📚 Bearings and Scale Drawings | 方位角与比例绘图
Bearings and scale drawings are used to describe direction and distance accurately on maps, construction plans and navigation charts. In Cambridge KS3 Mathematics, you need to measure, draw and calculate bearings, and you need to use scale factors to convert between drawing lengths and real lengths.
方位角和比例绘图用于在地图、建筑平面图和航海中精确描述方向与距离。在剑桥 KS3 数学中，你需要测量、绘制和计算方位角，并需要利用比例尺在图纸长度与实际长度之间进行换算。
1. What Are Bearings? | 什么是方位角？
A bearing is a way of describing direction. It is always measured clockwise from the north line, and it is written as a three-figure number, such as 045° or 120°.
方位角是描述方向的一种方法。它总是从正北线开始按顺时针方向测量，并写成三位数的形式，例如 045° 或 120°。
A bearing does not tell you how far to go; it only gives a direction. To describe a journey fully, you usually need both a bearing and a distance.
方位角只表示方向，并不表示距离。要完整描述一段行程，通常需要同时给出方位角和距离。
2. Three-Figure Bearing Rules | 三位数方位角规则
When you write or measure bearings, follow three rules: first, measure from the north line; second, measure clockwise; third, always write the angle using three digits.
在书写或测量方位角时，要遵循三条规则：第一，从正北线开始测量；第二，按顺时针方向测量；第三，始终用三位数字表示角度。
- Measure from North | 从正北方向测量
- Measure clockwise | 按顺时针方向测量
- Use three figures | 使用三位数表示
For example, a bearing of 60° is written as 060°, and a bearing of 8° is written as 008°.
例如，60° 的方位角应写为 060°，8° 的方位角应写为 008°。
3. Key Bearings: North, East, South and West | 基本方位角：北、东、南、西
The four cardinal directions have exact three-figure bearings. These are useful reference points when you estimate or check compass directions.
四个基本方向都有精确的三位数方位角。这些是你估算或检查罗盘方向时有用的参考点。
| Direction | 方向 | Bearing | 方位角 |
|---|---|
| North | 北 | 000° |
| East | 东 | 090° |
| South | 南 | 180° |
| West | 西 | 270° |
Halfway directions are also common: northeast is 045°, southeast is 135°, southwest is 225° and northwest is 315°.
中间方向也很常见：东北是 045°，东南是 135°，西南是 225°，西北是 315°。
4. Measuring a Bearing | 测量方位角
To measure a bearing from point A to point B, first draw a north line at A. Then draw a straight line from A to B. Place your protractor with its centre at A and 0° pointing north, and measure the angle clockwise from the north line to the line AB.
要测量从点 A 到点 B 的方位角，首先在点 A 画一条正北线。然后画一条从 A 到 B 的直线。将量角器的中心放在点 A，使 0° 指向正北，并从正北线沿顺时针方向测量到直线 AB 的角度。
Make sure you read the outer scale or inner scale on the protractor correctly. If the angle is less than 100°, add zeros at the front so it has three figures, such as 075°.
确保正确读取量角器的外圈或内圈刻度。如果角度小于 100°，需要在前面补零，使其成为三位数，例如 075°。
5. Drawing a Bearing | 绘制方位角
To draw a bearing such as 130° from a point P, start by drawing a vertical north line at P. Place your protractor so that 0° is on the north line. Mark a point at 130° measured clockwise from north, then draw a ray from P through that mark.
要从点 P 绘制 130° 的方位角，首先在点 P 画一条竖直的正北线。将量角器放置好，使 0° 位于正北线上。从正北方向顺时针找到 130° 并标记一个点，然后从 P 画一条经过该标记的射线。
Label the ray with its bearing, and if the question gives a distance, use a scale drawing to place the correct length along the ray.
在这条射线上标出方位角；如果题目给出距离，则利用比例绘图在射线上量出相应的长度。
6. Bearings and Return Journeys | 方位角与返程
If you travel from A to B on a bearing, the return bearing from B back to A is not simply the opposite compass point. It is found by adding 180° or subtracting 180°.
如果你从 A 到 B 沿某个方位角行进，那么从 B 返回 A 的返程方位角并不是简单地把角度反过来，而是加上 180° 或减去 180°。
Return bearing = Bearing + 180° if Bearing < 180°
Return bearing = Bearing − 180° if Bearing ≥ 180°
For example, if the outward bearing is 060°, the return bearing is 240°. If the outward bearing is 210°, the return bearing is 030°.
例如，如果去程方位角是 060°，则返程方位角是 240°。如果去程方位角是 210°，则返程方位角是 030°。
7. Scale Drawings and Maps | 比例绘图与地图
A scale drawing is an accurate picture where every length is reduced or enlarged by the same ratio. A map is a scale drawing of a real place, and the scale tells you how drawing lengths relate to actual lengths.
比例绘图是一幅按相同比例缩小或放大的精确图形。地图是真实地点的比例绘图，比例尺告诉你图纸长度与实际长度之间的关系。
Common scales include 1 cm to 5 m, 1 cm to 2 km, or written ratios such as 1 : 50 000. The units must be converted carefully before calculating.
常见的比例尺包括 1 cm 代表 5 m、1 cm 代表 2 km，或如 1 : 50 000 的比。计算前必须仔细换算单位。
8. Using Scales: Calculation Method | 比例尺的使用：计算方法
To convert a drawing length to the real length, multiply by the scale factor. To convert a real length to a drawing length, divide by the scale factor.
要将图纸长度转换为实际长度，请乘以比例因子；要将实际长度转换为图纸长度，请除以比例因子。
scale factor is 500. A drawing length of 3.5 cm represents 3.5 × 500 = 1750 cm, or 17.5 m.
对于 1 cm 代表 5 m 的比例尺，纸上 1 cm 代表实际 500 cm，因此比例因子是 500。图纸长度为 3.5 cm 时，实际长度为 3.5 × 500 = 1750 cm，即 17.5 m。
Actual length = Drawing length × Scale factor
Drawing length = Actual length ÷ Scale factor
9. Combining Bearings and Scale Drawings | 结合方位角与比例绘图
In navigation problems, you often draw a route using a bearing and a scale. For example, a ship sails from port P on a bearing of 060° for 8 km. Using a scale of 1 cm to 2 km, the ship’s path is 4 cm long on paper.
在航海问题中，你常常需要根据方位角和比例尺画出路线。例如，一艘船从港口 P 出发，沿 060° 的方位角航行 8 km。若比例尺为 1 cm 代表 2 km，则船在图纸上的路径长度为 4 cm。
Draw the north line at P, measure 060° clockwise, then mark a point Q exactly 4 cm from P along the ray. You can then measure the return bearing at Q by drawing a north line there.
在 P 处画正北线，顺时针量出 060°，然后在射线上距离 P 点 4 cm 处标记点 Q。之后你可以在 Q 处画正北线并测量返程方位角。
10. Common Mistakes to Avoid | 常见错误
One common mistake is measuring anticlockwise from north instead of clockwise. Another is writing 60° instead of 060°. Always check that your bearing has three digits and is measured the correct way around.
一个常见错误是从正北方向逆时针测量，而不是顺时针。另一个错误是把 060° 写成 60°。务必检查方位角是否为三位数，并检查测量方向是否正确。
Scale problems often go wrong when units are mixed. Convert all lengths to the same unit before multiplying or dividing, and state the final unit clearly.
比例尺问题常常因为单位混用而出错。在乘除之前，要把所有长度换算成相同单位，并清楚写出最终单位。
11. Exam-Style Worked Example | 考试风格例题
Question: A walker leaves point A on a bearing of 120° and walks 6 km to point B. Find the bearing from B back to A.
问题：一名步行者从点 A 出发，沿 120° 的方位角行走 6 km 到达点 B。求从 B 返回 A 的方位角。
Solution: The outward bearing is 120°. Since 120° < 180°, add 180° to find the return bearing.
解：去程方位角为 120°。因为 120° < 180°，所以将 120° 加上 180° 得到返程方位角。
120° + 180° = 300°
The return bearing is 300°, which is between west and northwest.
返程方位角是 300°，方向位于西和西北之间。
12. Practice Ideas | 练习建议
To improve, practise measuring bearings from diagrams where a north line is given. Then practise drawing routes with bearings and scale distances, such as 1 cm to 1 m or 1 cm to 10 km.
为了提高能力，可以练习在给出正北线的图中测量方位角。然后练习根据方位角和比例距离绘制路线，例如 1 cm 代表 1 m 或 1 cm 代表 10 km。
Check your answers by reversing the bearing: if the bearing from A to B is 045°, the return bearing should be 225°. If it is not, look again at your protractor position or your addition.
通过反转方位角来检查答案：如果从 A 到 B 的方位角是 045°，那么返程方位角应为 225°。如果不是，请重新检查量角器的位置或加法。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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