📚 Applications and interpretation: Geometry | 应用与解释：几何
The Geometry topic in IB Mathematics: Applications and Interpretation covers essential skills in coordinate geometry, trigonometry, spatial reasoning, and modelling. It equips students with tools to solve real-world problems involving distances, areas, volumes, and optimal shapes. The use of technology and applications such as Voronoi diagrams makes this a highly practical branch of the syllabus.
IB数学“应用与解释”课程中的几何主题涵盖了坐标几何、三角学、空间推理和建模的基础技能。它为学生提供了解决涉及距离、面积、体积和最优形状等实际问题的方法。沃罗诺伊图等科技工具和应用使这一板块具有高度实用性。
1. Coordinate Geometry Essentials | 坐标几何基础
The Cartesian coordinate system uses ordered pairs (x, y) to locate points on a plane. The horizontal axis is the x-axis and the vertical axis is the y-axis. Geometry problems in this topic often involve calculating distances, midpoints, gradients, and equations of lines.
笛卡尔坐标系使用有序对 (x, y) 在平面上定位点。横轴为x轴，纵轴为y轴。该主题中的几何问题通常涉及计算距离、中点、斜率和直线方程。
Understanding how to move between algebraic and geometric representations is fundamental for modelling constraints, boundaries, and paths.
理解如何在代数与几何表示之间切换，对于建模约束条件、边界和路径至关重要。
2. Distance and Midpoint | 距离与中点
Given two points A(x₁, y₁) and B(x₂, y₂), the distance d between them is given by the formula derived from Pythagoras' theorem.
给定两点 A(x₁, y₁) 和 B(x₂, y₂)，它们之间的距离 d 由基于毕达哥拉斯定理的公式给出。
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This distance represents the length of the straight-line segment AB and is always a non-negative value.
该距离表示直线段 AB 的长度，并且始终为非负值。
The midpoint M of the segment AB is the average of the coordinates of the two endpoints.
线段 AB 的中点 M 是两个端点坐标的平均值。
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
These formulas are used extensively in finding perpendicular bisectors and in Voronoi diagram construction.
这些公式广泛用于求垂直平分线以及构建沃罗诺伊图。
3. Gradient and Equations of Lines | 斜率与直线方程
The gradient (slope) m of a line passing through points (x₁, y₁) and (x₂, y₂) is the ratio of vertical change to horizontal change.
经过点 (x₁, y₁) 和 (x₂, y₂) 的直线的斜率 m 是垂直变化与水平变化的比值。
m = (y₂ − y₁) / (x₂ − x₁)
rcept. The point-gradient form is y − y₁ = m(x − x₁). The general form is ax + by + d = 0.
直线可用多种有用的形式表示。斜截式为 y = mx + c，其中 c 是y轴截距。点斜式为 y − y₁ = m(x − x₁)。一般式为 ax + by + d = 0。
radients (m₁ = m₂). Perpendicular lines have gradients whose product is −1 (m₁ × m₂ = −1).
平行线的斜率相同（m₁ = m₂）。垂直线的斜率乘积为 −1（m₁ × m₂ = −1）。
Choosing the appropriate form simplifies solving intersection, distance from a point to a line, and optimisation problems.
选择合适的直线形式可简化求交点、点到直线的距离以及优化问题。
4. Perpendicular Bisectors | 垂直平分线
The perpendicular bisector of a segment is the line of points equidistant from its two endpoints. It passes through the midpoint and is perpendicular to the segment.
线段的垂直平分线是到其两个端点距离相等的点的直线。它经过中点并与线段垂直。
To find its equation: (1) calculate the midpoint M, (2) determine the gradient of the segment m, then the perpendicular gradient is −1/m, (3) use the point-gradient form through M.
求其方程：(1) 计算中点 M，(2) 确定线段的斜率 m，则垂直线的斜率为 −1/m，(3) 利用过 M 的点斜式。
Perpendicular bisectors are the building blocks of Voronoi diagrams, where they define boundaries between adjacent sites.
垂直平分线是沃罗诺伊图的构建模块，它们定义了相邻站点之间的边界。
5. Voronoi Diagrams | 沃罗诺伊图
A Voronoi diagram partitions a plane into regions based on proximity to a given set of points called sites. Each region contains all points closer to its site than to any other site.
沃罗诺伊图根据到一组给定点（称为站点）的距离将平面划分为区域。每个区域包含所有离其站点比离其他站点更近的点。
The edges of the diagram consist of segments of perpendicular bisectors between neighbouring sites. Vertices are points equidistant from three or more sites.
图的边由相邻站点间的垂直平分线线段组成。顶点是到三个或更多站点等距的点。
Applications include modelling cell phone coverage, identifying nearest service locations, epidemiology mapping, and animal territory analysis.
应用包括模拟手机信号覆盖、识别最近服务点、流行病制图和动物领地分析。
When adding a new site, find the existing region it lies within, then compute perpendicular bisectors with nearby sites to redraw the affected boundaries.
当添加新站点时，先找到它所在的现有区域，然后计算与附近站点的垂直平分线，以重绘受影响的边界。
6. Trigonometry: Sine and Cosine Rules | 三角学：正弦与余弦定理
For any triangle with sides a, b, c opposite angles A, B, C respectively, the Sine Rule relates sides and angles.
对于任意三角形，边 a、b、c 分别对应角 A、B、C，正弦定理建立了边与角的关系。
a / sin A = b / sin B = c / sin C
This is used to find missing sides or angles when at least one side-angle pair is known. Be aware of the ambiguous case (SSA), where two different triangles may exist.
当至少已知一对边角时，可用该定理求未知边或角。注意歧义情况 (SSA)，此时可能存在两个不同的三角形。
The Cosine Rule generalises Pythagoras' theorem and is essential when dealing with non-right triangles with SAS or SSS given.
余弦定理推广了毕达哥拉斯定理，对于给出两边夹角 (SAS) 或三边 (SSS) 的非直角三角形至关重要。
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / (2bc)
Both rules are heavily used in navigation, surveying, and construction where direct measurement of heights or distances is impractical.
这两个定理广泛应用于导航、测量和建筑中，这些场合通常无法直接测量高度或距离。
7. Area of a Triangle | 三角形面积
When the perpendicular height is not known, the area of a triangle can be found using two sides and the included angle.
当不知道垂直高时，可以使用两边及其夹角来计算三角形的面积。
Area = ½ ab sin C
This formula works for any triangle and is extremely useful in trigonometrical applications such as finding the area of a non-right-angled plot of land.
该公式适用于任何三角形，在三角学应用中非常有用，例如计算一块非直角三角形土地的面积。
If all three sides are known, Heron' rovides an alternative: Area = √[s(s−a)(s−b)(s−c)], where semiperimeter s = (a + b + c)/2.
如果已知三边长，海伦公式提供了另一种方法：Area = √[s(s−a)(s−b)(s−c)]，其中半周长 s = (a + b + c)/2。
Choosing the appropriate area formula depends on the given data; both are common in IB questions that combine trigonometry with measurement.
选择适当的面积公式取决于所给数据；在IB考题中，两者常与三角函数和测量相结合。
8. Radians, Arcs and Sectors | 弧度、弧长与扇形面积
as the ratio of arc length to radius. One full revolution is 2π radians; 180° = π radians.
弧度将角量度化为弧长与半径之比。一整圈为 2π 弧度；180° = π 弧度。
To convert: multiply degrees by π/180 to obtain radians, and multiply radians by 180/π to obtain degrees.
转换方法：度数乘以 π/180 得出弧度值；弧度值乘以 180/π 得出度数。
For a circle with radius r and central angle θ measured in radians, the arc length and sector area are elegantly simple.
对于半径为 r、圆心角 θ 以弧度为单位时，弧长和扇形面积的公式非常简洁。
Arc length l = r θ
Sector area A = ½ r² θ
These formulas appear in problems involving circular motion, land irrigation systems, and designing curved structures.
这些公式出现在涉及圆周运动、土地灌溉系统和设计曲线结构的问题中。
9. Surface Area and Volume of 3D Solids | 立体图形的表面积与体积
Common three-dimensional solids examined in the course include prisms, cylinders, pyramids, cones, and spheres. Composite solids are formed by combining or subtracting these shapes
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