位置编号与该项值之间的关系。第 n 项通常可以写成一个含 n 的式子。
5. Geometry Basics: Angles & Lines | 几何基础：角与线
Vertically opposite angles are equal. ∠AOD = ∠BOC
When two lines cross, the angles opposite each other at the vertex are identical in size.
两条直线相交时，对顶角相等。
Angles on a straight line sum to 180°.
If several angles sit on a straight line, their measures add up to 180°.
直线上相邻的几个角之和为 180°。
Angles around a point sum to 360°.
The full turn at a point is 360°.
绕一个点的周角是 360°。
When a pair of parallel lines is cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and co‑interior (allied) angles sum to 180°.
当一条截线穿过一对平行线时，同位角相等，内错角相等，同旁内角互补（和为 180°）。
6. Triangles & Polygons | 三角形与多边形
Sum of interior angles of a triangle = 180°
No matter what shape a triangle has, its three interior angles always add up to 180°.
无论三角形的形状如何，三个内角之和恒为 180°。
Exterior angle = sum of the two opposite interior angles
An exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it.
三角形的一个外角等于与它不相邻的两个内角之和。
Isosceles triangle: base angles are equal.
In an isosceles triangle the two angles opposite the equal sides are congruent. An equilateral triangle has all three angles equal to 60°.
等腰三角形两底角相等；等边三角形的三个角都等于 60°。
Sum of interior angles of an n‑sided polygon = (n − 2) × 180°
For any polygon, divide it into (n − 2) triangles from one vertex; the angle sum follows. The exterior angles of any convex polygon add up to 360°.
对于 n 边形，从一个顶点连对角线可将其分为 (n−2) 个三角形，因此内角和为 (n−2)×180°。任何凸多边形的外角和都是 360°。
7. Pythagoras' Theorem | 毕达哥拉斯定理
a² + b² = c²
In a right‑angled triangle, c (the hypotenuse) is the longest side, opposite the right angle. The squares of the two shorter legs a and b add to the square of the hypotenuse.
在直角三角形中，斜边 c 是最长边，直角的对边。两直角边 a 和 b 的平方和等于斜边的平方。
This theorem is used both to find missing side lengths and to check whether a triangle is right‑angled. If three sides satisfy the equality, the triangle must contain a 90° angle.
该定理既可用于求未知边长，也可用于检验三角形是否为直角三角形。如果三边满足等式，则三角形必含一个 90° 角。
8. Area, Perimeter & Volume | 面积、周长与体积
Rectangle: A = l × w, P = 2(l + w)
The area of a rectangle is length times width; its perimeter is twice the sum of length and width.
矩形面积 = 长 × 宽；周长 = 2 × (长 + 宽)。
Triangle: A = ½ × base × height
The height must be perpendicular to the base. This formula works for all types of triangles.
高必须与底边垂直；该公式适用于所有三角形。
Circle: C = 2πr or πd, A = πr²
Circumference is π times the diameter. Area uses the radius squared multiplied by π.
圆周长 = π × 直径，或 2πr；圆面积 = π × 半径的平方。
Parallelogram: A = b × h
As with a rectangle, the perpendicular height is the key, not the sloping side.
与矩形类似，关键在于垂直高度，而非斜边。
Trapezium: A = ½ (a + b)h
Here a and b are the parallel sides; h is the perpendicular distance between them.
其中 a 和 b 是平行的两条边，h 是它们之间的垂直距离。
Volume of a prism: V = area of cross‑section × length
A prism has a uniform cross‑section. Multiply the area of that cross‑section by the length (or height) of the prism. For a cuboid this becomes V = l × w × h.
棱柱具有形状大小不变的横截面，体积 = 横截面积 × 棱柱长（或高）。对于长方体，即为 V = 长 × 宽 × 高。
9. Fractions, Percentages, Ratio & Proportion | 分数、百分数、比率和比例
Fraction addition: a/b + c/d = (ad + bc) / bd
Find a common denominator, convert each fraction, then add the numerators. Subtraction follows the same idea.
先通分，将每个分数转换为同分母，然后分子相加。减法遵循同样的思路。
Fraction multiplication: (a/b) × (c/d) = ac / bd
Multiply the numerators and multiply the denominators. Cancel common factors before multiplying to keep numbers small.
分子相乘作分子，分母相乘作分母。在相乘之前先约去公因数可以使数字变小。
Dividing fractions: a/b ÷ c/d = a/b × d/c
Turn the second fraction upside down and multiply. This 'keep, change, flip' rule always works.
将第二个分数上下颠倒后相乘，“保、变、倒”法则总是有效的。
Percentage change = (change ÷ original) × 100%
To calculate a percentage increase or decrease, divide the actual change by the original amount and multiply by 100%.
计算百分比增减时，用变化量除以原值再乘以 100%。
Ratio a : b = c : d implies ad = bc
In a proportion, the cross‑products are equal. This fact lets you solve for an unknown term.
在比例式中，内项积等于外项积，这一性质可以帮助求解未知项。
is the constant of proportionality. Doubling x doubles y. For inverse proportion, xy = k.
当两个量成正比例时，y = kx，k 为比例常数，x 加倍则 y 加倍。成反比例时满足 xy = k。
10. Probability & Statistics | 概率与统计基础
P(event) = number of favourable outcomes / total number of outcomes
Probability is always a number between 0 and 1. A probability of 0 means the event cannot happen; a probability of 1 means it is certain.
概率总是介于 0 和 1 之间的一个数。概率为 0 表示事件不可能发生，为 1 表示事件必然发生。
For mutually exclusive events: P(A or B) = P(A) + P(B)
When two events cannot occur at the same time, the probability that either happens is the sum of their individual probabilities.
两个互斥事件不可能同时发生，其至少一个发生的概率等于各自概率之和。
For independent events: P(A and B) = P(A) × P(B)
If one event does not affect the other, the probability that both happen is the product of their probabilities.
如果一个事件的发生不影响另一个事件的发生，则两者同时发生的概率是各自概率的乘积。
Mean = sum of data values ÷ number of data values
are arranged in order; the mode is the most frequent value. The range = maximum − minimum.
平均数是数据的算术平均值。将数据按大小排序后，位于中间的数据是中位数；出现次数最多的数据是众数；极差 = 最大值 − 最小值。
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