📚 Bearings and Scale Drawings for IGCSE Mathematics | IGCSE 数学：方位角与比例绘图
Bearings are used in navigation, surveying and map reading to describe direction accurately. In IGCSE Mathematics, you need to measure, draw and calculate bearings, and you must be able to combine them with scale drawings and right-angled trigonometry.
方位角用于导航、测量和地图判读，以精确描述方向。在 IGCSE 数学中，你需要掌握方位角的测量、绘图和计算，并能将方位角与比例绘图以及直角三角形三角学结合起来。
1. What Is a Bearing? | 什么是方位角？
A bearing is a way of giving direction. It is always measured from North, always measured clockwise, and always written as a three-digit angle such as 045° or 200°.
方位角是表示方向的一种方式。它始终从北方向开始测量，始终沿顺时针方向读取，并且始终写成三位数的角度，例如 045° 或 200°。
In IGCSE questions, you will see bearings in navigation problems, maps and scale drawings. A bearing does not tell you a distance by itself, so it is often used together with a distance to fix a position.
在 IGCSE 题目中，你会看到方位角出现在导航问题、地图和比例绘图中。方位角本身不能表示距离，因此它通常与距离一起使用，以确定一个位置。
2. Reading and Writing Bearings | 读取与书写方位角
To read a bearing, start at the North line, turn clockwise until you reach the direction you want. The angle you turn through is the bearing.
读取方位角时，从北方线开始，沿顺时针方向转动，直到到达所需方向。所转过的角度就是方位角。
For example, East is a turn of 090° from North. Southwest is a turn of 225° because it is halfway between South 180° and West 270°.
例如，东方是从北方顺时针转动 090°。西南方向是 225°，因为它位于南方 180° 和西方 270° 的中点。
A bearing must always have three digits. Do not write 45°; write 045°. Do not write 5°; write 005°.
方位角必须始终写成三位数。不要写成 45°，而应写成 045°。不要写成 5°，而应写成 005°。
3. The Three-Digit Rule | 三位数规则
The three-digit rule is one of the most common marking points in IGCSE exams. Even if the angle is less than 100°, you must add leading zeroes.
三位数规则是 IGCSE 考试中最常见的给分点之一。即使角度小于 100°，也必须在前面补零。
- North = 000° or 360° | 北方 = 000° 或 360°
- Northeast = 045° | 东北方向 = 045°
- East = 090° | 东方 = 090°
- Southeast = 135° | 东南方向 = 135°
- South = 180° | 南方 = 180°
- Southwest = 225° | 西南方向 = 225°
- West = 270° | 西方 = 270°
- Northwest = 315° | 西北方向 = 315°
Using a table like this can help you convert quickly between compass directions and bearings.
使用这样的表格可以帮助你在罗盘方向和方位角之间快速转换。
4. Compass Directions and Bearings | 罗盘方向与方位角
You should know the standard compass directions and their equivalent bearings without hesitation. These are often used as building blocks in more complicated navigation questions.
你应该毫不犹豫地掌握标准罗盘方向及其对应的方位角。它们经常作为更复杂的导航问题中的基础构件。
The four cardinal points are North 000°, East 090°, South 180° and West 270°. The four intercardinal points are Northeast 045°, Southeast 135°, Southwest 225° and Northwest 315°.
四个基本方位是北方 000°、东方 090°、南方 180° 和西方 270°。四个中间方位是东北 045°、东南 135°、西南 225° 和西北 315°。
If a direction is given as North 30° East, it means starting from North and turning 30° towards East, giving a bearing of 030°. This is not the same as East 30° North, which would be 060°.
如果方向表示为北偏东 30°，表示从北方开始向东转 30°，得到的方位角是 030°。这与东偏北 30° 不同，后者的方位角是 060°。
5. Scale Drawings | 比例绘图
A scale drawing shows a real object or distance at a reduced or enlarged size. The scale tells you the ratio between the drawing length and the real length.
比例绘图以缩小或放大的尺寸表示真实物体或距离。比例尺告诉你图上长度与实际长度之间的比率。
in real life. Since 25 000 cm = 250 m, 1 cm on the drawing represents 250 m on the ground.
例如，比例尺 1 : 25 000 表示图上 1 厘米代表实际 25 000 厘米。因为 25 000 厘米 = 250 米，所以图上 1 厘米代表地面 250 米。
To convert a drawing length to a real distance, multiply the drawing length by the scale factor. To convert a real distance to a drawing length, divide by the scale factor.
要将图上长度换算为实际距离，用图上长度乘以比例因子。要将实际距离换算为图上长度，除以比例因子即可。
6. Drawing a Bearing and Distance | 绘制方位角与距离
When a question asks you to draw a bearing and distance, follow a clear sequence. First draw a North line at the starting point. Then place your protractor with 000° along North and 090° along East.
当题目要求绘制方位角和距离时，请按照清晰的步骤进行。首先在起点处画出北方线。然后将量角器放置好，使 000° 对着北方，090° 对着东方。
Mark the required bearing clockwise from North. Draw a light construction line in that direction. Use the given scale to convert the real distance to a drawing length, and mark the final point.
从北方按顺时针方向标出所需方位角。沿该方向画一条轻构造线。使用给定的比例尺将实际距离换算为图上长度，并标出终点。
Label the bearing and distance clearly. In IGCSE exams, construction lines should be left visible because they may earn method marks.
清楚标注方位角和距离。在 IGCSE 考试中，构造线应保留可见，因为它们可能为你赢得方法分。
7. Using Trigonometry with Bearings | 方位角与三角学结合
Many bearing problems involve right-angled triangles. When a ship or walker changes direction by 090°, the two legs of the journey form the perpendicular sides of a right-angled triangle.
许多方位角问题涉及直角三角形。当船或行人改变方向 090° 时，两段行程构成直角三角形的两条直角边。
You can use Pythagoras' theorem to find the distance from the starting point, and basic trigonometric ratios to find a missing bearing or angle.
你可以使用勾股定理求出离起点的距离，并使用基本三角比来求缺失的方位角或角度。
c² = a² + b²
tan θ = opposite ÷ adjacent
direction is found from tan θ = 6 ÷ 8, so θ = 36.9°. The bearing is therefore about 037°.
例如，一艘船向正东航行 6 公里，再向正北航行 8 公里。北方线与最终方向之间的夹角由 tan θ = 6 ÷ 8 求得，因此 θ = 36.9°。所以方位角约为 037°。
8. Back Bearings | 反向方位角
A back bearing is the bearing of the starting point from the finishing point. It describes the opposite direction along the same line.
反向方位角是从终点看起点的方位角。它描述的是同一直线上的相反方向。
To find a back bearing, add 180° if the original bearing is less than 180°, and subtract 180° if the original bearing is 180° or greater.
求反向方位角时，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°。
Bback = (B + 180°) mod 360°
For example, the back bearing of 065° is 245°. The back bearing of 210° is 030°.
例如，065° 的反向方位角是 245°。210° 的反向方位角是 030°。
9. Worked Example: Bearing Problem | 例题：方位角问题
Two towns, A and B, are located from a harbour H. Town A is 8 km from H on a bearing of 035°. Town B is 6 km from H on a bearing of 125°. Find the distance between A and B.
两座城镇 A 和 B 相对于港口 H 定位。城镇 A 位于 H 的 035° 方位角上，距离 8 公里；城镇 B 位于 H 的 125° 方位角上，距离 6 公里。求 A 与 B 之间的距离。
ings is 125° − 035° = 090°, so triangle AHB is right-angled at H. Therefore, by Pythagoras' theorem, AB² = 8² + 6² = 64 + 36 = 100, so AB = √100 = 10 km.
在 H 处两个方位角之间的夹角为 125° − 035° = 090°，因此三角形 AHB 在 H 处为直角。根据勾股定理，AB² = 8² + 6² = 64 + 36 = 100，所以 AB = √100 = 10 公里。
This type of question often requires you to show the right angle first, then apply Pythagoras or trigonometry. Always write down the bearing subtraction clearly.
这类题目通常要求你先证明是直角，然后再应用勾股定理或三角学。始终要清楚写出方位角的减法过程。
10. Common Mistakes | 常见误区
- Measuring anticlockwise from North instead of clockwise. | 从北方逆时针测量，而不是顺时针测量。
- Writing 60° instead of 060°. | 写成 60° 而不是 060°。
- Using the wrong starting point for a bearing. | 方位角的起点搞错。
- Confusing the bearing of A from B with the bearing of B from A. | 混淆从 B 看 A 的方位角与从 A 看 B 的方位角。
- Forgetting to convert units in a scale drawing. | 在比例绘图中忘记换算单位。
Checking for these errors can save many marks in an IGCSE exam. Slow down when reading the words 'from' and 'of' in a bearing question.
检查这些错误可以在 IGCSE 考试中保住许多分数。阅读方位角题目中的 'from' 和 'of' 时要放慢速度。
11. Exam Tips | 考试技巧
Always draw a clear North line and label all given bearings. If a diagram is not provided, sketch one before doing any calculations.
始终画出清晰的北方线并标注所有已知方位角。如果没有给出图，先画一个草图再进行计算。
Show your working step by step. In bearing questions, method marks are given for correct use of 180°, Pythagoras' theorem or trigonometry, even if the final answer is slightly wrong.
逐步展示解题过程。在方位角题目中，即使最终答案略有错误，只要正确使用了 180°、勾股定理或三角学，仍可获得方法分。
When using a protractor, align it carefully with the North line. A drawing error of even 2° can change a bearing answer significantly.
使用量角器时，要仔细将其与北方线对齐。即使 2° 的绘图误差也可能显著改变方位角答案。
12. Summary | 总结
Bearings are measured clockwise from North and written with three digits. Scale drawings convert real distances into a smaller or larger representation by a fixed ratio.
方位角从北方顺时针测量，并写成三位数。比例绘图通过固定比率将实际距离转换为缩小或放大的表示。
To succeed in IGCSE, practise drawing bearings with a protractor, converting compass directions, finding back bearings, and solving right-angled triangle problems that involve bearings and distances.
要在 IGCSE 中取得成功，请多练习使用量角器绘制方位角、转换罗盘方向、求反向方位角，以及解决涉及方位角和距离的直角三角形问题。
Mastering bearings will also help you with vectors, trigonometry and real-world map questions later in the course.
掌握方位角还将帮助你后续学习向量、三角学和现实生活中的地图问题。
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