📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings are used in navigation and surveying to describe direction in a precise, unambiguous way. A bearing is always measured clockwise from the north line and is written as a three-digit angle, such as 045° or 210°. Together with scale drawings, bearings allow us to solve real-world problems involving distance and direction, from piloting ships to reading maps and planning flight paths.
方位角用于导航和测量中，以精确且无歧义的方式描述方向。方位角总是从正北方向顺时针测量，并写成三位数字的角度，例如 045° 或 210°。结合比例绘图，方位角可以帮助我们解决涉及距离和方向的现实问题，例如船舶航行、阅读地图和规划飞行路线。
1. What Are Bearings? | 什么是方位角？
A bearing is a way of describing the direction of one point from another. It is the angle measured clockwise from the north line to the line joining the two points. Bearings are commonly used in navigation because they remove any doubt about left or right, and they always use north as the fixed reference direction.
方位角是描述一个点相对于另一个点方向的方法。它是从正北方向顺时针测量到连接两点的直线的角度。方位角通常用于导航，因为它消除了左右方向上的歧义，并且始终以正北作为固定参考方向。
For example, if a ship is due east of a lighthouse, the bearing of the ship from the lighthouse is 090°. If the ship is due south-west, the bearing is 225°. Bearings are always given as three figures, so 5° is written as 005°, and 90° is written as 090°.
例如，如果一艘船在灯塔的正东方，那么从灯塔看船的方向角为 090°。如果船在西南方向，则方位角为 225°。方位角总是写成三位数字，因此 5° 写作 005°，90° 写作 090°。
2. Three-Figure Bearings and Rules | 三位数方位角及其规则
When working with bearings, there are four key rules to remember. First, always measure from the north line. Second, always measure clockwise. Third, always write the answer as a three-digit number. Fourth, bearings never include a degree sign if written as three digits in a worded answer, but in calculations we still use the degree symbol.
在处理方位角时，有四条关键规则需要记住：第一，总是从正北方向测量；第二，总是顺时针测量；第三，总是将答案写成三位数字；第四，在文字表达中方位角通常写为三位数而不带度符号，但在计算中我们仍然使用度符号。
| North | 000° | 正北 |
| East | 090° | 正东 |
| South | 180° | 正南 |
| West | 270° | 正西 |
These four cardinal directions form the basis of the bearing system. Between them, angles are measured clockwise in degrees, so north-east is 045°, south-east is 135°, south-west is 225°, and north-west is 315°.
这四个基本方向构成了方位角系统的基础。它们之间的角度按顺时针方向以度为单位测量，因此东北方向为 045°，东南方向为 135°，西南方向为 225°，西北方向为 315°。
3. Measuring a Bearing with a Protractor | 用量角器测量方位角
To measure the bearing of point B from point A, first draw a clear north line at point A, pointing straight up the page. Place the centre of the protractor at point A and align its 0° mark with the north line. Then read the angle clockwise from north to the line AB.
要测量点 B 从点 A 出发的方位角，首先在点 A 处画一条清晰的正北线，指向页面正上方。将量角器的中心放在点 A，并将量角器的 0° 刻度线与北线对齐。然后从正北方向顺时针读出到线段 AB 的角度。
If the angle is greater than 180°, you can either measure the smaller anticlockwise angle and subtract it from 360°, or use a protractor with a full 360° scale. Always double-check that you have measured clockwise and that your final bearing has three digits.
如果角度大于 180°，你可以测量较小的逆时针角，然后用 360° 减去它，或者使用带有完整 360° 刻度的量角器。始终仔细检查你是否沿顺时针方向测量，以及最终方位角是否为三位数字。
4. Drawing an Accurate Bearing | 精确绘制方位角
To draw a bearing from a point, start by marking the point clearly and drawing a vertical north line upwards. Place the protractor centred on the point with 0° on the north line. Mark the required angle at the edge of the protractor, then remove the protractor and draw a straight line through the point and the mark. Label the bearing if needed.
要从一个点绘制方位角，首先清楚标出该点，并向上画一条垂直的正北线。将量角器中心放在该点上，使 0° 对准北线。在量角器边缘标出所需角度，然后移开量角器，通过该点和标记画一条直线。如有需要，标注方位角。
For angles above 180°, it is often easier to calculate the smaller clockwise angle from south, or the acute angle between the line and a vertical reference. For example, a bearing of 230° is 50° clockwise from south, so you can draw the south line and measure 50° from it.
对于大于 180° 的角，通常更容易计算从正南方向开始的较小顺时针角，或者直线与垂直参考线之间的锐角。例如，230° 的方位角是从正南方向顺时针 50°，因此你可以画出正南线，并从它开始测量 50°。
5. Back Bearings | 反向方位角
A back bearing is the bearing of point A from point B when you already know the bearing of point B from point A. If the original bearing is less than 180°, add 180° to get the back bearing. If the original bearing is greater than 180°, subtract 180° instead.
反向方位角是指当你已知点 B 从点 A 出发的方位角时，点 A 从点 B 出发的方位角。如果原始方位角小于 180°，则加 180° 得到反向方位角。如果原始方位角大于 180°，则减 180°。
Back bearing = bearing ± 180°
80° = 130°. Always ensure the result is still a three-digit bearing between 000° and 360°.
例如，如果 B 从 A 出发的方位角为 075°，那么 A 从 B 出发的方位角为 075° + 180° = 255°。如果 C 从 D 出发的方位角为 310°，那么 D 从 C 出发的方位角为 310° − 180° = 130°。始终确保结果仍是 000° 到 360° 之间的三位方位角。
6. Scale Drawings and Map Ratios | 比例绘图与地图比例尺
A scale drawing represents a real object or area with all dimensions reduced or enlarged by the same ratio. The scale can be written as a ratio such as 1 : 50 000, which means 1 cm on the drawing represents 50 000 cm, or 0.5 km, in real life. Scale drawings are essential for using bearings to find actual distances.
比例绘图是用相同的比例将实际物体或区域的所有尺寸缩小或放大的表示方法。比例尺可以写成比值，例如 1 : 50 000，表示图上 1 cm 代表实际 50 000 cm，即 0.5 km。比例绘图对于利用方位角求实际距离至关重要。
Scale = drawing distance ÷ actual distance
000 = 100 000 cm, which is 1000 m or 1 km. Always show the conversion clearly in working.
要使用比例尺，首先将所有单位转换为相同类型。例如，在比例为 1 : 25 000 的地图上，4 cm 的距离代表 4 × 25 000 = 100 000 cm，即 1000 m 或 1 km。计算过程中务必清楚写出单位换算。
7. Constructing a Scale Diagram | 构建比例图
When a question asks you to find the location of a point using bearings and distances, you need to construct an accurate scale diagram. Start by choosing a suitable scale that fits the page and gives clear measurements. Then plot the starting points, draw north lines, and use a protractor and ruler to draw the given bearings and distances.
当题目要求使用方位角和距离确定某点的位置时，你需要构建一个准确的比例图。首先选择一个适合页面并便于清晰测量的比例尺。然后标出起点，画出正北线，并使用量角器和直尺画出给定的方位角和距离。
- Choose a scale such as 1 cm = 10 km, and write it on the diagram.
- Draw all north lines parallel to each other; they must never be slanted differently.
- Use a sharp pencil and a fine ruler so that distances are accurate.
- Convert all real distances to drawing lengths before drawing.
- 选择比例尺，例如 1 cm = 10 km，并标注在图上。
- 所有正北线必须互相平行；绝不能画成不同的倾斜方向。
- 使用削尖的铅笔和精细的直尺，以确保距离准确。
- 在绘图前把所有实际距离转换为图上长度。
8. Finding Unknown Distances | 求未知距离
sured distance between two points is 3.6 cm, then the actual distance is 3.6 × 15 = 54 km.
比例图完成后，你可以通过测量图上长度并乘以比例因子来求未知的实际距离。例如，如果比例尺为 1 cm = 15 km，两点之间的图上距离为 3.6 cm，那么实际距离为 3.6 × 15 = 54 km。
When the scale is given as a ratio, convert it into a simple statement first. A scale of 1 : 200 000 means 1 cm on the drawing is 200 000 cm in real life, which is 2 km. Then multiply the measured length in cm by 2 to get the actual distance in km.
当比例尺以比值给出时，首先将其转换为简单陈述。比例 1 : 200 000 表示图上 1 cm 等于实际 200 000 cm，即 2 km。然后将测量的厘米长度乘以 2，即可得到以 km 为单位的实际距离。
9. Worked Example: Boat Journey | 例题：小船航行
Problem: A boat leaves a harbour H and sails 40 km on a bearing of 140° to reach a buoy B. From B it sails on a bearing of 230° for 25 km to reach a lighthouse L. Find the bearing of L from H and the distance from H to L using a scale drawing.
题目：一艘船从港口 H 出发，沿 140° 方位角航行 40 km 到达浮标 B。然后从 B 沿 230° 方位角航行 25 km 到达灯塔 L。利用比例图求 L 从 H 出发的方位角以及 H 到 L 的距离。
. Draw point H with a north line. Measure 140° clockwise and draw a line 4 cm long to locate B. At B, draw a north line parallel to the first north line, measure 230° clockwise, and draw a line 2.5 cm long to locate L. Finally, join H to L.
解：选择比例尺 1 cm = 10 km。画出点 H 和正北线。顺时针量出 140°，并画一条 4 cm 长的线段定位 B。在 B 处画一条与第一条正北线平行的正北线，顺时针量出 230°，画一条 2.5 cm 长的线段定位 L。最后连接 H 和 L。
angle from north at H to the line HL; if it is 205°, then the bearing of L from H is 205°. Answers may vary slightly depending on drawing accuracy.
测量图上 HL 的长度。如果 HL 约为 4.8 cm，那么实际距离为 4.8 × 10 = 48 km。测量从 H 点正北方向到线段 HL 的角度；如果是 205°，那么 L 从 H 出发的方位角为 205°。根据绘图精度，答案可能略有差异。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake is measuring anticlockwise instead of clockwise, which gives a completely wrong bearing. Another error is forgetting to write the bearing as three digits, such as writing 90 instead of 090. Students also sometimes place the protractor incorrectly or use north lines that are not parallel at different points.
一个常见错误是逆时针测量而不是顺时针测量，这会导致完全错误的方位角。另一个错误是忘记将方位角写成三位数字，例如写 90 而不是 090。学生有时还会错误地放置量角器，或在不同点使用不平行的正北线。
In the exam, always show your scale clearly and include it on your diagram. Use a sharp pencil, draw north lines parallel, and label all points with capital letters. Check that each bearing is between 000° and 360°, and that distances have been converted correctly. If you use a scale drawing, state that drawings are approximate and measurements may vary by a small amount.
在考试中，务必清楚标出比例尺，并在图上注明。使用削尖的铅笔，画平行的正北线，并用大写字母标注所有点。检查每个方位角是否在 000° 到 360° 之间，距离是否换算正确。如果使用比例图，应说明绘图是近似的，测量结果可能有少量差异。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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