📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings and scale drawings are core topics in KS3 geometry that link to real-world navigation and map reading. A bearing is an angle measured clockwise from the north direction, and scale drawings allow us to represent large distances accurately on paper using a chosen ratio. Understanding these concepts is essential for solving problems involving direction, distance, and position.
方位角和比例绘图是KS3几何中的核心主题，它们与现实世界中的导航和地图阅读紧密相关。方位角是从正北方向顺时针测量的角度，比例绘图则允许我们按照选定的比例将实际距离精确地绘制到纸上。理解这些概念对于解决涉及方向、距离和位置的问题至关重要。
1. Understanding Bearings | 理解方位角
A bearing is a measurement of direction used in navigation. It is always measured from the north line, turning clockwise, and written as a three-digit number (e.g., 045°). The abbreviation for degrees is °.
方位角是一种用于导航的方向度量。它总是从指北线出发，顺时针转动测量，并以三位数形式书写（例如045°）。度的简写符号为°。
Key rules for bearings:
- Always measure from North (000°).
- Measure clockwise.
- Express the bearing as three digits (e.g., 005°, not 5°).
方位角的关键规则：
- 总是从北（000°）开始测量。
- 顺时针测量。
- 用三位数字表示方位角（例如005°，而不是5°）。
2. Measuring Bearings from North | 从北开始测量方位角
To find the bearing of point B from point A, draw a north line at A, then measure clockwise from this north line to the line AB. Use a protractor to measure the angle. If the angle is less than 100°, pad with zeros to make three digits (e.g., 060°).
要从A点找到B点的方位角，首先在A点画出一条指北线，然后从该指北线顺时针测量到线段AB的角度。用量角器测量该角。如果角度小于100°，用零补足至三位数（例如060°）。
3. True Bearings and Three-Digit Notation | 真方位角与三位数表示法
True bearings are always given three digits. For example, north is 000° (or 360°), east is 090°, south is 180°, and west is 270°. This standard avoids confusion when communicating directions.
真方位角始终为三位数字。例如，北是000°（或360°），东是090°，南是180°，西是270°。这一标准避免在交流方向时产生混淆。
A summary of compass points:
罗盘点总结：
| Compass Point | Bearing |
|---|---|
| N | 000° |
| NE | 045° |
| E | 090° |
| SE | 135° |
| S | 180° |
| SW | 225° |
| W | 270° |
| NW | 315° |
4. Scale Drawings: Converting Real Distances | 比例绘图：转换实际距离
A scale drawing uses a ratio to represent real distances. For example, a scale of 1 : 50 000 means 1 cm on the drawing represents 50 000 cm (or 0.5 km) in real life. To convert a map distance to actual distance, multiply by the scale factor.
比例绘图使用比例表示实际距离。例如，比例1 : 50000意味着图上的1厘米代表实际中的50000厘米（即0.5公里）。要将地图上的距离转换为实际距离，乘以比例尺因子。
Actual distance = Map distance × Scale factor
实际距离 = 图上距离 × 比例因子
If a map has a scale of 1 cm to 2 km, a 7.5 cm line on the map represents 7.5 × 2 = 15 km.
示例：如果地图的比例是1厘米代表2公里，那么地图上一条7.5厘米的线代表7.5 × 2 = 15公里。
5. Drawing a Scale Diagram with Bearings | 绘制带方位角的比例图
rk a start point, draw a north line, measure the required bearing with a protractor, then draw a line to the correct length according to the scale. Always label distances and bearings on the diagram.
绘制比例图时，首先选择合适的比例（例如1厘米 = 10米）。标出起点，画出指北线，用量角器量出所需的方位角，然后按照比例画出正确长度的线段。记得在图上标注距离和方位角。
Step-by-step: draw north line -> align protractor -> mark angle -> draw line to scaled length.
步骤：画指北线 -> 对齐量角器 -> 标记角度 -> 根据比例画出线段长度。
6. Calculating Bearings from Given Angles | 根据已知角计算方位角
Sometimes a diagram shows an angle from the south or another direction. To find the bearing, you may need to use the fact that angles around a point sum to 360° and angles on a straight line sum to 180°. Always measure from North clockwise.
有时图形中给出的是从南或其他方向的角度。计算方位角时，可能需要利用一点周围的角度和为360°，以及平角为180°的性质。始终从北顺时针测量。
rom 180° (180° - 40° = 140°). Always sketch a north line and visualise the clockwise turn.
例如，方向描述为南偏东40°时，对应的方位角是140°，因为你用180°减去该角度（180° - 40° = 140°）。始终画出指北线并想象顺时针旋转。
7. Back Bearings (Reciprocal Bearings) | 反向方位角
The bearing from B to A is called the back bearing or reciprocal bearing of the bearing from A to B. To find the back bearing, add 180° if the bearing is less than 180°, or subtract 180° if the bearing is 180° or greater.
从B到A的方位角叫做从A到B的方位角的反向方位角或互逆方位角。求反向方位角的方法是：如果原方位角小于180°，则加上180°；如果原方位角大于等于180°，则减去180°。
Back bearing = Bearing + 180° (if Bearing < 180°) or Bearing – 180° (if Bearing ≥ 180°)
反向方位角 = 方位角 + 180°（若方位角 < 180°） 或 方位角 – 180°（若方位角 ≥ 180°）
ng = 075° + 180° = 255°. If bearing from A to C = 210°, back bearing = 210° - 180° = 030°.
示例：从A到B的方位角是075°，反向方位角 = 075° + 180° = 255°。如果从A到C的方位角是210°，反向方位角 = 210° - 180° = 030°。
8. Applications of Bearings and Scale Drawings | 方位角与比例绘图的应用
Bearings and scale drawings are used in navigation, map making, surveying, and architecture. Pilots, sailors, and hikers rely on bearings to set their compass course. Scale drawings help engineers plan buildings and bridges.
方位角和比例绘图在导航、地图绘制、测量和建筑中都有应用。飞行员、水手和徒步旅行者依靠方位角设置指南针航向。比例绘图帮助工程师规划建筑和桥梁。
In exams, you may be asked to interpret a map with bearings, draw a route using given bearings and distances, or calculate the actual distance or area from a scaled diagram.
在考试中，你可能需要解读带有方位角的地图，根据给定的方位角和距离绘制路线，或者从比例图计算实际距离或面积。
9. Common Mistakes and Tips | 常见错误与技巧
- Forgetting to use three-digit form (e.g., writing 30° instead of 030°).
- Measuring the angle anticlockwise or from a line other than North.
- Using the wrong scale conversion (e.g., mixing cm and m).
- Not drawing a clear north line before measuring bearings.
- Forgetting to add or subtract 180° correctly for back bearings.
常见错误列表：
- 忘记使用三位数格式（例如写30°而非030°）。
- 逆时针测量或者从非指北线开始测量。
- 使用错误的比例换算（比如混淆厘米与米）。
- 在测量方位角之前未画出清晰的指北线。
- 反向方位角时错误地加减180°。
Tip: Always sketch a small north symbol and double-check your protractor alignment. When in doubt, visualise the compass points.
提示：总是画一个小的指北符号，并仔细检查量角器的对齐。当有疑问时，想象罗盘点的位置。
10. Summary and Key Points | 总结与要点
Bearings are clockwise angles from North, written as three digits. Scale drawings use a ratio to connect drawing lengths to real lengths. To solve problems, combine accurate angle measurement with careful scale conversion. Practice drawing diagrams and converting between bearings and back bearings to build confidence.
方位角是从北顺时针旋转的角度，以三位数字表示。比例绘图利用比例将图纸长度与实际长度关联起来。解决相关问题时，需要结合精确的角度测量与仔细的比例换算。通过练习绘制示意图以及方位角与反向方位角的转换，可以增强解题信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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