Bearings and Scale Drawings | 方位与比例图

📚 Bearings and Scale Drawings | 方位与比例图

In KS3 Mathematics, bearings and scale drawings are essential topics that combine geometry, measurement, and real-world applications. Understanding how to describe direction using three-figure bearings and how to represent distances accurately with scale drawings prepares students for more advanced work in navigation, engineering, and trigonometry.

在 KS3 数学中，方位和比例图是至关重要的主题，它们结合了几何、测量和现实世界应用。理解如何使用三位数方位角描述方向，以及如何通过比例图准确表示距离，为学生进一步学习导航、工程和三角学等高阶内容打下坚实基础。

1. What Are Bearings? | 什么是方位角？

Bearings are a way of describing the direction from one point to another, measured in degrees clockwise from North. They are used in navigation, map reading, and surveying to give precise directions without ambiguity.

方位角是一种描述从一点到另一点方向的方法，以度为单位，顺时针从正北方向测量。它们常用于导航、地图阅读和测量中，以便无歧义地给出精确方向。

Unlike compass points like North or South, bearings provide an exact numeric direction, which is essential when plotting routes or sharing location data.

与北、南等罗盘点不同，方位角提供精确的数值方向，这在规划路线或共享位置数据时必不可少。

Bearings are always measured in a clockwise direction from North. This convention ensures consistency and prevents mistakes when interpreting directions.

方位角总是从正北顺时针测量。这一惯例确保了一致性，并防止解释方向时出现错误。

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2. Three-Figure Bearings | 三位数方位角

A bearing is always written with three digits. For example, a direction of 60° is written as 060°, and a direction of 5° is written as 005°. This standard avoids confusion when reading and communicating directions.

方位角总是用三位数字书写。例如，60°的方向写为060°，5°的方向写为005°。这一标准在阅读和传达方向时避免了混淆。

The full circle of 360° means that bearings range from 000° (due North) through 090° (East), 180° (South), 270° (West), and back to 360°.

完整的360°圆周意味着方位角范围从000°（正北）经过090°（东），180°（南），270°（西），再回到360°。

Note that 360° is equivalent to 000° for a bearing exactly North. In practice, you will see 000° used to indicate North, not 360°.

注意，对于正北方向，360°等同于000°。在实际中，你会看到用000°表示正北，而不是360°。

Writing bearings with leading zeros may seem unnatural at first, but it quickly becomes second nature and is critical for clarity in technical work.

起初写带前导零的方位角可能不自然，但很快就会习以为常，并且在技术工作中对清晰度至关重要。

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

To measure a bearing, place the protractor so that its centre is on the point from which the bearing is taken. Align the 0° mark with North, then read the angle clockwise to the line representing the direction of the target point.

要测量方位角，将量角器中心放在起始点上，将0°刻度与正北对齐，然后顺时针读取指向目标点方向线的角度。

Always measure clockwise from the North line. If the angle goes past 180°, you can measure the smaller angle between the line and South, then add 180° to find the bearing.

始终顺时针从正北线测量。如果角度超过180°，你可以测量该线与南向之间的较小角度，然后加上180°得到方位角。

Use a sharp pencil to draw a clear North line at the starting point. Place the protractor accurately—any shift in the protractor will cause errors in the measured bearing.

使用尖铅笔在起始点清晰地画出北线。准确放置量角器——量角器的任何偏移都会导致测量的方位角出错。

Practice with diagrams where you are given the position of two points; draw the North line, the connecting line, and then use the protractor to determine the three-figure bearing.

通过给出两点位置的图表进行练习；画出北线、连接线，然后用量角器确定三位数方位角。

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4. Bearings Always from North | 方位角始终从正北起算

A bearing is always measured from North, regardless of the observer’s orientation. This means you must draw a North line at the point you are ‘taking the bearing from’ every single time.

方位角总是从正北起算，无论观察者的朝向如何。这意味着每次都必须在你作为“出发方位”的点上画出北线。

Even if a second point is directly South of the first, the bearing is still measured clockwise from North, giving a value of 180°. For a point to the south-west, the bearing would be between 180° and 270°, such as 225°.

即使第二点在第一点正南，方位角仍然从北顺时针测量，得到180°。对于西南方的一点，方位角将在180°到270°之间，例如225°。

This ‘always from North’ rule removes subjectivity. It is one of the first things to check when you are unsure about a bearing problem—draw the North line first.

这条“始终从北”的规则消除了主观性。当你对方位问题不确定时，首先要检查的就是——先画出北线。

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5. Back Bearings | 反向方位角

A back bearing is the bearing from the destination point back to the starting point. It is found by adding or subtracting 180° from the original bearing, keeping the result as a three-digit number between 000° and 360°.

反向方位角是从终点返回起点的方位角。通过原始方位角加上或减去180°得到，结果保持为000°到360°之间的三位数。

For example, if the bearing from A to B is 060°, the back bearing from B to A is 060° + 180° = 240°.

例如，若从A到B的方位角是060°，则从B到A的反向方位角是060° + 180° = 240°。

If the original bearing is 200°, adding 180° gives 380°, which is greater than 360°, so subtract 360° to obtain 020°. For bearings greater than 180°, it is often easier to subtract 180° directly: 200° – 180° = 020°.

如果原始方位角是200°，加180°得到380°，大于360°，所以减去360°得到020°。对于大于180°的方位角，直接减去180°通常更简单：200° – 180° = 020°。

Back bearings are extremely useful for checking your work or when you need to navigate a return journey.

反向方位角在检查作业或需要规划返程路线时非常有用。

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6. Introduction to Scale Drawings | 比例图简介

A scale drawing represents a real object or distance at a reduced or enlarged size, using a scale ratio. Maps are the most common example: 1 cm on a map might represent 1 km in reality.

比例图以缩小或放大的尺寸表示真实物体或距离，使用比例比。地图是最常见的例子：地图上1厘米可能代表实际1公里。

The scale is written as a ratio, such as 1 : 100 000, meaning 1 unit on the drawing equals 100 000 of the same units in reality. It is vital that the units match on both sides of the ratio.

比例写为比率，如1:100 000，表示图上1单位等于实际100 000同一单位。必须保证比率两边的单位一致。

Scale drawings are essential for architects, engineers, and when solving problems that combine bearings and distances, allowing you to visualise the situation on paper.

比例图对建筑师、工程师以及解决涉及方位和距离的问题至关重要，它能让你在纸上将情况可视化。

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7. Understanding Scale Factors | 理解比例因子

The scale factor is the number by which you multiply or divide to convert between measurements on the drawing and the real world. For a scale of 1 : 50, the real length is 50 times the drawing length.

比例因子是你乘或除以进行图纸与真实世界转换时所用的数字。对于1:50的比例，真实长度是图上长度的50倍。

To find the real distance from a scale drawing, measure the length on the drawing and multiply by the scale factor. To draw a real length to scale, divide the real length by the scale factor.

要从比例图求实际距离，测量图上的长度并乘以比例因子。要按比例绘制实际长度，将实际长度除以比例因子。

If a diagram says 1 cm represents 2 m, first convert 2 m into 200 cm, then the scale factor is 200. A 3 cm line on the diagram would represent 3 × 200 = 600 cm, or 6 m in reality.

如果图上标注1厘米代表2米，先将2米转换为200厘米，那么比例因子是200。图上3厘米的线代表实际3×200 = 600厘米，即6米。

Always work in the same unit before calculating. Confusing metres and centimetres is a common source of errors.

计算前始终使用相同单位。混淆米和厘米是常见的错误来源。

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8. Constructing a Scale Drawing | 构建比例图

Start by choosing an appropriate scale that will fit your drawing neatly onto the paper. Convert all real measurements using the scale, and employ a ruler and protractor to draw lines accurately.

首先选择适合纸张、能让图形整洁放置的比例尺。使用比例转换所有实际测量值，并借助尺子和量角器准确地画线。

When a question gives you a bearing and a distance, first draw a North line at the starting point, measure the bearing with a protractor, then scale the distance to find the length on paper.

当题目给出方位角和距离时，先在起点画出北线，用量角器测量方位角，然后按比例缩放距离得到图纸上的长度。

For instance, if the scale is 1 cm = 0.5 km and the distance is 2.8 km, the drawing length is 2.8 ÷ 0.5 = 5.6 cm. Use a sharp pencil and measure precisely; small errors can lead to an inaccurate final position.

例如，如果比例是1厘米=0.5公里，距离为2.8公里，则图纸长度为2.8÷0.5 = 5.6厘米。使用尖铅笔并精确测量；微小的误差会导致最终位置不准确。

Always label the scale on your drawing and include a North arrow to make your diagram self-explanatory.

始终在图上标注比例，并画上北箭头，使你的图示一目了然。

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9. Bearings and Scale Drawings Combined | 方位与比例图结合应用

Many practical problems require you to take a bearing and distance from one point to another, then use a scale drawing to locate an unknown point. For example, a ship sails on a bearing of 075° for 6 km; you can draw this leg using a suitable scale.

许多实际问题要求你从一个点出发，根据方位和距离使用比例图找到未知点。例如，一艘船沿075°方位航行6公里；你可以用合适的比例画出这段路径。

From that new point, you might take a second bearing and distance to reach a final destination. The scale drawing then shows the relative positions of all points clearly.

从那个新点出发，你可能再取第二个方位和距离到达最终目的地。然后比例图清晰地显示所有点的相对位置。

This method, used in navigation and surveying, lets you calculate straight-line distance between the start and finish, or the bearing from start to finish, simply by measuring on your accurate diagram.

这种方法用于导航和测量，你可以通过在你的精确图上测量，轻松计算出起点到终点的直线距离，或起点到终点的方位角。

Remember that each stage must be drawn carefully; an error in one leg affects all subsequent positions. Always measure both the angle and the distance with great care.

请记住，每一阶段都必须仔细绘制；一段中的错误会影响所有后续位置。务必非常小心地测量角度和距离。

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10. Common Errors and Tips | 常见错误与提示

A frequent mistake is writing a bearing with fewer than three digits, e.g., 60° instead of 060°. Always add the leading zero if the angle is less than 100°.

一个常见错误是方位角书写少于三位数字，如60°而不是060°。如果角度小于100°，务必添加前导零。

When measuring with a protractor, ensure it is perfectly aligned with the North line and you read the scale clockwise. Many protractors have two rows of numbers, so check that you are using the correct set.

用量角器测量时，确保其与北线完全对齐，并按顺时针读数。许多量角器有两圈数字，所以要检查你是否在使用正确的数列。

For scale drawings, using inconsistent units is a serious pitfall. Convert all distances to the same unit—preferably centimetres for drawings—before applying the scale factor.

对于比例图，使用不一致的单位是一个严重的陷阱。应用比例因子之前，将所有距离转换为同一单位——图纸上最好用厘米。

Another tip: draw a rough sketch before making the accurate scale drawing. This helps you visualise the problem and catch obvious mistakes in bearing or scale choice.

另一个提示：在绘制精确的比例图之前先画一个粗略草图。这有助于你想象问题，并发现方位或比例选择中的明显错误。

Finally, always label your final diagram with the scale used, a North arrow, and all given bearings. Clear labelling makes checking your work much easier.

最后，始终在最终图上标出所用的比例、北箭头和所有给定的方位角。清晰的标注将使检查工作容易得多。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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