📚 IGCSE Maths Teacher's Book 200: Algebra & Geometry Practice | IGCSE数学教师用书200：代数与几何练习
This revision article reviews the algebra and geometry skills most commonly tested in IGCSE Mathematics, based on the mixed practice in Teacher's Book page 200. It covers linear equations, inequalities, simultaneous equations, quadratics, Pythagoras, trigonometry, polygons, circle theorems, mensuration and exam technique. Read each English point first, then study the paired Chinese explanation to build both language and mathematical confidence.
本文复习 IGCSE 数学中常考的代数与几何核心技能，内容取材于教师用书第200页的混合练习。涵盖一元一次方程、不等式、联立方程、二次式、勾股定理、三角函数、多边形、圆定理、测量以及应试技巧。请先阅读英文要点，再学习配对的汉语讲解，逐步提升语言能力与数学信心。
1. Solving Linear Equations | 解一元一次方程
A linear equation contains variables only to the power 1. To solve it, collect like terms first, then isolate the unknown by using inverse operations. Keep the equation balanced by doing the same thing to both sides.
一元一次方程的变量指数为 1。解法是首先合并同类项，然后用逆运算把未知数单独留在等号一边。每一步都要对等式两边同时进行相同运算，保持等式平衡。
3x − 7 = 2x + 5
ain x = 12. Check by substitution: 3(12) − 7 = 29 and 2(12) + 5 = 29, so both sides agree.
两边同时减去 2x，得到 x − 7 = 5；再同时加上 7，得到 x = 12。代入检验：3(12) − 7 = 29，2(12) + 5 = 29，两边相等。
2. Linear Inequalities on a Number Line | 数轴上的线性不等式
Linear inequalities are solved in the same way as linear equations, with one important exception: when multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
线性不等式的解法与一元一次方程类似，但有一个重要区别：当两边同时乘或除以一个负数时，不等号方向必须改变。
−2x < 8
open circle for strict inequalities such as < or >, and a closed circle for ≤ or ≥.
两边除以 −2 并改变方向：x > −4。在数轴上，严格不等式 < 或 > 用空心圆表示，≤ 或 ≥ 用实心圆表示。
3. Simultaneous Equations by Elimination | 消元法解联立方程
Simultaneous equations can be solved by elimination or substitution. Elimination is usually faster when the coefficients of one unknown can be made equal, so the unknown can be removed by addition or subtraction.
联立方程可以用消元法或代入法求解。当某个未知数的系数可以化为相同或相反时，消元法通常更快，此时可通过相加或相减消去该未知数。
2x + y = 7; x − y = 2
2 to obtain 3 − y = 2, hence y = 1. Write the solution as an ordered pair (x, y) = (3, 1).
两式相加得到 3x = 9，所以 x = 3。把 x = 3 代入 x − y = 2，得 3 − y = 2，因此 y = 1。最后把解写成有序数对 (x, y) = (3, 1)。
4. Factorising Quadratic Expressions | 二次三项式因式分解
When a = 1, factorisation reduces to finding two numbers that multiply to c and add to b.
二次三项式的一般形式为 ax² + bx + c。当 a = 1 时，因式分解简化为找出两个数，使它们的积为 c，和为 b。
x² + 7x + 12
4). If a ≠ 1, use splitting the middle term or systematically try factor pairs of a and c.
3 和 4 的积是 12，和是 7，所以 x² + 7x + 12 = (x + 3)(x + 4)。如果 a ≠ 1，可以用拆中项法，或系统地尝试 a 与 c 的因数组合。
5. Pythagoras' Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. The hypotenuse is always the longest side, opposite the right angle.
在直角三角形中，斜边的平方等于两条直角边的平方和。斜边总是最长边，其位置在直角的对面。
a² + b² = c²
36 + 64 = 100, so c = 10 cm. To find a shorter side, rearrange the formula: a² = c² − b².
如果两条直角边分别为 6 cm 和 8 cm，则 c² = 6² + 8² = 36 + 64 = 100，所以斜边 c = 10 cm。求直角边时移项：a² = c² − b²。
6. Trigonometry in Right-Angled Triangles | 直角三角形中的三角函数
θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent.
使用 SOHCAHTOA 记忆三个三角函数关系：sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边。
sin θ = O ÷ H
n angle, use the inverse function: θ = sin⁻¹(O ÷ H), θ = cos⁻¹(A ÷ H) or θ = tan⁻¹(O ÷ A).
先从已知角出发标出对边、邻边和斜边。求未知边时，选择能把已知边和未知边联系起来的三角函数。求角度时使用反函数：θ = sin⁻¹(O ÷ H)、θ = cos⁻¹(A ÷ H) 或 θ = tan⁻¹(O ÷ A)。
7. Angles in Polygons | 多边形内角与外角
s formula works for convex polygons and is essential for solving regular polygon problems.
n 边形内角和为 (n − 2) × 180°。该公式适用于凸多边形，是解决正多边形问题的关键。
Sum = (n − 2) × 180°
For a regular polygon, each interior angle equals the total sum divided by n. The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.
正多边形的每个内角等于内角和除以 n。任意凸多边形的外角和总是 360°，与边数无关。
8. Circle Theorems | 圆定理
The angle at the centre of a circle is twice the angle at the circumference standing on the same arc. This theorem is frequently tested in IGCSE geometry questions.
同弧所对的圆心角是圆周角的两倍。该定理在 IGCSE 几何题中出现频率很高。
∠AOB = 2 × ∠APB
Angles in the same segment are equal, and the angle in a semicircle is always 90°. A tangent meets a radius at 90°, and opposite angles in a cyclic quadrilateral add to 180°.
同弓形内的圆周角相等，半圆所对的圆周角是 90°。切线与过切点的半径垂直，圆内接四边形的对角互补，即相加为 180°。
9. Perimeter, Area and Volume | 周长、面积与体积
s of the parallel sides and h is the perpendicular height. Do not confuse height with side length.
梯形面积公式为 (a + b) × h ÷ 2，其中 a 和 b 是两条平行边的长度，h 是垂直高度。注意不要将高度与斜边长度混淆。
Area = (a + b)h ÷ 2
ays convert units before substituting, and give area in square units and volume in cubic units.
棱柱体积 = 横截面积 × 长度。圆柱的横截面是圆，所以 V = πr²h。代入前先统一单位，面积用平方单位表示，体积用立方单位表示。
10. Exam Technique for Mixed Questions | 混合题应试技巧
Show all working, even for simple arithmetic. In IGCSE Mathematics, method marks are awarded for correct steps even when the final answer is wrong, so clear working protects your score.
写出完整步骤，即使是简单计算也不要省略。在 IGCSE 数学中，即使最终答案错误，正确步骤仍可获得方法分，因此清晰的书写能保护你的得分。
For geometry questions, draw diagrams and label known values before calculating. In algebra, check solutions by substitution. Manage time by attempting all parts; if stuck, move on and return later.
几何题要先画图并标出已知量再计算。代数题要把解代回原式检验。合理安排时间，尽量完成所有小题；遇到卡壳先跳过，最后再回来思考。
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