Bearings and Scale Drawings | 方位角与比例尺

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

Bearings and scale drawings are practical tools used in navigation, map reading and construction. In the Cambridge KS3 mathematics course, you need to measure and draw three-figure bearings, use back bearings, and solve problems involving map scales and scale diagrams.

方位角和比例图是用于导航、地图阅读和制图的实用工具。在剑桥 KS3 数学课程中，你需要掌握测量和绘制三位数方位角、使用反向方位角，以及解决涉及地图比例尺和比例图的问题。

1. What Is a Bearing? | 什么是方位角？

A bearing is an angle measured clockwise from the north direction. It tells you the exact direction from one point to another. Bearings are used when the direction must be stated clearly and without ambiguity.

方位角是从正北方向顺时针测量的角度。它表示从一个点到另一个点的精确方向。当方向必须被清晰且无歧义地表达时，就会使用方位角。

Bearings are always measured from north, not from east, south or west. This rule is agreed so that everyone reads and writes a bearing in the same way.

方位角始终从北方开始测量，而不是从东、南或西方。这一规则是统一约定的，以便每个人都能用相同的方式读取和书写方位角。

For example, if a point is exactly east of you, its bearing is 090°. If a point is exactly southwest, its bearing is 225°. The north direction itself is written as 000°.

例如，如果某点在你的正东方，它的方位角是 090°。如果某点在你的西南方向，它的方位角是 225°。正北方向本身写作 000°。

2. Measuring Bearings from North | 从正北方向量测方位角

To measure a bearing, first draw a clear north line at the starting point. Then place a protractor so that 0° is on the north line and the centre of the protractor is on the point.

要测量方位角，首先在起点画一条清晰的正北线。然后将量角器的 0° 刻度放在北线上，并使量角器的中心对准该点。

You must measure clockwise from the north line. This means if the direction is to the right of north, the angle increases from 0° through 90° to 180° and eventually to 360°.

你必须从北线开始顺时针测量。这意味着如果方向在北线的右侧，角度会从 0° 增加到 90°、180°，直到 360°。

The main compass directions can be expressed as bearings:

主要罗盘方向可以表示为方位角：

North = 000°
East = 090°
South = 180°
West = 270°

北 = 000°，东 = 090°，南 = 180°，西 = 270°。

Using a protractor accurately is essential. Make sure the protractor does not slip while you read the angle.

准确使用量角器非常重要。读取角度时，确保量角器不会滑动。

3. Three-Figure Bearings | 三位数方位角

Bearings are always written with three digits. If the angle is less than 100°, you must include leading zeros. This avoids confusion between bearings such as 5°, 50° and 500°.

方位角始终写成三位数。如果角度小于 100°，必须加上前导零。这样可以避免混淆像 5°、50° 和 500° 这样的角度。

For example, a bearing of 7° is written as 007°. A bearing of 45° is written as 045°. A bearing of 135° already has three digits, so it stays as 135°.

例如，7° 的方位角写作 007°。45° 的方位角写作 045°。135° 的方位角已经是三位数，所以保持为 135°。

Writing bearings in three-figure form is important in navigation and examinations. Always check that your answer has exactly three digits.

在导航和考试中，使用三位数形式书写方位角非常重要。请始终检查你的答案是否恰好是三位数字。

4. Drawing Bearings and Distances | 绘制方位角与距离

To draw a bearing from a point, start by drawing a vertical north line at that point. Use a protractor to measure the required angle clockwise from the north line, then make a small mark.

要从某点绘制方位角，先在该点画一条竖直的北线。使用量角器从北线顺时针量出所需角度，然后做一个小标记。

Draw a straight line through the mark and the starting point. This line shows the required direction. If a distance is given, measure it along this line using a suitable scale.

画一条穿过标记和起点的直线。这条线表示所需方向。如果给出了距离，则沿此线用合适的比例尺量出距离。

Always place the centre of the protractor exactly on the starting point and keep the 0° mark exactly on the north line. A sharp pencil produces more accurate diagrams.

始终将量角器的中心精确放在起点上，并确保 0° 刻度精确对齐北线。使用削尖的铅笔可以画出更准确的图形。

5. Back Bearings | 反向方位角

A back bearing is the bearing from the second point back to the first point. If you travel from A to B on a certain bearing, the return bearing from B to A is called the back bearing.

反向方位角是从第二个点回到第一个点的方位角。如果你从 A 点按某一方位角走到 B 点，那么从 B 点返回 A 点的方位角就叫作反向方位角。

To find a back bearing, add 180° if the original bearing is less than 180°. If the original bearing is 180° or more, subtract 180° from it.

求反向方位角的方法是：如果原方位角小于 180°，则加上 180°；如果原方位角大于或等于 180°，则减去 180°。

Back bearing = original bearing + 180° (when original bearing < 180°)

反向方位角 = 原方位角 + 180°（当原方位角小于 180° 时）

Back bearing = original bearing − 180° (when original bearing ≥ 180°)

反向方位角 = 原方位角 − 180°（当原方位角大于或等于 180° 时）

For example, a bearing of 060° has a back bearing of 240°. A bearing of 240° has a back bearing of 060°. The two bearings always differ by 180°.

例如，060° 的方位角，其反向方位角为 240°。240° 的方位角，其反向方位角为 060°。两个方位角总是相差 180°。

6. Scale Drawings and Map Scales | 比例图与地图比例尺

A scale drawing shows a real object or distance reduced or enlarged in a fixed ratio. A map scale tells you how a distance on the map is related to the actual distance on the ground.

比例图按固定比例缩小或放大真实物体或距离。地图比例尺告诉你地图上的距离与实地距离之间的对应关系。

Map scales are written as ratios such as 1:25,000. The first number is the distance on the map, and the second number is the corresponding real distance in the same units.

地图比例尺写成比值形式，例如 1:25,000。第一个数字表示地图上的距离，第二个数字表示相同单位下对应的实际距离。

Map scale	1 cm on map represents
1:10,000	100 m
1:25,000	250 m
1:50,000	500 m
1:100,000	1 km

For example, 1 cm on a 1:25,000 map represents 25,000 cm on the ground. Since 25,000 cm = 250 m, every centimetre on the map stands for 250 metres in real life.

例如，在 1:25,000 的地图上，1 厘米代表实地 25,000 厘米。因为 25,000 厘米 = 250 米，所以地图上的每厘米代表实地 250 米。

7. Converting Real Distances Using a Scale | 利用比例尺换算实际距离

To convert a map distance to a real distance, multiply the map distance by the scale factor. The scale factor is the second number in the ratio when the first number is 1.

要将地图距离换算为实际距离，将地图距离乘以比例系数。比例系数是比例尺中第一个数为 1 时的第二个数。

real distance = map distance × scale factor

实际距离 = 地图距离 × 比例系数

For example, on a 1:25,000 map, two villages are 6.4 cm apart. The real distance is 6.4 × 25,000 cm = 160,000 cm = 1600 m = 1.6 km.

例如，在 1:25,000 的地图上，两个村庄相距 6.4 厘米。实际距离为 6.4 × 25,000 厘米 = 160,000 厘米 = 1600 米 = 1.6 公里。

To convert a real distance to a map distance, divide the real distance by the scale factor. Always use the same units before calculating.

要将实际距离换算为地图距离，将实际距离除以比例系数。计算前必须使用相同的单位。

map distance = real distance ÷ scale factor

地图距离 = 实际距离 ÷ 比例系数

For example, a real distance of 3 km on a 1:50,000 map gives a map distance of 300,000 cm ÷ 50,000 = 6 cm.

例如，实际距离为 3 公里，在 1:50,000 地图上，地图距离为 300,000 厘米 ÷ 50,000 = 6 厘米。

8. Constructing a Scale Drawing from Bearings | 根据方位角构建比例图

In navigation problems, you often need to combine bearings and distances to show a journey. First choose a sensible scale, such as 1 cm = 1 km or 1 cm = 10 km, so the drawing fits on the page.

在导航问题中，你经常需要结合方位角和距离来表示一段行程。首先选择一个合适的比例尺，例如 1 厘米 = 1 公里或 1 厘米 = 10 公里，使图形能放在页面上。

Draw a north line at the starting point and plot the first leg of the journey using its bearing and distance. At the end of this leg, draw a new north line before plotting the next bearing.

在起点画一条北线，并根据方位角和距离绘制行程的第一段。在第一段的终点，画一条新的北线，然后再绘制下一个方位角。

Example: A ship sails 5 km from harbour H on a bearing of 060°, then sails 3 km on a bearing of 150°. Using a scale of 1 cm = 1 km, draw the first path 5 cm at 060° and the second path 3 cm at 150° from the new position.

示例：一艘船从港口 H 出发，沿 060° 方位角航行 5 公里，然后沿 150° 方位角航行 3 公里。使用 1 厘米 = 1 公里的比例尺，先画 5 厘米、060° 的第一段，再在新位置画 3 厘米、150° 的第二段。

After completing the journey, you can measure the direct distance and bearing from the starting point to the final position. This gives the return route or the single equivalent journey.

完成行程后，你可以测量从起点到终点位置的直线距离和方位角。这就得出了返程路线或等效的单段行程。

9. Solving Navigation Problems | 解决导航问题

Navigation problems often ask you to find the direct distance and bearing from the start to the finish after two or more legs. Use a scale drawing and a protractor to measure the missing route.

导航问题通常要求你在经过两段或多段行程后，求出从起点到终点的直线距离和方位角。使用比例图和量角器来测量缺失的路线。

In some cases, the two legs form a right angle. Then you can use Pythagoras’ theorem to calculate the direct distance exactly.

在某些情况下，两段行程会形成直角。此时你可以使用勾股定理精确计算直线距离。

For example, a person walks 5 km due east and then 3 km due north. The east and north legs are perpendicular, so the direct distance from the start is:

例如，某人先向正东走 5 公里，再向正北走 3 公里。正东和正北两段互相垂直，因此从起点出发的直线距离为：

distance = √(5² + 3²) = √(25 + 9) = √34 ≈ 5.83 km

距离 = √(5² + 3²) = √(25 + 9) = √34 ≈ 5.83 公里

Even when the legs are not perpendicular, a careful scale drawing will give you a good estimate of the direct distance and bearing.

即使两段行程不互相垂直，仔细绘制比例图也能帮助你较好地估算出直线距离和方位角。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is writing a bearing with fewer than three digits, such as 45° instead of 045°. Always check the number of digits in your final answer.

一个常见错误是方位角没有写成三位数，例如写成 45° 而不是 045°。务必检查最终答案的数字位数。

Another mistake is measuring anticlockwise from north. Remember that bearings are measured clockwise, so the angle opens towards the east first.

另一个错误是从北方逆时针测量。记住方位角是顺时针测量的，所以角度首先向东侧打开。

When converting scale distances, students sometimes mix centimetres and metres. Convert all lengths to the same unit before multiplying or dividing.

换算比例尺距离时，学生有时会混淆厘米和米。在乘法或除法之前，应先将所有长度转换为相同的单位。

When drawing a journey with more than one leg, draw a fresh north line at each new point. Do not reuse the original north line for the second bearing.

绘制多段行程时，应在每个新点处重新画一条北线。不要继续使用原来的北线来绘制第二个方位角。

Finally, use a sharp pencil, place your protractor carefully, and double-check your scale before measuring. Accuracy in construction leads to better marks in examinations.

最后，使用锋利的铅笔，仔细放置量角器，并在测量前再次检查比例尺。精确作图能在考试中取得更好的成绩。

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Bearings and Scale Drawings | 方位角与比例尺