C4.1, C4.3; E4.1, E4.3 - Scale drawings and bearings

Reading and drawing to scale

You travel from a harbour to a buoy on a bearing of $070^\circ$. Why is the bearing back $250^\circ$, rather than $290^\circ$? A drawing will make the turn clear. First decide how much real distance each length on the drawing represents.

A scale drawing represents every length using the same multiplier. If $1\,\mathrm{cm}$ represents $20\,\mathrm{m}$, a $3\,\mathrm{cm}$ line represents $60\,\mathrm{m}$. Doubling the drawing length doubles the real length. A scale of $1:2000$ expresses both lengths in the same unit: $1\,\mathrm{cm}$ represents $2000\,\mathrm{cm}=20\,\mathrm{m}$.

To draw a real length, divide by the amount represented by one drawing unit. To interpret a drawing length, multiply by that amount. Label the scale, rule straight edges, and keep enough room on the paper.

[DIAGRAM: asset_name: Scaled rectangular plan; asset_slug: scale_drawing_plan; caption: All lengths use the same scale; a scale bar survives screen resizing.; description: An exact rectangular plan is labelled actual length 80 metres and width 50 metres, with drawing instructions 4 centimetres by 2.5 centimetres at 1 centimetre to 20 metres. A proportional scale bar marked 0 20 and 40 metres is shown, and the note says use the labels or bar, not physical screen centimetres.]

Keep the same path but change only the scale. More metres per centimetre means a shorter drawing.

For a full plan, convert every side using the same scale and preserve the required angles. For example, an $80\,\mathrm{m}$ by $50\,\mathrm{m}$ rectangular enclosure at $1\,\mathrm{cm}$ to $20\,\mathrm{m}$ becomes $4\,\mathrm{cm}$ by $2.5\,\mathrm{cm}$, with right angles drawn using a protractor or set square.

Use a labelled plan measurement, so there is no screen-size assumption.

Three-figure bearings from north

A bearing fixes the direction of a line by measuring a turn from north at the starting point. The words “from $A$” tell you where to put the protractor.

[DIAGRAM: asset_name: Bearings measured clockwise from north; asset_slug: bearings_clockwise_from_north; caption: Both bearings start from north; the western bearing needs the longer clockwise turn.; description: Two exact compass diagrams show a ray at bearing 065 degrees and a ray at bearing 295 degrees. North is vertical in each, and clockwise arrows from north follow the correct short eastern turn and long western turn. The latter has a separate small counterclockwise angle of 65 degrees labelled not the bearing.]

For a bearing below $180^\circ$, a semicircular protractor can measure the clockwise turn directly on the eastern side of the north line. For a bearing above $180^\circ$, either use a full-circle protractor or measure the smaller anticlockwise angle from north and subtract it from $360^\circ$. This subtraction gives the required clockwise turn; it is not a reverse bearing.