📚 Bearings and Scale Drawings | 方位角与比例图
Bearings and scale drawings are essential tools for describing direction and distance in real-world navigation, map reading and construction. In the Cambridge KS3 Mathematics curriculum, students learn to measure and draw three-figure bearings, use back bearings, and interpret scale drawings to solve practical problems.
方位角和比例图是描述方向与距离的重要工具，广泛用于现实生活中的导航、地图阅读和施工测量。在剑桥 KS3 数学课程中，学生需要学会测量和绘制三位数方位角、使用反向方位角，并解读比例图来解决实际问题。
1. Three-Figure Bearings | 三位数方位角
A bearing is a direction measured clockwise from the north line. It is always written as a three-digit angle. For example, a bearing of 65 degrees is written as 065°.
方位角是从北线按顺时针方向测量的角。它总是写成三位数的角度。例如，65 度的方位角写成 065°。
Bearings range from 000° to 360°. North is 000°, east is 090°, south is 180° and west is 270°. This convention removes ambiguity when giving directions.
方位角的取值范围是 000° 到 360°。北方是 000°，东方是 090°，南方是 180°，西方是 270°。这种规定可以避免在描述方向时产生歧义。
Three-figure bearings are especially important when a direction does not follow a simple compass point. For instance, 115° means the direction lies between east and southeast, rotated 25° clockwise from east.
三位数方位角在方向不正好是罗盘主方向时特别重要。例如，115° 表示该方向位于东和东南之间，即从东顺时针再转 25°。
2. Measuring Bearings | 测量方位角
To measure the bearing of point B from point A, draw a vertical north line at A. Place the protractor with its centre at A and 0° pointing north. Read the angle clockwise from the north line to the line AB.
要测量从点 A 到点 B 的方位角，先在点 A 画一条竖直的北线。将量角器的中心放在点 A，并使 0° 指向北方。沿顺时针方向读取从北线到直线 AB 的角度。
Protractors often have two scales. Always check that you are reading the outer or inner scale that starts from 0° at north. If the angle is greater than 180°, you may need to measure the remaining clockwise part carefully.
量角器通常有两圈刻度。务必确认你读的是从北线 0° 开始的那一圈刻度，无论读外圈还是内圈。如果角度大于 180°，你可能需要仔细测量剩余的顺时针部分。
In a diagram, the north line is usually shown as a dashed or solid vertical line with an arrow at the top. Always draw this line yourself if it is not given.
在图中，北线通常画成一条顶部带箭头的虚线或实线。如果题目没有给出北线，一定要自己画出。
3. Compass Directions and Bearings | 罗盘方向与方位角
The main compass directions correspond to these bearings:
主要罗盘方向对应的方位角如下：
| Compass direction | Bearing | 中文方向 |
| North | 000° | 北 |
| Northeast | 045° | 东北 |
| East | 090° | 东 |
| Southeast | 135° | 东南 |
| South | 180° | 南 |
| Southwest | 225° | 西南 |
| West | 270° | 西 |
| Northwest | 315° | 西北 |
For intermediate directions such as NNE or WSW, the bearing can be calculated in steps of 22.5°, but at KS3 the eight-point compass is most common. Being able to convert quickly between compass language and bearings is a key exam skill.
对于像北东北或西西南这样的中间方向，方位角可以按 22.5° 的步长计算，但在 KS3 阶段最常用的是八方位罗盘。能够在罗盘语言和方位角之间快速转换是一项重要的考试技能。
4. Drawing Bearings | 绘制方位角
To draw a bearing of 120° from point P, first draw a north line through P. Place the protractor with 0° on this north line and mark a point at 120° on the clockwise scale. Remove the protractor and draw a straight line from P through the mark.
要从点 P 绘制 120° 的方位角，首先过点 P 画一条北线。将量角器的 0° 放在这条北线上，在顺时针刻度的 120° 处标记一点。移开量角器，从 P 出发过该标记画一条直线。
Label the line with the bearing and, if needed, the distance using a scale. Always use a sharp pencil for accuracy, because errors in angle measurement can lead to large errors in final position.
在这条线上标出方位角，如果题目有要求，再按比例尺标出距离。作图时始终使用削尖的铅笔，因为角度测量误差会导致最终位置出现较大偏差。
If the bearing is greater than 180°, your protractor may only show angles up to 180°. In that case, you can measure the smaller anticlockwise angle from north instead, then convert it to the correct clockwise bearing.
如果方位角大于 180°，你的量角器可能只显示到 180°。此时你可以先量出从北线逆时针方向的较小角度，再换算成正确的顺时针方位角。
5. Back Bearings | 反向方位角
The back bearing is the bearing taken from the second point back to the first point. If the bearing from A to B is θ°, then the back bearing from B to A is θ + 180° if θ is less than 180°, or θ − 180° if θ is 180° or more.
反向方位角是从第二个点返回第一个点时的方位角。如果从 A 到 B 的方位角是 θ°，那么从 B 到 A 的反向方位角为：当 θ 小于 180° 时是 θ + 180°；当 θ 等于或大于 180° 时是 θ − 180°。
Back bearing = θ + 180° (if θ < 180°) ; Back bearing = θ − 180° (if θ ≥ 180°)
is 075 + 180 = 255°. A common check is that the forward and back bearings differ by 180°.
例如，如果 B 在 A 的 075° 方位角上，那么 A 在 B 的方位角就是 075 + 180 = 255°。一个常用的检验方法是正反方位角相差 180°。
Back bearings are useful when a journey involves a return leg or when a ship needs to travel back to its starting point. They also help check whether a bearing has been measured or drawn correctly.
反向方位角在行程包含返程段或船只需要返回起点时非常有用。它们也有助于检查方位角是否被正确测量或绘制。
6. Scale Drawings and Scale Factors | 比例图与比例因子
A scale drawing is an accurate drawing where lengths are reduced or enlarged by a fixed ratio. A scale such as 1 : 50 000 means that 1 cm on the drawing represents 50 000 cm, or 500 m, in real life.
比例图是按固定比例缩小或放大长度的精确图形。比例尺如 1 : 50 000 表示图上 1 厘米代表实际 50 000 厘米，即 500 米。
To convert a real distance to a drawing distance, divide by the scale factor. To convert a drawing distance to a real distance, multiply by the scale factor. Always convert both distances to the same unit before multiplying or dividing.
将实际距离换算成图上距离时，除以比例因子；将图上距离换算成实际距离时，乘以比例因子。在乘除之前，始终将两个距离换算成相同单位。
Example: On a 1 : 20 000 map, 5 cm represents
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