📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings and scale drawings are core skills in the Cambridge KS3 mathematics curriculum. They connect geometry, proportional reasoning and real-world navigation, and they appear again in IGCSE and A Level work on vectors and trigonometry. This revision note explains every rule and method you need for questions such as Exercise 2 on page 195, and it shows you how to draw, measure and check your answers with confidence.
方位角与比例绘图是剑桥 KS3 数学课程中的核心技能，它们把几何、比例推理与现实导航联系起来，并会在 IGCSE 与 A Level 的向量和三角学中再次出现。本复习笔记将系统讲解第 195 页练习 2 这类题目所需的全部规则与方法，并教你如何自信地绘图、测量和检验答案。
1. What Are Bearings? | 什么是方位角
In navigation and map work, a bearing is a direction measured as an angle from north. It tells you the direction of one point from another, and it is the standard language that sailors, pilots and surveyors use because it never depends on which way you happen to be facing.
在导航和地图作业中，方位角是从正北方向测量的角度，用来表示一个点相对于另一个点的方向。它是水手、飞行员和测量员通用的标准语言，因为它的数值不取决于你本人面朝哪个方向。
Bearings are always measured clockwise from the north line. The angle is written using three digits, such as 045°, 090° or 210°, and the degree symbol is always included. A bearing is a pure direction, so it describes where to go but never how far to travel.
方位角始终从正北线沿顺时针方向测量，并使用三位数字表示，例如 045°、090° 或 210°，且必须带上度数符号。方位角是纯粹的"方向"，它只说明往哪里走，从不说明要走多远。
A bearing answers the question: 'From A, in which direction should I travel to reach B?' It does not give a distance, only a direction. If a question asks for both, you must be given or calculate a length as well, usually through a scale drawing or by using the distance on a map.
方位角回答的问题是："从 A 点出发，我应向哪个方向行进才能到达 B 点？"它只给出方向，不给出距离。如果题目同时要求距离，你必须额外获得或计算出一个长度，通常依靠比例绘图或地图上的实际距离来完成。
Because a bearing is measured from north rather than from a previous direction, it is an absolute measurement. Turning left or right does not change a bearing, but moving to a different starting point usually does, which is why every bearing must state clearly where it is measured from.
由于方位角是从正北而不是从上一段方向测量的，它是一种绝对量。向左转或向右转都不会改变方位角，但换一个起点通常就会改变它，因此每个方位角都必须明确说明是从哪一点测量的。
2. Three-Figure Bearings Rules | 三位数方位角规则
There are three essential rules for writing bearings correctly, and examiners check them in almost every question. Break any one of them and you lose the mark, even if your angle is numerically correct.
正确书写方位角有三条基本规则，考官在几乎每道题中都会检查它们。只要违反其中任何一条，即使角度数值正确，你也会失分。
- Always measure from the north line. / 始终从正北线开始测量。
- Always rotate clockwise. / 始终沿顺时针方向旋转。
- Always give the answer as three digits, from 000° to 360°. / 始终用三位数字表示答案，范围是 000° 到 360°。
For example, due east is 090°, due south is 180°, due west is 270°, and due north is 000° or 360°. Notice that 45° written on its own is not a bearing, because the third digit is missing and the reader cannot tell whether you meant 045° or 145°.
例如，正东为 090°，正南为 180°，正西为 270°，正北为 000° 或 360°。注意，只写 45° 不能算作方位角，因为缺少了第三位数字，读者无法判断你想表达的是 045° 还是 145°。
| Direction | Bearing | 中文方向 |
|---|---|---|
| North | 000° | 正北 |
| East | 090° | 正东 |
| South | 180° | 正南 |
| West | 270° | 正西 |
| South-west | 225° | 西南 |
The upper limit of the scale is 360°, which is the same direction as 000°. A bearing is never written as a four-digit number such as 370°, because 370° is simply 010° after one complete turn.
方位角的量程上限是 360°，它与 000° 是同一个方向。方位角永远不会写成 370° 这样的四位数，因为转满一整圈后 370° 其实就是 010°。
3. Compass Points and Standard Bearings | 罗盘方位与标准方位角
It is worth memorising the eight principal compass points as bearings, because they let you check instantly whether an answer is sensible. If a diagram clearly shows a direction to the north-east but your calculation gives 300°, something has gone wrong.
把八个主要罗盘方位记成对应的方位角是很值得的，因为它们能让你立刻判断答案是否合理。如果图中明显指向东北方向，而你的计算结果却是 300°，那一定出了问题。
| Compass point | Bearing | 中文 |
|---|---|---|
| North | 000° | 北 |
| North-east | 045° | 东北 |
| East | 090° | 东 |
| South-east | 135° | 东南 |
| South | 180° | 南 |
| South-west | 225° | 西南 |
| West | 270° | 西 |
| North-west | 315° | 西北 |
These benchmarks also split the full turn into four quadrants: 000° to 090° points somewhere between north and east, 090° to 180° between east and south, 180° to 270° between south and west, and 270° to 360° between west and north.
这些基准值也把一整圈分成四个象限：000° 到 090° 指向北与东之间，090° 到 180° 指向东与南之间，180° 到 270° 指向南与西之间，270° 到 360° 指向西与北之间。
4. Measuring and Drawing a Bearing Accurately | 准确测量与绘制方位角
To measure an existing bearing on a diagram, place the centre of the protractor on the point where the bearing starts. Line up the 0° mark with the north line drawn through that point, then read the scale that increases clockwise from that zero.
要测量图上已有的方位角，先把量角器的中心对准方位角的起点，把 0° 刻度与该点处的正北线对齐，然后沿从零开始顺时针递增的那一圈刻度读数。
- Draw a north line at the starting point, always pointing straight up. / 在起点画一条正北线，方向永远竖直向上。
- Place the protractor centre exactly on the point. / 把量角器中心精确对准该点。
- Read the clockwise scale, not the anticlockwise one. / 读顺时针刻度，不要读逆时针刻度。
- Mark the angle with a small dot, then join the dot to the starting point with a ruler. / 用一个小点标出角度，再用直尺把该点与起点连起来。
- Label the bearing clearly next to the line. / 在线旁清楚标注方位角。
The most common reading error is using the wrong scale on the protractor. Bearings from 0° to 180° are read on the outer scale if the protractor is held the standard way, and bearings from 180° to 360° are read on the inner scale, so always check which numbers are increasing clockwise.
最常见的读数错误是看错了量角器的刻度圈。按常规拿法，0° 到 180° 的方位角读外圈刻度，180° 到 360° 读内圈刻度，所以一定要确认哪一圈数字是顺时针递增的。
5. Back Bearings | 反方位角
A back bearing is the bearing that takes you from B back to A, given the bearing from A to B. Because you have turned through half a revolution, the back bearing differs from the original by exactly 180°.
反方位角是指在已知"从 A 到 B"方位角的情况下，从 B 返回 A 的方位角。由于你正好转了半圈，反方位角与原方位角恰好相差 180°。
Back bearing = θ + 180° if θ < 180°, and θ − 180° if θ ≥ 180°
ring of A from B is 250° − 180° = 070°. The answer must always stay between 000° and 360°.
例如，若 B 相对于 A 的方位角是 060°，则 A 相对于 B 的方位角为 060° + 180° = 240°。若 B 相对于 A 的方位角是 250°，则 A 相对于 B 的方位角为 250° − 180° = 070°。答案必须始终落在 000° 与 360° 之间。
The rule is only valid when both points are connected by a single straight line. If the journey has more than one leg, you must work out the back bearing for each leg separately, because each leg has its own north line.
这条规则只在两点由一条直线相连时才成立。如果行程包含不止一段，你必须对每一段分别求反方位角，因为每一段都有自己的正北线。
6. Scale Drawings and Scale Notation | 比例绘图与比例尺的表示
A scale drawing is a diagram in which every length is a fixed fraction of the real length, so the shape on paper matches the real shape exactly. Scale is usually written as a ratio such as 1 : 50 000, which means 1 unit on the drawing represents 50 000 of the same units in real life.
比例图是一种图上的每条长度都是真实长度某一固定倍数的图形，因此纸上的形状与真实形状完全相似。比例尺通常写成 1 : 50 000 这样的比，意思是图上 1 个单位代表现实中 50000 个相同单位。
Actual distance = Drawing length × Scale factor
f 1 : 50 000 therefore means 1 cm on the drawing represents 50 000 cm, which is 500 m, or 0.5 km.
单位换算是失分最多的地方。请记住 1 km = 1000 m，1 m = 100 cm，所以 1 km = 100 000 cm。因此 1 : 50 000 的比例尺表示图上 1 cm 代表现实中的 50 000 cm，即 500 m，也就是 0.5 km。
| Scale | 1 cm represents | 中文含义 |
|---|---|---|
| 1 : 1000 | 10 m | 图上 1 cm = 实际 10 m |
| 1 : 25 000 | 250 m | 图上 1 cm = 实际 250 m |
| 1 : 50 000 | 500 m | 图上 1 cm = 实际 500 m |
| 1 : 200 000 | 2 km | 图上 1 cm = 实际 2 km |
The larger the second number, the smaller the drawing. A 1 : 200 000 map is a road atlas covering a whole county, while a 1 : 1000 plan might show a single school building in detail.
第二个数字越大，图形就越小。1 : 200 000 的地图是覆盖整个郡的公路图册，而 1 : 1000 的平面图可能只详细展示一所学校的一栋楼。
7. Building an Accurate Scale Drawing | 绘制精确比例图的步骤
Scale drawing questions are marked on method as well as on the final answer, so work through the steps in a fixed order and keep your construction lines visible. Rubbing out the north lines or the arc marks can cost you marks even when the answer is right.
比例绘图题既看最终答案，也看方法，所以要按固定顺序操作，并保留作图线。即使用擦掉了正北线或弧线痕迹，即使答案正确也可能失分。
- Step 1: Write down the scale and convert all real distances into drawing lengths. / 第一步：写下比例尺，把所有真实距离换算成图上长度。
- Step 2: Mark the first point and draw a north line through it. / 第二步：标出第一个点，并过该点画一条正北线。
- Step 3: Measure the first bearing with a protractor and draw the first leg to the correct length. / 第三步：用角器量出第一个方位角，按正确长度画出第一段。
- Step 4: Draw a new, parallel north line at the end of that leg. / 第四步：在该段的端点再画一条平行的正北线。
- Step 5: Repeat for every remaining leg of the journey. / 第五步：对行程中剩余的每一段重复上述步骤。
- Step 6: Join the start to the finish, then measure the required distance and bearing from the drawing. / 第六步：连接起点与终点，再从图上量出所求的距离和方位角。
- Step 7: Convert the measured drawing length back into a real distance using the scale. / 第七步：用比例尺把量得的图上长度换算回真实距离。
Use a sharp pencil, a ruler with a clear edge and a protractor of at least 10 cm diameter. Draw the longest leg first if the figure becomes crowded, and always label the diagram so a reader can follow your reasoning without asking questions.
使用削尖的铅笔、刻度清晰的直尺和直径至少 10 cm 的量角器。如果图形显得拥挤，先画最长的一段；并且一定要给图标上标签，让别人不必追问就能看懂你的思路。
8. Parallel North Lines and Angle Facts | 平行正北线与角度关系
In every bearing diagram the north line at one point is parallel to the north line at every other point, because all meridians point to the same distant north. These parallel lines allow you to transfer angles around the diagram using alternate angles and corresponding angles.
在每一张方位角图中，某一点的正北线都与其它所有点的正北线平行，因为它们都指向同一个遥远的北方。借助这些平行线，你可以用内错角和同位角把角度"搬运"到图中的其它位置。
When two north lines are parallel and a straight path cuts across them, the alternate angles are equal. This means an angle measured between a route and one north line reappears between the same route and a parallel north line elsewhere, which is exactly how you calculate bearings inside triangles without measuring.
当两条正北线平行、一条直线路径横穿它们时，内错角相等。也就是说，某条路线与一条正北线之间的夹角，会在该路线与另一条平行正北线之间再次出现，这正是你在三角形内部不靠测量就能算出方位角的原理。
0°, this gives you two independent ways to check a bearing calculation, and examiners expect to see at least one of them used on longer multi-leg problems.
截线同旁的内角互补，和为 180°。结合 ±180° 的反方位角规则，你就有了两种独立的检验方法，而在较长的多段行程问题中，考官期望你至少用到其中一种。
9. Worked Example 1: A Two-Leg Journey | 例题一：两段行程
A boat leaves harbour A and sails 5 km on a bearing of 075° to point B. It then sails 4 km on a bearing of 150° to point C. Using a scale of 1 cm : 1 km, find the distance AC and the bearing of C from A.
一艘船从港口 A 出发，沿 075° 方位角航行 5 km 到达 B 点，再沿 150° 方位角航行 4 km 到达 C 点。按 1 cm : 1 km 的比例尺，求 AC 的距离以及 C 相对于 A 的方位角。
First convert the distances: 5 km becomes 5 cm and 4 km becomes 4 cm. Draw a north line at A, measure 075° clockwise with the protractor, and draw AB 5 cm long. Then draw a fresh north line at B, measure 150° clockwise, and draw BC 4 cm long.
先换算距离：5 km 变成 5 cm，4 km 变成 4 cm。在 A 点画正北
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