IGCSE Maths: Coordinate Geometry and Straight-Line Graphs (G-K Student Book 400) | IGCSE 数学：坐标几何与直线图像（G-K 学生用书 400）

📚 IGCSE Maths: Coordinate Geometry and Straight-Line Graphs (G-K Student Book 400) | IGCSE 数学：坐标几何与直线图像（G-K 学生用书 400）

Coordinate geometry connects algebra and geometry on the xy-plane. In IGCSE Maths you must be able to find midpoints, distances, gradients and equations of straight lines, and then combine these skills to solve real problems. This article works through the key ideas in G-K Student Book 400 with clear methods, standard formulas and fully worked examples, together with the exam habits that turn a correct method into full marks.

坐标几何把代数与几何联系在 xy 平面上。在 IGCSE 数学中，你必须掌握中点、距离、斜率和直线方程的求法，并能把这些技能组合起来解决实际问题。本文围绕 G-K 学生用书 400 的核心知识点，给出清晰的方法、标准公式与完整例题，并附上能把会做变成拿满分的考试习惯。

1. Coordinates and the xy-Plane | 坐标与 xy 平面

A point on the plane is written as an ordered pair (x, y). The first number, the x-coordinate, tells you how far the point lies to the right (positive) or to the left (negative) of the y-axis. The second number, the y-coordinate, tells you how far the point lies above (positive) or below (negative) of the x-axis. The two axes meet at the origin (0, 0) and divide the plane into four quadrants.

平面上的点写作有序数对 (x, y)。第一个数即 x 坐标，表示该点在 y 轴右侧（正）或左侧（负）的距离；第二个数即 y 坐标，表示该点在 x 轴上方（正）或下方（负）的距离。两条坐标轴相交于原点 (0, 0)，并把平面分成四个象限。

) and (5, 2) are two completely different points. When you plot a graph in an exam, label both axes, mark the scale clearly, and draw points as small crosses rather than heavy dots so that accuracy can be seen.

点通常用大写字母标注，所以 A(2, 5) 就表示 x = 2、y = 5。顺序非常重要：(2, 5) 与 (5, 2) 是两个完全不同的点。考试作图时，要标注两条坐标轴、写清刻度，并用小叉而不是粗圆点来描点，这样评分老师才能看出你的作图精度。

2. Midpoint of a Line Segment | 线段的中点

The midpoint is the point exactly halfway between two given points. To find it, average the x-coordinates and average the y-coordinates separately. The same rule works whether the segment is horizontal, vertical or sloping, and it never requires Pythagoras.

中点就是恰好位于两个已知点正中间的点。求法是把横坐标取平均值、纵坐标取平均值。无论线段是水平、竖直还是倾斜，规则都相同，而且完全不需要用到勾股定理。

M = ( (x₁ + x₂) ÷ 2 , (y₁ + y₂) ÷ 2 )

• Example: for A(2, 5) and B(8, 11), the midpoint is ((2 + 8) ÷ 2, (5 + 11) ÷ 2) = (5, 8).
  示例：对于 A(2, 5) 和 B(8, 11)，中点为 ((2 + 8) ÷ 2, (5 + 11) ÷ 2) = (5, 8)。
• same way: the midpoint of (−2, 7) and (6, −1) is ((−2 + 6) ÷ 2, (7 + (−1)) ÷ 2) = (2, 3).
  坐标含负数时方法完全一样：(−2, 7) 与 (6, −1) 的中点是 ((−2 + 6) ÷ 2, (7 + (−1)) ÷ 2) = (2, 3)。
• This is useful when you need the centre of a line segment or a point of symmetry.
  当你需要线段的中点或对称中心时，这个方法非常有用。
• g backwards: if the midpoint M and one endpoint A are known, the other endpoint is B = (2x_M − x_A, 2y_M − y_A).
  反向求解：若已知中点 M 和一个端点 A，则另一个端点为 B = (2x_M − x_A, 2y_M − y_A)。

3. Distance Between Two Points | 两点间距离

The distance between two points comes directly from Pythagoras' theorem. It is the length of the hypotenuse of a right-angled triangle whose legs are the horizontal change and the vertical change between the points.

两点间距离直接来自勾股定理。它是直角三角形的斜边长度，两条直角边分别是两点之间的水平变化量和竖直变化量。

d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]

• Example: for A(1, 2) and B(4, 6), d = √[(4 − 1)² + (6 − 2)²] = √9 + 16 = √25 = 5.
  示例：对于 A(1, 2) 和 B(4, 6)，d = √[(4 − 1)² + (6 − 2)²] = √9 + 16 = √25 = 5。
• Always subtract the coordinates in the same order for both x and y inside the bracket; because the differences are squared, swapping the order still gives the correct length.
  括号内 x 和 y 的坐标相减顺序必须一致；由于差值要平方，即使全部颠倒顺序，结果依然正确。
• If the answer is not a perfect square, leave it as a simplified surd unless the question asks for a decimal. For example, the distance between (0, 0) and (2, 3) is √13.
  如果结果不是完全平方数，除非题目要求小数，否则要保留为最简根式。例如 (0, 0) 与 (2, 3) 之间的距离是 √13。
• Recognising common Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17 lets you check answers instantly.
  熟记常见勾股数如 3-4-5、5-12-13、8-15-17，可以让你立刻检验答案。

4. Gradient of a Straight Line | 直线的斜率

The gradient measures how steep a line is. It is the ratio of the vertical change to the horizontal change between any two points on the line, and it is the same value no matter which pair of points you choose.

广告 / Sponsored

斜率衡量一条直线的倾斜程度。它是直线上任意两点之间竖直变化量与水平变化量的比值；无论你选哪一对点，这个比值都相同。

m = (y₂ − y₁) ÷ (x₂ − x₁) = rise ÷ run

• A positive gradient rises from left to right; a negative gradient falls from left to right; a zero gradient is horizontal.
  正斜率从左到右上升；负斜率从左到右下降；斜率为零表示水平线。
• xample: a line through (2, 3) and (6, 11) has gradient m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
  示例：过 (2, 3) 与 (6, 11) 的直线斜率为 m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2。
• Reading a gradient from a graph: pick two points that lie exactly on grid intersections, then count squares up (or down) and across.
  从图像上读斜率：选两个正好落在格点上的点，然后数竖直方向上升（或下降）的格数与水平方向横移的格数。
• hat a steeper line has a larger numerical gradient, and a line at 45° has m = 1 or m = −1.
  一个实用的检验方法：线越陡，斜率的绝对值越大；与水平方向成 45° 的直线斜率是 1 或 −1。

5. The Equation of a Straight Line: y = mx + c | 直线方程 y = mx + c

m is the gradient and c is the y-intercept, that is the y-value where the line crosses the y-axis. This single equation contains everything you need to draw the line or to test whether a given point lies on it.

每一条不竖直的直线都可以写成 y = mx + c 的形式，其中 m 是斜率，c 是 y 轴截距，也就是直线与 y 轴交点的纵坐标。这一个方程就包含了画出直线或判断某点是否在直线上的全部信息。

• Example: the line y = 3x − 5 has gradient 3 and crosses the y-axis at (0, −5).
  示例：直线 y = 3x − 5 的斜率是 3，与 y 轴交于 (0, −5)。
• To test a point, substitute it in: (2, 1) lies on y = 3x − 5 because 3 × 2 − 5 = 1. ✓
  检验某点是否在直线上：把坐标代入即可。(2, 1) 在直线 y = 3x − 5 上，因为 3 × 2 − 5 = 1。✓
• rm, rearrange it first. For example, 2y + 4x = 6 becomes y = −2x + 3, so m = −2 and c = 3.
  如果方程不是这个形式，先变形。例如 2y + 4x = 6 可化为 y = −2x + 3，于是 m = −2、c = 3。
• efully: y = 4x + 7 crosses the y-axis above the origin, while y = 4x − 7 crosses it below.
  务必注意 c 的符号：y = 4x + 7 与 y 轴交于原点上方，而 y = 4x − 7 交于原点下方。

6. Finding the Equation of a Line Through Two Points | 求过两点的直线方程

e the gradient and the coordinates of one point into y = mx + c and solve for c. Step 3: write out the final equation. Step 4: check by substituting the other point.

标准的四步法既快捷又可靠，而且能拿满方法分。第一步：用 m = (y₂ − y₁) ÷ (x₂ − x₁) 求斜率。第二步：把斜率和其中一点的坐标代入 y = mx + c，解出 c。第三步：写出最终方程。第四步：代入另一点进行检验。

• Worked example: find the equation of the line through (2, 3) and (6, 11).
  例题：求过 (2, 3) 与 (6, 11) 的直线方程。
• Step 1: m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2.
  第一步：m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2。
• Step 2: using (2, 3), 3 = 2 × 2 + c, so c = −1.
  第二步：代入 (2, 3)，得 3 = 2 × 2 + c，所以 c = −1。
• Step 3: the equation is y = 2x − 1. Step 4: check with (6, 11): 2 × 6 − 1 = 11. ✓
  第三步：方程为 y = 2x − 1。第四步：用 (6, 11) 检验：2 × 6 − 1 = 11。✓
• the formula (y − y₁) ÷ (x − x₁) = m, which gives the same answer with less rearrangement.
  当坐标比较麻烦时，可以用公式 (y − y₁) ÷ (x − x₁) = m，结果相同但变形更少。

7. Parallel and Perpendicular Lines | 平行线与垂直直线

Two lines are parallel when they have exactly the same gradient, so they never meet. Two lines are perpendicular when their gradients multiply to −1, which means each gradient is the negative reciprocal of the other. Recognising these relationships can save a lot of time in exam questions about shapes on the coordinate plane.

两条直线平行时斜率完全相同，因此永不相交。两条直线垂直时，它们的斜率之积等于 −1，也就是说一个斜率是另一个的负倒数。识别这些关系可以在坐标系图形题中节省大量时间。

Parallel 平行: m₁ = m₂   Perpendicular 垂直: m₁ × m₂ = −1 ⇔ m₂ = −1 ÷ m₁

• Example: y = 3x + 1 and y = 3x − 7 are parallel because both gradients are 3.
  示例：y = 3x + 1 与 y = 3x − 7 平行，因为斜率都是 3。
• Example: y = 3x + 1 and y = −x ÷ 3 + 4 are perpendicular because 3 × (−1 ÷ 3) = −1.
  示例：y = 3x + 1 与 y = −x ÷ 3 + 4 垂直，因为 3 × (−1 ÷ 3) = −1。
• Be careful with negative signs: the negative reciprocal of −2 ÷ 5 is 5 ÷ 2, not −5 ÷ 2.
  小心负号：−2 ÷ 5 的负倒数是 5 ÷ 2，而不是 −5 ÷ 2。
• A common exam task is to find the line through a given point parallel or perpendicular to a given line; find the gradient first, then use the method from Section 6 to find c.
  常见考题是求过某点且与已知直线平行或垂直的直线；先求出斜率，再用第 6 节的方法求 c。

8. Horizontal and Vertical Lines | 水平线与竖直线

rtical lines have equations of the form x = k, where every point has the same x-coordinate and the gradient is undefined, since the run is zero and division by zero is not allowed.

水平线的方程为 y = k 的形式，其中 k 是常数。线上每一点的纵坐标都相同，斜率为零。竖直线的方程为 x = k 的形式，线上每一点的横坐标都相同，斜率为未定义，因为水平变化量为零，而除以零是不允许的。

• Example: the line through (4, −1) and (4, 9) is vertical, so its equation is x = 4.
  示例：过 (4, −1) 与 (4, 9) 的直线是竖直线，方程为 x = 4。
• Example: the line through (−3, 6) and (7, 6) is horizontal, so its equation is y = 6.
  示例：过 (−3, 6) 与 (7, 6) 的直线是水平线，方程为 y = 6。
• ays check whether the two x-coordinates are equal before you start calculating a gradient.
  竖直线无法写成 y = mx + c，所以在计算斜率之前，一定要先检查两点的横坐标是否相等。
• passing through the origin have c = 0 and take the simple form y = mx, for example y = 5x.
  经过原点的直线 c = 0，形式最简单，为 y = mx，例如 y = 5x。

9. Intercepts and Sketching Graphs | 截距与图像草图

t, set x = 0; the answer is simply c. To find the x-intercept, set y = 0 and solve for x, giving x = −c ÷ m as long as m is not zero.

截距是直线与坐标轴的交点，通常是画出直线图像最快的方法。求 y 轴截距时令 x = 0，答案就是 c；求 x 轴截距时令 y = 0 解出 x，只要 m 不为零，就得到 x = −c ÷ m。

• Example: for y = 2x − 6, the y-intercept is (0, −6
  广告 / Sponsored

  【现货】剑桥国际高中教材 Cambridge IGCSE & O Level Additional Mathemati…

  数学 备考教辅 · 正版现货 · 与本文内容相关

  查看详情 / View on Taobao参考价 ¥198.0

  更多咨询请联系16621398022（同微信）

  备考资讯每日更新，关注不错过

  考纲变化 · 科目分析 · 学习方法

  

  在微信里长按二维码识别关注
  或截图后用「扫一扫 → 相册」选择本图

  公众号：tutorhao