📚 Bearings and Scale Drawings | 方位与比例绘图
Bearings are a way of describing the direction from one point to another. They are measured in degrees, always starting from the north direction and moving clockwise. This topic also includes drawing and interpreting scale diagrams to represent real-life distances and positions. In this revision guide, you will learn how to measure and write three-figure bearings, construct accurate scale drawings, and solve problems using these skills.
方位是描述从一点到另一点的方向的一种方法。它以度为单位测量，总是从正北方向开始，沿顺时针方向移动。本主题还包括绘制和解释比例图，以表示现实生活中的距离和位置。在本复习指南中，你将学习如何测量和书写三位数的方位角，构建精确的比例图，并使用这些技能解决问题。
1. What Are Bearings? | 什么是方位？
In mathematics, a bearing shows the exact direction you must travel from one location to another. It is always measured from north in a clockwise direction, and is given as a three‑digit number (e.g. 045°). Even if the angle measures less than 100°, we write a zero in front so the bearing has three digits. The angle cannot be negative and cannot exceed 360°.
在数学中，方位表示你必须从一个位置移动到另一个位置的确切方向。它总是从正北沿顺时针方向测量，并以三位数表示（例如045°）。即使角度小于100°，我们也要在前面写一个零，使得方位有三位数字。角度不能为负值，也不能超过360°。
Bearings are used by pilots, ship captains, and hikers to navigate without landmarks. They provide a common reference direction – true north – which everyone can understand. On a map, the north line is usually drawn vertically up the page.
飞行员、船长和徒步旅行者使用方位在没有地标的情况下导航。它们提供了一个共同的参考方向——真北——每个人都能理解。在地图上，北向线通常垂直向上绘制。
| Direction | Bearing |
| North | 000° |
| East | 090° |
| South | 180° |
| West | 270° |
| Northeast | 045° |
2. Three-Figure Bearings Rule | 三位数方位规则
The most important rule for bearings is that they must always be written using three digits. This means an angle of 30° is recorded as 030°. Similarly, 5° becomes 005°, and 345° remains 345°. This avoids confusion between angles that could be confused with two‑digit directions, and sets a standard format for all bearings.
方位最重要的规则是必须始终使用三位数字书写。这意味着30°的角度要记录为030°。同样，5°变成005°，345°仍然是345°。这避免可能与两位数方向混淆的角度，并为所有方位设定了标准格式。
When measuring the bearing of B from A, you must draw a north line at point A, then measure clockwise until the line points towards B. Use a protractor carefully: align the centre with A and the 0° mark with north, then read the angle at the line joining A to B. This angle is the bearing, and you should immediately convert it to a three‑digit number.
当测量A点相对于B点的方位时，必须在A点画一条北向线，然后顺时针测量直到那条线指向B。小心使用量角器：将中心对准A，0°标记对准北方，然后读取连接A到B的线的角度。这个角度就是方位，你应该立即将其转换为三位数字。
3. Measuring Bearings from Maps or Diagrams | 从地图或图表中测量方位
To find the bearing of point Y from point X on a diagram, follow these steps: (1) Place your protractor so that its centre is exactly on X. (2) Make sure the 0° line points directly north – you may need to draw a faint north line through X parallel to the given north arrow. (3) Draw a straight line from X to Y if it is not already there. (4) Read the outer scale of the protractor clockwise from north to the line XY. (5) Write the reading as a three‑digit bearing.
要在图表中找到Y点相对于X点的方位，请遵循以下步骤：(1) 将量角器放置，使其中心恰好位于X上。(2) 确保0°线直接指向北方——你可能需要通过X画一条与给定北方箭头平行的虚北向线。(3) 如果还没有连线，从X到Y画一条直线。(4) 从北顺时针读取量角器的外圈刻度，直到XY线。(5) 将读数写为三位数的方位。
Always check which scale you are using on the protractor – most protractors have two scales, an inner and an outer. For a clockwise measurement from north, you will normally use the outer scale if the protractor is placed facing the correct direction. Practise with different diagrams until you are confident reading angles both less than 180° and greater than 180°.
始终检查你正在使用量角器上的哪个刻度——大多数量角器有两个刻度，内圈和外圈。对于从北开始的顺时针测量，如果量角器放置方向正确，通常会使用外圈刻度。在能够自信地读取小于180°和大于180°的角度之前，请使用不同的图表进行练习。
4. Scale Drawings – Representing Real Distances | 比例图——表示实际距离
A scale drawing is an accurate diagram where lengths are drawn in proportion to the real distances they represent. The scale tells you how much each unit on the paper corresponds to in reality. For example, a scale of 1 cm : 5 km means that every 1 cm on the drawing equals 5 km in the real world. Scale drawings allow us to solve navigation and construction problems without needing to measure huge lengths directly.
比例图是一种精确的图表，其中长度按比例绘制，代表它们所表示的真实距离。比例告诉你纸上每一单位在现实中对应多少。例如，比例1 cm : 5 km表示图上的每1厘米等于现实中的5公里。比例图使我们无需直接测量巨大的长度就能解决导航和建筑问题。
will be 24 ÷ 3 = 8 cm. Use a ruler to draw lines with millimetre accuracy where possible.
选择比例时，确保整个图画能舒适地容纳在你的页面上。在除以比例因子之前，将所有距离转换为相同的单位。如果实际距离是24公里，比例是1厘米:3公里，那么图上的长度将是24 ÷ 3 = 8厘米。尽可能使用尺子以毫米精度画线。
5. Constructing a Scale Drawing Using Bearings | 使用方位构建比例图
e the first leg: if you choose 1 cm : 4 km, the first line will be 20 ÷ 4 = 5 cm. Mark point B. (4) Draw a new north line at B, measure 140° clockwise, and draw the second leg 12 km → 3 cm to point C. (5) Finally, you can measure the direct bearing and distance from C back to A if required.
假设一艘船从港口A出发，沿着075°方位航行20公里到达浮标B，然后转向140°方位航行12公里到达灯塔C。要比例绘制这次航程：(1) 在A处画一条北向线。(2) 使用量角器，从北顺时针测量075°，并通过A画一条直的构造线。(3) 为第一段航程定比例：如果你选择1厘米:4公里，第一条线将为20 ÷ 4 = 5厘米。标记B点。(4) 在B处画一条新的北向线，顺时针测量140°，并画出第二段12公里→3厘米通往C点。(5) 最后，如果需要，你可以测量从C返回A的直接方位和距离。
Always use a sharp pencil, and keep construction lines faint. The bearings at each point must be measured from a north line through that point. Be careful: the north direction always points the same way, so all north lines must be parallel. If your drawing is accurate, you can measure the return bearing with a protractor and the distance with a ruler, then convert back using your scale.
始终使用尖细的铅笔，并保持构造线浅淡。每个点处的方位必须从通过该点的北向线测量。注意：北向总是指向相同的方向，因此所有北向线必须平行。如果你的图画准确，你可以用量角器测量返回方位，用尺子测量距离，然后使用比例换算回去。
6. Back Bearings (Reverse Bearings) | 反向方位
0° → back bearing = 060° + 180° = 240°. Bearing of D from C is 200° → back bearing = 200° − 180° = 020°.
如果已知Y相对于X的方位，我们可以轻松找到X相对于Y的方位，称为反向方位或互逆方位。规则是：如果原始方位小于180°，则加上180°得到反向方位。如果原始方位等于或大于180°，则减去180°。例如，B相对于A的方位是060° → 反向方位 = 060° + 180° = 240°。D相对于C的方位是200° → 反向方位 = 200° − 180° = 020°。
Back bearings are useful when you need to describe the direction of the starting point from the ending point without redrawing the whole diagram. They are also used to check the accuracy of a scale drawing: the angle you measure from the end point back to the start should match the calculated back bearing closely.
当你需要从终点描述起点的方向而不必重绘整个图表时，反向方位非常有用。它们也用于检查比例图的准确性：你从终点量回起点的角度应该与计算出的反向方位非常接近。
7. Solving Problems with Bearings and Parallel Lines | 使用方位与平行线解决问题
Many bearing problems combine the skill of measuring angles with properties of parallel lines. In a diagram, all north lines are parallel. This creates alternate angles and co‑interior angles that can help you calculate unknown bearings without a protractor. For instance, if you know the bearing from A to B and the angle the path turns at B, you can work out the new bearing using facts about angles on a straight line and around a point.
许多方位问题将测量角度的技能与平行线的性质结合起来。在图表中，所有北向线都是平行的。这就产生了内错角和同旁内角，可以帮助你在不使用量角器的情况下计算出未知的方位。例如，如果你知道从A到B的方位以及路径在B处的转弯角度，你可以利用直线上的角和围绕一点的角的事实计算出新的方位。
Typical KS3 questions might give a diagram with some bearings marked and some angles needed. Remember: angles around a point add up to 360°. If a north line at B goes straight up, and the bearing from A to B is 120°, then the angle between the line AB extended backwards and the north line at B can be linked using corresponding angles or alternate angles. These geometric links are the key to solving more complex questions.
典型的KS3问题可能会给出一个图，其中标注了一些方位和需要计算的角度。请记住：围绕一个点的角的总和为360°。如果在B处的北向线垂直向上，并且从A到B的方位是120°，那么AB反向延长线与B处北向线之间的角度可以通过同位角或内错角联系起来。这些几何联系是解决更复杂问题的关键。
8. Common Mistakes When Using Bearings | 使用方位时的常见错误
• Forgetting three digits: Writing 30° instead of 030° is a very common slip. Always check if you need to add leading zeros.
• Measuring anticlockwise or from south: Bearings are always measured clockwise from north. Double‑check your protractor orientation.
• Using the wrong scale on the protractor: If the angle is over 180°, make sure you read the correct scale that continues beyond 180°.
• Neglecting to draw north lines: For each new point you must draw a fresh north line; do not use the same north line as the first point unless they coincide.
• Misapplying the back bearing rule: Adding or subtracting 180° works only when the bearing is given correctly; if you have an answer like 400°, you should subtract 360° to bring it back into range.
• 忘记三位数：将30°写作030°是非常常见的疏忽。始终检查是否需要添加前导零。
• 逆时针测量或从南方测量：方位始终是从北方顺时针测量的。仔细检查你的量角器方向。
• 使用量角器上错误的刻度：如果角度超过180°，确保你读取延续超过180°的正确刻度。
• 忽略画北向线：每到一个新的点，你必须画一条新的北向线；不要使用与第一个点相同的北向线，除非它们重合。
• 错误应用反向方位规则：只有在给定方位正确的情况下，加或减180°才有效；如果你得到像400°这样的结果，应该减去360°使它回到范围内。
9. Real‑Life Applications | 实际应用
Bearings and scale drawings are not just textbook topics – they are used every day by pilots, sailors, architects, and even in video game design. When an air traffic controller says, 'Turn to heading 270',' they are giving a bearing. When delivery drivers plan routes using digital maps, the software uses bearings to calculate the shortest path. Ordnance Survey maps also contain grid bearings for walkers to follow.
方位和比例图不仅仅是课本上的话题——它们每天被飞行员、水手、建筑师，甚至在电子游戏设计中用到。当空中交通管制员说“转向航向270”时，他们就是在给定一个方位。当送货司机使用数字地图规划路线时，软件会使用方位来计算最短路径。英国地形测量局的地图也包含供步行者遵循的网格方位。
Even in emergency rescue operations, coastguards take a bearing from a distress call and plot it on a chart to locate a vessel. Understanding how to interpret and use bearings accurately can therefore be a vital skill, not just a mathematical exercise.
即使在紧急救援行动中，海岸警卫队也会从求救信号中获取方位，并在海图上标出以定位船只。因此，理解如何准确解读和使用方位不仅是一项数学练习，而且是一项至关重要的技能。
10. Practice Example Walkthrough | 练习示例讲解
Question: A helicopter leaves base H and flies 36 km on a bearing of 110° to reach checkpoint P. Then it flies 28 km on a bearing of 245° to reach checkpoint Q. Using a scale of 1 cm : 6 km, make a scale drawing and find the bearing and distance of the return journey from Q back to H.
问题：一架直升机离开基地H，沿110°方位飞行36公里到达检查点P。然后它沿245°方位飞行28公里到达检查点Q。使用1厘米:6公里的比例，制作一个比例图，并求出从Q返回H的返回方位和距离。
Solution approach:
• Scale convert: 36 km ÷ 6 = 6 cm; 28 km ÷ 6 ≈ 4.7 cm (to 1 decimal place).
• Draw north line at H, measure 110° clockwise, draw line HP 6 cm long, label P.
• At P, draw a new north line (parallel to the first). Measure 245° clockwise and draw line PQ 4.7 cm, label Q.
• line. Measure the length of QH with a ruler (e.g. 8.2 cm) → distance = 8.2 × 6 = 49.2 km.
• value close to the theoretical answer from trigonometry: distance ≈ 49 km, bearing ≈ 322°.
求解步骤：
• 比例转换：36公里 ÷ 6 = 6厘米；28公里 ÷ 6 ≈ 4.7厘米（保留一位小数）。
• 在H处画北向线，顺时针测量110°，画6厘米长的线段HP，标记P。
• 在P处，画一条新的北向线（与第一条平行）。顺时针测量245°，并画4.7厘米长的线段PQ，标记Q。
• 用直线连接Q和H。用尺子测量QH的长度（例如8.2厘米） → 距离 = 8.2 × 6 = 49.2公里。
• 在Q处画一条北向线。将量角器放在Q上，将0°对准北方，顺时针测量到线段QH。如果读数在320°左右，那么H相对于Q的方位是320°（三位数）。你应该得到一个接近通过三角学计算得出的理论答案：距离 ≈ 49公里，方位 ≈ 322°。
Always check that your measured bearing is reasonable. A return bearing should be roughly the original bearing adjusted by the back bearing rule if the path was straight, but with two legs it will differ. Practising with such multi‑leg journeys will sharpen both your bearing measuring and scale drawing accuracy.
始终检查你测量出的方位是否合理。如果路径是直的，且根据反向方位规则进行调整，返回方位应该大致等于原始方位，但在有两段航程的情况下会有所不同。练习这种多段航程将同时提高你的方位测量和比例图准确性。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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