📚 Coordinate Geometry for IGCSE WJEC Mathematics | IGCSE WJEC 数学：坐标几何 考点精讲
Coordinate geometry brings algebra and geometry together by plotting points, lines, and shapes on the Cartesian plane. For IGCSE WJEC Mathematics, you will be expected to calculate distances, midpoints, and gradients, find equations of straight lines, and use these to solve problems involving intersections, parallel lines, and perpendicular lines. This guide walks you through every key topic with clear examples and exam tips so you can approach coordinate geometry questions with confidence.
坐标几何将代数与几何结合在一起，通过在笛卡尔平面上绘制点、线和图形来解决问题。在IGCSE WJEC数学考试中，你需要掌握如何计算两点间的距离、中点坐标和斜率，求出直线方程，并利用这些知识解决关于交点、平行线与垂直线的问题。本指南将带你梳理每一个关键知识点，并配以清晰的示例和考试技巧，让你自信面对坐标几何考题。
1. The Cartesian Plane and Plotting Points | 笛卡尔平面与点的绘制
The Cartesian plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Points are written as ordered pairs (x, y), where x gives the horizontal position and y the vertical position. In the WJEC exam, you may need to plot points accurately or read coordinates from a diagram, so familiarity with the four quadrants is essential.
笛卡尔平面由两条互相垂直的数轴构成：x轴（水平）和y轴（垂直）。点的坐标写作有序对 (x, y)，其中x表示水平位置，y表示垂直位置。在WJEC考试中，你可能需要准确地描点或从图中读取坐标，因此熟悉四个象限是必不可少的。
- The first quadrant has both x and y positive. | 第一象限中x和y均为正。
- The second quadrant has x negative, y positive. | 第二象限中x为负，y为正。
- The third quadrant has both negative. | 第三象限中两者均为负。
- The fourth quadrant has x positive, y negative. | 第四象限中x为正，y为负。
2. Distance Between Two Points | 两点间的距离
The distance d between two points (x₁, y₁) and (x₂, y₂) is found using Pythagoras’ theorem: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. This formula appears frequently in WJEC questions, often as part of a larger problem such as proving a triangle is right-angled or finding the perimeter of a shape.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的距离 d 可以用勾股定理求得：d = √[(x₂ − x₁)² + (y₂ − y₁)²]。该公式在WJEC试题中频繁出现，通常作为更大题目的一部分，例如证明一个三角形是直角三角形，或求一个图形的周长。
Example: Find the distance between A(1, 2) and B(4, 6). | 示例：求点A(1, 2)与点B(4, 6)之间的距离。
d = √[(4 − 1)² + (6 − 2)²] = √[3² + 4²] = √(9 + 16) = √25 = 5
3. Midpoint of a Line Segment | 线段的中点
The midpoint M of the segment joining (x₁, y₁) and (x₂, y₂) has coordinates M = ((x₁ + x₂)/2, (y₁ + y₂)/2). It is simply the average of the x-coordinates and the y-coordinates. WJEC often includes this in problems about line segments or in finding the centre of a circle given the endpoints of a diameter.
连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点 M 坐标为 M = ((x₁ + x₂)/2, (y₁ + y₂)/2)，也就是x坐标和y坐标的平均值。WJEC常将其纳入关于线段的问题中，或在已知直径端点时求圆心的问题中出现。
Example: Find the midpoint of (−2, 5) and (4, −1). | 示例：求 (−2, 5) 和 (4, −1) 的中点。
M = ((−2 + 4)/2, (5 + (−1))/2) = (2/2, 4/2) = (1, 2)
4. Gradient of a Straight Line | 直线的斜率
The gradient (or slope) m of a line tells you how steep it is. Given two points (x₁, y₁) and (x₂, y₂), m = (y₂ − y₁)/(x₂ − x₁). A positive gradient means the line goes uphill from left to right; a negative gradient means downhill. Horizontal lines have gradient 0, and vertical lines have undefined gradient. Being able to calculate gradient quickly is crucial for WJEC exam success.
直线的斜率 m 告诉你它的倾斜程度。给定两点 (x₁, y₁) 和 (x₂, y₂)，斜率 m = (y₂ − y₁)/(x₂ − x₁)。斜率为正表示直线从左向右上升；斜率为负表示下降。水平线的斜率为0，垂直线的斜率无定义。能迅速计算斜率对在WJEC考试中取得成功至关重要。
Example: The gradient of the line through (2, 3) and (5, 11) is: | 示例：经过点 (2, 3) 和 (5, 11) 的直线的斜率为：
m = (11 − 3) / (5 − 2) = 8 / 3
5. Equation of a Straight Line: y = mx + c | 直线方程：y = mx + c
The most common form of a straight-line equation in WJEC is y = mx + c, where m is the gradient and c is the y-intercept (the value of y when x = 0). If you know the gradient and one point on the line, you can substitute to find c. Alternatively, if you are given two points, calculate m first, then use one point to find c.
在WJEC考试中，直线方程最常见的形式是 y = mx + c，其中 m 是斜率，c 是y轴截距（即当 x = 0 时 y 的值）。如果你知道斜率和直线上的一个点，就可以代入求得c。如果给出两个点，则先计算m，再使用其中一个点求c。
Example: A line has gradient 2 and passes through (3, 7). Find its equation. | 示例：一条直线的斜率为2，且经过点 (3, 7)。求其方程。
y = 2x + c → 7 = 2(3) + c → 7 = 6 + c → c = 1 → y = 2x + 1
6. Equation of a Line Given Two Points | 已知两点求直线方程
When two points are given, first find the gradient using m = (y₂ − y₁)/(x₂ − x₁). Then substitute one point into y = mx + c to solve for c. This method is heavily tested in WJEC, sometimes in context of real-life graphs or as part of geometry proofs.
当给出两个点时，首先用 m = (y₂ − y₁)/(x₂ − x₁) 求出斜率，然后将其中一个点代入 y = mx + c 求出c。此方法在WJEC考试中反复出现，有时会结合现实生活图像或作为几何证明的一部分。
Example: Find the equation of the line through (−1, 4) and (3, −2). | 示例：求经过点 (−1, 4) 和 (3, −2) 的直线方程。
m = (−2 − 4) / (3 − (−1)) = −6 / 4 = −3/2
Using (−1, 4): 4 = (−3/2)(−1) + c → 4 = 3/2 + c → c = 5/2
Equation: y = −3/2 x + 5/2 or multiply by 2: 2y = −3x + 5 → 3x + 2y = 5
7. Parallel Lines and Their Gradients | 平行线及其斜率
Two distinct lines are parallel if and only if their gradients are equal: m₁ = m₂. In WJEC problems, you may be asked to find the equation of a line parallel to a given line and passing through a specific point. Simply use the same gradient and substitute the point to find c.
两条不重合的直线平行，当且仅当它们的斜率相等：m₁ = m₂。在WJEC问题中，可能会要求你求与已知直线平行且通过某特定点的直线方程。只需使用相同的斜率，并代入该点求c即可。
Example: Find the equation of the line parallel to y = 4x − 7 that passes through (2, 1). | 示例：求平行于直线 y = 4x − 7 且经过点 (2, 1) 的直线方程。
Gradient m = 4, so y = 4x + c. Sub (2,1): 1 = 4(2) + c → c = −7. Equation: y = 4x − 7? Wait, that’s the same line. Actually, distinct parallel lines have same gradient but different intercepts. Check: The given line has intercept −7, our new line has intercept −7 as well? Let’s recalculate: 1 = 8 + c → c = −7. So both lines are identical, meaning (2,1) lies on the original line. If the point was (2, 0), then c = −8. Point (2,1) gives identical line. In exam, if the point lies on the line, the line itself is the answer.
8. Perpendicular Lines and Negative Reciprocals | 垂直线与负倒数
Two lines are perpendicular if the product of their gradients is −1: m₁ × m₂ = −1. This means m₂ = −1/m₁. This concept is extremely common in WJEC, especially in questions about altitudes of triangles or finding the normal to a curve (though curve normals are more IGCSE additonal maths, but basic perpendicularity is core).
两条直线垂直的条件是它们的斜率之积为 −1：m₁ × m₂ = −1，即 m₂ = −1/m₁。这一概念在WJEC中极为常见，尤其出现在关于三角形的高或求曲线的法线等问题中（虽然曲线法线多属于IGCSE附加数学，但基础的垂直关系是核心内容）。
Example: Find the gradient of a line perpendicular to y = 2/3 x + 5. | 示例：求与直线 y = 2/3 x + 5 垂直的直线的斜率。
m₁ = 2/3, so m₂ = −1 / (2/3) = −3/2
9. Intersection of Two Lines | 两条直线的交点
To find where two lines meet, solve their equations simultaneously. For WJEC, this typically involves two linear equations. You can use substitution or elimination. The solution (x, y) is the point of intersection. This skill is also used to find vertices of polygons or points of contact in geometric problems.
要求两条直线的交点，需联立它们的方程并求解。WJEC考试中，这通常涉及两个线性方程，可以使用代入法或消元法。解出的 (x, y) 即为交点坐标。这一技巧也用于求多边形的顶点或几何问题中的接触点。
Example: Find the intersection of y = 2x + 1 and y = −x + 4. | 示例：求直线 y = 2x + 1 与 y = −x + 4 的交点。
2x + 1 = −x + 4 → 3x = 3 → x = 1. Then y = 2(1) + 1 = 3. Intersection: (1, 3).
10. Using Coordinate Geometry to Prove Properties | 运用坐标几何证明性质
WJEC exam questions often demand that you use coordinate geometry to prove that a triangle is right-angled, isosceles, or that a quadrilateral is a parallelogram, rectangle, or rhombus. This involves calculating lengths, midpoints, and gradients, then interpreting the results: right angle if product of gradients = −1; isosceles if two sides have equal length; parallelogram if both pairs of opposite sides are parallel (gradients equal) and diagonals bisect each other (same midpoint).
WJEC考试常要求运用坐标几何证明一个三角形是直角三角形、等腰三角形，或者证明一个四边形是平行四边形、矩形或菱形。这就需要计算长度、中点和斜率，然后解读结果：若两线斜率乘积为−1，则为直角；若两边长度相等，则为等腰；若两组对边分别平行（斜率相等）且对角线互相平分（中点相同），则为平行四边形。
Example: Show that A(2,1), B(6,4), C(5,8) form a right-angled triangle. | 示例：证明点A(2,1), B(6,4), C(5,8)构成直角三角形。
Gradient AB = (4−1)/(6−2) = 3/4. Gradient BC = (8−4)/(5−6) = 4/−1 = −4. Product = (3/4) × (−4) = −3, not −1. Check AC: gradient AC = (8−1)/(5−2) = 7/3. AB and AC product = (3/4)×(7/3) = 7/4. BC and AC product = (−4)×(7/3) = −28/3. None give −1, so not right-angled by those pairs. Wait, let’s test lengths: AB² = (3²+4²)=25, BC² = ((−1)²+4²)=17, AC² = (3²+7²)=58. 25+17=42 ≠ 58, so not right-angled. So the points do not form a right triangle. That’s okay; the method is what matters.
11. Equation of a Line in Other Forms | 直线的其他形式方程
Although y = mx + c is the most common, WJEC also expects you to work with ax + by = c or ax + by + c = 0. Rearranging between forms is a key algebraic skill. Sometimes the intercept form x/a + y/b = 1 is useful for quickly identifying x- and y-intercepts.
虽然 y = mx + c 是最常见的形式，WJEC也要求你能够处理 ax + by = c 或 ax + by + c = 0 的形式。在不同形式之间灵活转换是一项关键的代数技能。有时截距式 x/a + y/b = 1 能够快速识别x轴和y轴的截距。
Example: Convert 3x + 2y = 6 into gradient-intercept form. | 示例：将 3x + 2y = 6 化为斜截式。
2y = −3x + 6 → y = −3/2 x + 3. Gradient = −3/2, y-intercept = 3.
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
When tackling WJEC coordinate geometry questions, always label your points clearly, show all working, and double-check arithmetic. Common pitfalls include confusing the gradient formula (subtracting y-coordinates in the wrong order), forgetting that parallel gradients are equal and perpendicular gradients multiply to −1, and mixing up x and y in midpoint calculations. Practising past papers under timed conditions will help you avoid these errors.
在解答WJEC坐标几何题目时，一定要清晰地标注各点，展示所有计算步骤，并仔细检查算术。常见错误包括：混淆斜率公式（y坐标相减的顺序出错），忘记平行斜率相等而垂直斜率乘积为−1，以及在中点计算中混淆x和y。在限时条件下练习历年真题有助于避免这些错误。
Remember: if a question involves a diagram, extract coordinates carefully. If no diagram is given, sketching one can prevent sign errors. Finally, always check if your final answer makes sense in the context of the question—does a negative gradient make sense if the line is sloping upwards? Simple checks can save marks.
记住：如果题目有示意图，仔细提取坐标。如果没有图，画一个草图可以避免符号错误。最后，一定要检查最终答案是否符合题意——如果直线向右上方倾斜，负斜率合理吗？简单的检查就能保住分数。
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