Bearings and Scale Drawings | 方位角与比例图

📚 Bearings and Scale Drawings | 方位角与比例图

In this KS3 Cambridge Mathematics revision guide, we will explore the concept of bearings and how they are used in scale drawings to find distances and directions. Bearings provide a precise way to describe movement between locations, forming a key part of geometry and practical map skills.

在这份KS3剑桥数学复习指南中，我们将探究方位角的概念，以及如何在比例图中使用方位角来求距离和方向。方位角提供了一种精确描述位置间移动的方式，是几何和实际地图技能的重要组成部分。

1. What Is a Bearing? | 什么是方位角？

A bearing is a measurement of direction, always taken clockwise from the north line. It tells you the angle you need to turn from north to face a target point. Bearings are used by pilots, sailors, and in many mapping applications.

方位角是对方向的度量，始终从正北线开始顺时针测量。它告诉你从正北方向需要转动多少角度才能朝向目标点。飞行员、海员以及许多地图应用中都会使用方位角。

All bearings are given as three-figure numbers, for example 025°, 134°, or 300°. This three-digit rule makes bearings unambiguous and easy to communicate.

所有方位角都表示为三位数，例如 025°、134° 或 300°。这个三位数规则使得方位角没有歧义，也便于交流。

The three essential rules for bearings are: (i) always measure from the north line, (ii) always measure clockwise, and (iii) always write the angle with three digits.

方位角的三个基本规则是：（i）始终从正北线开始测量，（ii）始终顺时针测量，（iii）始终用三位数字写出角度。

2. Measuring Bearings with a Protractor | 用量角器测量方位角

To measure the bearing of a point B from a point A, first draw a straight north line at A. Place the centre of the protractor on A, aligning the 0° mark with the north line. Then read the angle clockwise from north to the line AB. This gives the bearing.

要测量点B相对于点A的方位角，首先在A点画一条正北方向的直线。将量角器的中心放在A点上，让0°刻度与正北线对齐。然后顺时针从北方向读取到直线AB的角度，就得到了方位角。

If the angle is, for example, 75° clockwise from north, the bearing is 075°. If AB points exactly east, the protractor shows 90°, so the bearing is 090°.

如果这个角度是，例如从北方向顺时针75°，方位角就是075°。如果AB指向正东，量角器显示90°，那么方位角就是090°。

When the point B is located to the west, you may need to measure a reflex angle on the protractor. For example, if the direction is northwest, the bearing might be 315°. It is always helpful to sketch a diagram.

当点B位于西侧时，你可能需要用量角器测量一个优角。例如，如果方向是西北，方位角可能是315°。画一个草图总是很有帮助。

3. Three-Figure Bearings | 三位数方位角

Writing bearings with three digits is a strict convention. An angle of 8° becomes 008°, 67° becomes 067°, and 180° remains 180°. This format avoids confusion between, for instance, a bearing of 5° and 50°.

用三位数写方位角是一项严格的约定。8° 要写成 008°，67° 写成 067°，180° 仍写成 180°。这种格式避免了诸如 5° 和 50° 之间的混淆。

The table below shows some common directions and their bearings:

下表列出了一些常见方向及其方位角：

Direction (English) / 方向（中文）	Bearing / 方位角
North / 北	000°
East / 东	090°
South / 南	180°
West / 西	270°
Northeast / 东北	045°
Southeast / 东南	135°
Southwest / 西南	225°
Northwest / 西北	315°

Memorising these cardinal and intercardinal bearings can help you quickly estimate bearings when working with maps or scale drawings.

记住这些基本方位和中间方位，可以帮助你在使用地图或比例图时快速估计方位角。

4. Reverse Bearings | 反向方位角

The bearing of B from A is generally different from the bearing of A from B. These two bearings differ by 180°. If you know one bearing, you can find the reverse bearing by adding or subtracting 180°.

从A到B的方位角通常不同于从B到A的方位角。这两个方位角相差180°。如果你知道其中一个，就可以通过加或减180°来求出反向方位角。

For example, if the bearing from lighthouse L to ship S is 052°, then the bearing from the ship back to the lighthouse is 052° + 180° = 232°. If the sum exceeds 360°, subtract 360° instead.

例如，如果从灯塔L到船只S的方位角是052°，那么从船只返回灯塔的方位角就是 052° + 180° = 232°。如果和超过了360°，就减去360°。

Another example: the bearing from village X to village Y is 290°. The reverse bearing from Y to X is 290° – 180° = 110°. Always check that your answer is a three-figure bearing.

另一个例子：从村庄X到村庄Y的方位角是290°。从Y到X的反向方位角就是 290° – 180° = 110°。始终检查你的答案是否为三位数的方位角。

5. Drawing Bearings | 绘制方位角

To draw a bearing from a point, start by drawing a vertical north line through the point. Then place your protractor with the centre on the point and 0° pointing north. Measure the required angle clockwise, mark the direction, and draw a ray.

要绘制从某点出发的方位角，首先过该点画一条垂直的正北线。然后将量角器的中心放在该点上，0° 指向北方。顺时针量取所需角度，标记方向，然后画出一条射线。

If you are drawing an bearing of 120°, you would measure 120° clockwise from north, which points into the southeast quadrant. Label the ray clearly with the bearing in three digits.

如果你要绘制120°的方位角，你将从北方向顺时针量出120°，指向东南象限。清晰地用三位数标记该射线。

When drawing bearings from multiple points, keep all north lines parallel to each other. This is essential because bearings are always measured from local north lines.

当从多个点绘制方位角时，要让所有正北线互相平行。这一点至关重要，因为方位角始终是从当地的正北线测量的。

6. Scale Drawings and Bearings | 比例图与方位角

A scale drawing is a diagram in which real distances are reduced or enlarged by a fixed ratio, such as 1 cm representing 1 km. We often combine scale drawings with bearings to find unknown distances and directions accurately.

比例图是一种按照固定比例（如1厘米代表1千米）缩小或放大实际距离的图示。我们经常将比例图与方位角结合使用，以精确求出未知的距离和方向。

To construct a scale drawing from bearing information, first choose a suitable scale. For example, if the actual distance is 50 km, and the scale is 1 cm to 10 km, you draw a line 5 cm long.

要根据方位角信息构建比例图，首先要选择合适的比例尺。例如，如果实际距离是50千米，比例尺是1厘米代表10千米，你就画一条5厘米长的线。

The steps are: (i) mark your starting point, (ii) draw the north line, (iii) use a protractor to draw the bearing, (iv) measure the distance along that line using the scale, and (v) mark the end point.

步骤如下：（i）标记起点，（ii）画正北线，（iii）用量角器画出方位角方向，（iv）根据比例尺沿该线量取距离，（v）标记终点。

7. Finding Distances Using Scale and Bearings | 利用比例尺和方位角求距离

Once a scale drawing is complete, you can measure the distance between two points on the diagram and convert it back to real-world units using the scale factor.

一旦完成比例图，你就可以测量图上两点之间的距离，然后用比例因子将其转换回现实世界的单位。

For instance, if the scale is 1 : 50,000 and the measured length on paper is 4.2 cm, the actual distance is 4.2 × 50,000 = 210,000 cm, which is 2.1 km.

例如，如果比例尺是 1 : 50,000，纸上量得的长度是 4.2 厘米，那么实际距离就是 4.2 × 50,000 = 210,000 厘米，即 2.1 千米。

Always check that you are using the same units for measurement and conversion. If the scale is given as ‘1 cm represents 5 m’, convert the measured centimetres to metres by multiplying by 5.

务必确保测量和换算时使用相同的单位。如果比例尺给定为“1厘米代表5米”，就把量得的厘米数乘以5换算成米。

You can also find the bearing of one point from another by measuring the angle at the appropriate north line, just as you did when constructing the diagram.

你还可以通过在相应的正北线上测量角度，来求出一个点相对于另一个点的方位角，就像构建图表时做的那样。

8. Worked Example | 例题解析

Problem: A ship sails from port P on a bearing of 135° for 60 km to point Q. It then sails on a bearing of 060° for 80 km to reach point R. Using a scale of 1 cm = 10 km, find the bearing and distance from R back to P.

问题：一艘船从港口P出发，沿135°方位角航行60千米到达点Q。然后沿060°方位角航行80千米到达点R。使用比例尺1厘米=10千米，求从R返回P的方位角和距离。

Solution: Draw a north line at P. Draw the first leg: bearing 135°, length 60 km → 6 cm on the diagram. Mark Q. At Q, draw a north line parallel to the first. From Q, draw the second leg: bearing 060°, length 80 km → 8 cm. Mark R.

解答：在P点画正北线。画出第一段航线：方位角135°，长度60千米 → 图上6厘米。标记点Q。在Q点画一条与第一条平行的正北线。从Q点画出第二段航线：方位角060°，长度80千米 → 8厘米。标记点R。

Now, to find the return bearing and distance from R to P, simply join R and P. Measure the length RP on the diagram. Suppose RP = 7.4 cm. The actual distance is 7.4 × 10 = 74 km.

现在，要求从R到P的返程方位角和距离，只需连接R和P。在线段RP上测量长度。假设RP = 7.4厘米，实际距离就是 7.4 × 10 = 74 千米。

To find the bearing of P from R, draw a north line at R. Measure the angle clockwise from this north line to the line RP. Suppose the measurement is 300°. Then the bearing from R to P is 300°.

要求从R到P的方位角，在R点画一条正北线。沿顺时针方向测量从这条正北线到直线RP的角度。假设测量结果是300°，那么从R到P的方位角就是300°。

You can check your result by constructing the reverse journey or using accurate drawing tools.

你可以通过构建反向航线或使用精确的绘图工具来验证你的结果。

9. Common Mistakes to Avoid | 需避免的常见错误

Mistake 1: Measuring the angle anticlockwise instead of clockwise. Always remember that bearings rotate clockwise from north.

错误1：逆时针而不是顺时针测量角度。请始终记住，方位角是从北方向顺时针旋转的。

Mistake 2: Forgetting to write the bearing with three digits. Writing 45° instead of 045° may lose marks in examinations.

错误2：忘记用三位数写方位角。把045°写成45°可能会在考试中丢分。

Mistake 3: Using a north line that is not parallel to other north lines on the diagram. Bearings measured from non-parallel north lines will be incorrect.

错误3：使用的正北线与图上其他正北线不平行。从不平行的正北线测量的方位角将是不正确的。

Mistake 4: Confusing the bearing from A to B with the bearing from B to A. Always use the reverse bearing rule carefully.

错误4：混淆从A到B的方位角和从B到A的方位角。务必谨慎使用反向方位角规则。

Mistake 5: Applying the wrong scale factor or forgetting to convert units. A measurement in centimetres must be converted correctly to kilometres or metres.

错误5：应用了错误的比例因子或忘记换算单位。以厘米为单位的测量必须正确转换为千米或米。

10. Applications of Bearings and Scale Drawings | 方位角与比例图的应用

Bearings and scale drawings are not just textbook exercises; they are used in aviation, marine navigation, surveying, and orienteering. Pilots use bearings to navigate between waypoints, while architects use scale drawings to plan buildings.

方位角和比例图不仅仅是课本上的练习；它们被用于航空、海洋导航、测量和定向越野。飞行员使用方位角在航点间导航，而建筑师则使用比例图来规划建筑。

In real-life map reading, the north line is usually the grid north or true north, and distances are often measured along curves using a piece of string. The same principles of bearings and scale apply.

在实际的地图阅读中，正北线通常是网格北或真北，而距离往往用细绳沿曲线测量。方位角和比例尺的原理同样适用。

Understanding these skills also builds a strong foundation for trigonometry, where you will calculate bearings and distances without drawing diagrams to scale.

理解这些技能也为三角学打下了坚实的基础，在三角学中你将无需按比例绘图即可计算方位角和距离。

11. Checklist for Solving Bearing Problems | 解决方位角问题的核对清单

Use the following checklist when tackling any bearing and scale drawing question:

在处理任何方位角和比例图问题时，请使用以下核对清单：

1. Read the problem carefully and identify the starting point and all legs of the journey.

1. 仔细阅读题目，确定起点和航程的所有航段。

2. Decide on an appropriate scale and note it down clearly (e.g. 1 cm = 20 km).

2. 选择一个合适的比例尺并清晰地记录下来（例如 1厘米 = 20千米）。

3. Draw a clear north line at each point where a bearing is taken, ensuring all north lines are parallel.

3. 在需要测量方位角的每个点上画出清晰的正北线，确保所有正北线互相平行。

4. Use a protractor to accurately draw each bearing.

4. 使用量角器准确地画出每个方位角。

5. Measure the scaled distance accurately with a ruler.

5. 用直尺精确量取比例距离。

6. Label all points and bearings clearly.

6. 清晰地标记所有点和方位角。

7. When finding a return bearing, add or subtract 180°, and adjust if necessary to keep the bearing between 000° and 359°.

7. 求反向方位角时，加或减180°，并在必要时进行调整，使方位角保持在000°到359°之间。

8. Convert the measured diagram distance back to actual units using the scale.

8. 利用比例尺将图上量得的距离转换回实际单位。

9. Finally, check your answer makes sense – for example, the return bearing should be roughly opposite the original bearing.

9. 最后，检查你的答案是否合理——例如，反向方位角应该大致与原方位角方向相反。

12. Summary | 总结

Bearings are a key tool for describing direction, always measured clockwise from north and written as three-figure angles. Scale drawings allow us to represent real distances accurately on paper. By combining bearings with scale, we can solve practical problems involving navigation, maps, and construction plans.

方位角是描述方向的关键工具，始终从正北方向顺时针测量，并写成三位数的角度。比例图使我们能在纸上精确地表示实际距离。通过将方位角与比例尺结合，我们可以解决涉及导航、地图和施工计划的实际问题。

Remember the three rules: measure from north, go clockwise, and use three digits. Practise drawing and measuring bearings with a protractor until it becomes second nature.

记住三条规则：从北开始量，顺时针方向，使用三位数。练习用量角器画图和测量方位角，直到变得像第二天性一样自然。

These skills will not only help you in your KS3 Cambridge Mathematics studies but also in real-life situations where direction and distance matter.

这些技能不仅对你的KS3剑桥数学学习有帮助，而且在方向和距离至关重要的现实生活中也非常有用。

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Bearings and Scale Drawings | 方位角与比例图