📚 Essential Maths Book 9C Answers: Key Concepts Explained | 精华数学9C答案解析：核心知识点精讲
This article breaks down the most important topics covered in the Essential Maths Book 9C, using the answer key as a guide to reinforce understanding. Each section pairs a clear explanation in English with a matching Chinese translation, helping both international and bilingual learners master Key Stage 3 extension mathematics with confidence.
本文基于《精华数学》9C教材配套答案，提炼出核心知识点进行精讲。每个知识点都采用中英双语对照的方式，清晰解释概念与应用，帮助国际课程学生和双语学习者扎实掌握KS3进阶数学内容。
1. Solving Quadratic Equations by Factorising | 因式分解法解二次方程
To solve a quadratic equation like x² – 5x + 6 = 0, first factorise the expression into (x – 2)(x – 3) = 0. Then apply the zero-product property: if the product is zero, at least one factor must be zero. Set x – 2 = 0 to get x = 2, or x – 3 = 0 to get x = 3. Always check your solutions by substituting back into the original equation.
解二次方程如 x² – 5x + 6 = 0 时，首先将表达式因式分解为 (x – 2)(x – 3) = 0。然后利用零乘积性质：如果乘积为零，至少一个因子为零。令 x – 2 = 0 得 x = 2，或 x – 3 = 0 得 x = 3。务必将解代回原方程验算。
x² – 5x + 6 = (x – 2)(x – 3) = 0 → x = 2 or x = 3
2. Completing the Square and the Quadratic Formula | 配方法与求根公式
When factorising is not straightforward, use completing the square or the quadratic formula. For ax² + bx + c = 0, the formula is x = (-b ± √(b² – 4ac)) / 2a. The expression under the square root, b² – 4ac, is called the discriminant and determines the nature of the roots: two real roots if positive, one repeated root if zero, and no real roots if negative.
当因式分解不便时，可使用配方法或求根公式。对于 ax² + bx + c = 0，求根公式为 x = (-b ± √(b² – 4ac)) / 2a。根号下的表达式 b² – 4ac 称为判别式，决定根的性质：正数则有两个实根，零则有一个重根，负数则无实根。
x = (-b ± √(b² – 4ac)) / 2a
3. Simultaneous Equations (Linear and Quadratic) | 一次与二次联立方程组
To solve a pair of equations where one is linear and the other quadratic, substitute the linear expression into the quadratic. For example, if y = 2x + 1 and y = x² + x – 3, replace y in the second equation: 2x + 1 = x² + x – 3. Rearrange to form a quadratic, solve for x, then find the corresponding y values. Always write the solutions as coordinate pairs.
求解一个一次方程与一个二次方程联立的方程组时，将一次表达式代入二次方程。例如已知 y = 2x + 1 和 y = x² + x – 3，把第二个方程中的 y 替换为 2x + 1：2x + 1 = x² + x – 3。整理成二次方程，解出 x，再求出对应的 y 值。解集务必写作坐标对形式。
4. Trigonometry in Right-Angled Triangles | 直角三角形中的三角函数
In any right-angled triangle, the three trigonometric ratios link angles and side lengths: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Use the mnemonic SOH CAH TOA to remember them. When finding an angle, apply the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ to the ratio.
在任意直角三角形中，三个三角函数比值将角度与边长联系起来：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用口诀 “SOH CAH TOA” 帮助记忆。求角时，对比值使用反函数 sin⁻¹、cos⁻¹、tan⁻¹。
sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent
5. Sine and Cosine Rules | 正弦定理与余弦定理
For non-right-angled triangles, use the sine rule: a / sin A = b / sin B = c / sin C for sides and angles, or the cosine rule: a² = b² + c² – 2bc cos A to find a side, and cos A = (b² + c² – a²) / 2bc to find an angle. The sine rule is used when you know two angles and a side, or two sides and a non-included angle. The cosine rule applies when you know three sides or two sides and the included angle.
对于非直角三角形，使用正弦定理：a / sin A = b / sin B = c / sin C（边与角对应），或余弦定理：a² = b² + c² – 2bc cos A（求边），以及 cos A = (b² + c² – a²) / 2bc（求角）。已知两角一边或两边及一对角时用正弦定理；已知三边或两边及其夹角时用余弦定理。
a² = b² + c² – 2bc cos A
6. Arc Length and Sector Area | 弧长与扇形面积
In a circle of radius r, the arc length for an angle θ (in degrees) is L = (θ/360) × 2πr. The area of the corresponding sector is A = (θ/360) × πr². For angles in radians, the formulas simplify to L = rθ and A = ½ r² θ. Remember to use the correct unit for θ and to leave answers in terms of π unless otherwise stated.
半径为 r 的圆中，圆心角 θ（度数）所对的弧长为 L = (θ/360) × 2πr。对应扇形面积为 A = (θ/360) × πr²。若角度以弧度表示，公式简化为 L = rθ，A = ½ r² θ。注意使用正确的角度单位，除非另有说明，答案可保留 π。
Arc Length = (θ/360) × 2πr, Sector Area = (θ/360) × πr²
7. Similarity and Congruence | 相似与全等
Two shapes are congruent if they are identical in shape and size; corresponding sides and angles are equal. Congruence conditions for triangles include SSS, SAS, ASA, and RHS (right angle-hypotenuse-side). Shapes are similar if they have the same shape but may differ in size. In similar triangles, corresponding angles are equal and corresponding sides are in proportion. The scale factor k links lengths, areas (k²), and volumes (k³).
两个图形全等意味着形状和大小完全相同，对应边和角分别相等。三角形的全等条件有 SSS、SAS、ASA 以及 RHS（直角-斜边-边）。形状相同但大小可以不同的图形称为相似。相似三角形中，对应角相等，对应边成比例。比例因子 k 用于关联长度、面积（k²）和体积（k³）。
8. Circle Theorems | 圆定理
Key circle theorems include: the angle at the centre is twice the angle at the circumference subtended by the same arc; angles in the same segment are equal; the angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral sum to 180°; the tangent is perpendicular to the radius at the point of contact; and tangents from an external point are equal in length. Apply these theorems to find missing angles, giving reasons in logical steps.
重要圆定理包括：圆心角等于同弧所对圆周角的两倍；同弧上的圆周角相等；半圆所对的圆周角是直角；圆内接四边形对角互补；切线与过切点的半径垂直；从圆外一点引圆的两条切线长相等。应用这些定理时，需逐步给出推理过程和理由来求未知角。
9. Probability Trees and Conditional Probability | 概率树与条件概率
Probability tree diagrams display the outcomes of two or more successive events. Multiply probabilities along branches for ‘and’ scenarios, and add probabilities of distinct paths for ‘or’ scenarios. For conditional probability, the outcome of the first event affects the probability of the second. The notation P(A|B) means the probability of A given B has occurred. Use the formula P(A|B) = P(A and B) / P(B) when a tree is not available.
概率树状图展示连续两个或多个事件的结果。沿分支相乘得到“且”情形的概率；将不同路径的概率相加得到“或”情形的概率。条件概率中，第一个事件的结果影响第二个事件发生的概率。记号 P(A|B) 表示在 B 发生的条件下 A 发生的概率。若无树状图，可使用公式 P(A|B) = P(A 且 B) / P(B) 计算。
P(A | B) = P(A and B) / P(B)
10. Functions and Inverse Functions | 函数与反函数
A function f(x) maps every input x to exactly one output. The inverse function f⁻¹(x) reverses this mapping, returning the original input. To find an inverse, write y = f(x), swap x and y, then solve for y. For example, if f(x) = 2x + 3, then f⁻¹(x) = (x – 3)/2. The graphs of f and f⁻¹ are reflections in the line y = x. Composite functions such as fg(x) mean apply g first, then f.
函数 f(x) 将每个输入 x 映射到唯一的输出。反函数 f⁻¹(x) 逆转这个映射，返回原始输入。求反函数时，写出 y = f(x)，交换 x 和 y，再解出 y。例如 f(x) = 2x + 3，则 f⁻¹(x) = (x – 3)/2。函数与其反函数的图像关于直线 y = x 对称。复合函数如 fg(x) 表示先作用 g，再作用 f。
11. Inequalities and Regions | 不等式与区域
Solving linear inequalities follows the same rules as equations, except that multiplying or dividing by a negative number reverses the inequality sign. Represent solutions on a number line using open or closed circles. For graphical inequalities, shade the region that satisfies all given inequalities. A solid line means ≤ or ≥, while a dashed line means < or >. The unshaded region is often the required one in exam questions – read the instruction carefully.
解一元一次不等式与解方程类似，但当两边同乘或同除以一个负数时，不等号方向要改变。解集在数轴上用空心或实心圆点表示。对于不等式组，在坐标平面内将满足所有不等式的区域涂上阴影。实线表示“≤”或“≥”，虚线表示“<”或“>”。考题中有时要求标记未涂色区域，务必仔细审题。
12. Statistical Diagrams and Cumulative Frequency | 统计图表与累积频率
Cumulative frequency graphs show the running total of frequencies against the upper class boundaries. To draw the curve, plot cumulative frequency at the end of each interval, then join the points with a smooth curve. The median and quartiles can be estimated from the graph. Box plots (box-and-whisker diagrams) display the minimum, lower quartile, median, upper quartile, and maximum, giving a clear picture of spread and skewness.
累积频率图将逐步累加的频率对应于组距上限绘制。先标出各区间的累计频率点，再用平滑曲线连接。从图中可估算出中位数和四分位数。箱形图（箱线图）展示最小值、下四分位数、中位数、上四分位数和最大值，清晰呈现数据的分散程度和偏态。
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