📚 Mastering Bearings and Scale Drawings | 精通方位角与比例绘图
Bearings and scale drawings are essential tools for describing direction and distance on maps, in navigation, and in construction problems. In the Cambridge KS3 mathematics curriculum, students are expected to measure, draw, and interpret bearings as three-figure angles from north, and to use scale factors to convert between plan distances and real-life distances.
方位角与比例绘图是描述地图、航海和建筑问题中方向与距离的重要工具。在剑桥 KS3 数学课程中，学生需要会测量、绘制和解读方位角（从北方向顺时针起算的三位数角度），并会使用比例尺将图纸距离与实际距离相互转换。
1. Understanding Bearings | 理解方位角
A bearing is a way of describing the direction from one point to another. It is always measured in degrees, clockwise, starting from the north direction. Bearings are written as three-digit numbers, so an angle of 5° is written as 005° and an angle of 70° is written as 070°.
方位角是描述从一个点到另一个点的方向的方法。它总是从北方向开始，按顺时针方向测量，以度为单位。方位角写成三位数字，因此 5° 要写成 005°，70° 要写成 070°。
A bearing of 000° means due north, 090° means due east, 180° means due south, and 270° means due west. These fixed directions are useful reference points when checking that a measured bearing is reasonable.
方位角 000° 表示正北，090° 表示正东，180° 表示正南，270° 表示正西。这些固定方向是检查测量出的方位角是否合理的有用参考点。
2. Measuring Bearings with a Protractor | 使用量角器测量方位角
To measure the bearing of B from A, place a protractor with its centre at point A and align its 0° line with the north direction. Read the angle clockwise from north to the line segment AB. If the angle is less than 100°, add leading zeros so the bearing has three digits.
要测量从 A 到 B 的方位角，将量角器的中心对准 A 点，并将 0° 刻度线与北方向对齐。从北方向顺时针读出到线段 AB 的角度。如果角度小于 100°，需要补前导零，使方位角成为三位数。
When measuring, make sure the protractor is not turned upside down. Draw a faint north line if one is not already given, and always place the correct end of the protractor at the starting point.
测量时，确保量角器没有放反。如果没有给出北方向线，先画一条浅色的北方向线，并始终将量角器正确的端点对准起点。
3. Three-Figure Bearings | 三位数方位角
Bearings must always be given using three digits. For example, 90° is written as 090°, 8° as 008°, and 135° remains 135°. This convention prevents confusion between numbers like 15° and 150° when reading maps or giving navigation instructions.
方位角必须始终使用三位数字表示。例如，90° 写成 090°，8° 写成 008°，而 135° 保持为 135°。这一规定可以避免阅读地图或发出导航指令时混淆 15° 和 150° 等数字。
If you measure an angle of 1°, write it as 001°. If you measure 99°, write it as 099°. Do not write a bearing with a decimal point or a direction letter such as N or E.
如果量得的角度是 1°，写成 001°。如果量得 99°，写成 099°。不要在小数点或方向上加上 N 或 E 等字母。
4. Drawing Bearings from a Point | 从一点绘制方位角
To draw a bearing from a point, draw a north line through the point. Place the protractor on the point with 0° facing north. Mark the required clockwise angle, then draw a line from the point through the mark. Label the line with the bearing and the real direction if known.
要从一点绘制方位角，先过该点画一条北方向线。将量角器放在该点上，使 0° 朝北。标记所需的顺时针角度，然后从该点过标记处画一条直线。标注方位角和已知的实际方向。
For example, to draw a bearing of 135° from point P, draw north upwards, place the protractor, count 135° clockwise from north, make a small mark, and use a ruler to draw a straight line from P through the mark.
例如，要从点 P 绘制 135° 的方位角，先向上画北方向，放大量角器，从北方向顺时针数 135°，做一个小标记，然后用直尺从 P 点过标记处画一条直线。
5. Scale Drawings and Maps | 比例绘图与地图
A scale drawing is a drawing in which all lengths are reduced or enlarged by the same ratio. On a map, the scale tells you how map distance relates to real distance. A scale of 1 : 25000 means that 1 cm on the map represents 25000 cm in real life, which is 250 m or 0.25 km.
比例绘图是指所有长度都按同一比率缩小或放大的图。在地图上，比例尺告诉你图上距离与实际距离的关系。比例尺 1 : 25000 表示地图上的 1 厘米代表实际中的 25000 厘米，即 250 米或 0.25 公里。
Scales can be written in three ways: as a ratio (1 : 50000), as a statement (1 cm represents 500 m), or as a scale bar. In the Cambridge KS3 course, ratio and statement forms are the most common.
比例尺可以用三种方式表示：比（1 : 50000）、文字说明（1 厘米代表 500 米）或比例条。在剑桥 KS3 课程中，比和文字说明形式最为常见。
6. Using Scales to Find Real Distances | 利用比例尺求实际距离
ometres: 200000 cm = 2000 m = 2 km. Always check that your final units match the question.
要将图上距离转换为实际距离，乘以比例因子。例如，比例尺为 1 : 50000 的地图上两点相距 4 厘米，实际距离为 4 × 50000 = 200000 厘米。换算为公里：200000 厘米 = 2000 米 = 2 公里。始终检查最终单位是否与题目一致。
Real distance = Map distance × Scale factor
When working with different units, first write the scale as a fraction with the same unit. For example, 1 cm represents 500 m becomes 1 cm : 50000 cm, which is 1 : 50000.
处理不同单位时，先写出具有相同单位的比例。例如，1 厘米代表 500 米变为 1 厘米 : 50000 厘米，即 1 : 50000。
7. Constructing a Scale Drawing | 构造比例图
To construct a scale drawing, choose a suitable scale that makes the drawing fit on the page. Convert every real distance to a drawing length using the chosen scale. Draw the first line with a ruler, measure the bearing with a protractor, and continue joining points in order. Label all important points and distances clearly.
要构造比例图，选择一个合适的比例尺，使图形能容纳在纸上。利用所选比例将每个实际距离转换为绘图长度。用直尺画第一条线，用量角器测出方位角，然后按顺序连接各点。清楚标注所有重要点和距离。
For example, if 1 cm represents 10 m, then a real distance of 35 m is drawn as 3.5 cm. A scale that is too large will not fit on the page, while a scale that is too small will make the drawing difficult to measure accurately.
例如，如果 1 厘米代表 10 米，那么实际距离 35 米应绘制为 3.5 厘米。比例尺太大可能导致图形超出页面，比例尺太小则会使图形难以精确测量。
8. Bearings in Navigation Problems | 航海问题中的方位角
Navigation problems often describe a journey using a series of bearings and distances. For example, a ship may sail 5 km on a bearing of 060°, then 4 km on a bearing of 150°. A scale drawing can show the final position relative to the starting point. The final bearing and distance can then be measured from the drawing.
航海问题通常用一系列方位角和距离描述一段旅程。例如，一艘船以 060° 的方位角航行 5 公里，然后以 150° 的方位角航行 4 公里。比例绘图可以显示终点相对于起点的位置。然后可以从图中测量最终方位角和距离。
When using a scale drawing, draw the first leg from the start point, then draw a new north line at the end of that leg. Measure the second bearing from this new north line, not from the original north line.
使用比例绘图时，从起点画第一段航线，然后在第一段终点处画一条新的北方向线。从这条新的北方向线测量第二个方位角，而不是从原来的北方向线测量。
9. Combining Bearings and Distances | 结合方位角与距离
When drawing a route, always start each new leg from the point you have reached, not from the original start. Draw a fresh north line at each new point, because the bearing is measured from north at that location. Use a sharp pencil and an accurate ruler and protractor for reliable results.
绘制路线
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