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Enlargement, reduction and scale drawings

Use one scale factor for every length, keep angles unchanged, and move between real lengths and a scale drawing.

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Enlarged diagram

Maps, plans and model cars are scale drawings: every length is multiplied by the same factor, and every angle stays the same. This lesson finds and uses scale factors for enlargements and reductions, and converts between a drawing and the real object with a ratio scale.

One scale factor for every length

An enlargement or reduction multiplies every length of a figure by the same scale factor k. The angles do not change, so the image is similar to the original.

Scale factorEffect on lengths
k > 1Enlargement: every length gets longer
0 < k < 1Reduction: every length gets shorter

The scale factor from the original to the image is

k = (image length)/(matching original length).

Worked example 1

Reduce a triangle

A triangle has sides 4 cm, 5 cm and 6 cm, and the angle opposite the 6 cm side is 82.8°, to one decimal place. It is reduced with scale factor 0.6. Find the sides of the image and the matching angle. Then give the scale factor that turns the image back into the original.

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  1. Multiply every length

    Method

    Multiply each side by 0.6.

    Reason

    A reduction uses one factor for all lengths, so the shape stays the same.

    Working

    Answer

    0.6 × 4 = 2.4 cm,; 0.6 × 5 = 3 cm,; 0.6 × 6 = 3.6 cm.

    The matching angle, opposite the 3.6 cm side, is still 82.8°.

  2. Reverse the reduction

    Method

    Divide an original length by its image length.

    Reason

    Going back uses the reciprocal factor.

    Working

    Answer

    4/2.4 = 5/3 = 1/0.6.

Check your understanding 1

Add or multiply?

A learner “enlarges” a 3 cm by 5 cm rectangle by adding 2 cm to each side, giving a 5 cm by 7 cm rectangle. Is the new rectangle an enlargement of the original? Give the factor for each pair of sides.

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No. The short sides have factor 5/3 and the long sides have factor 7/5. These are different, so the shape has changed. A true enlargement with factor 5/3 would give a 5 cm by 8 1/3 cm rectangle.

A scale drawing uses one reduction

A scale drawing is a reduction of a real object. The scale 1 : 300 means that 1 cm on the drawing represents 300 cm, or 3 m, in real life. Angles on the drawing are the real angles.

A garden drawn at 1 : 300

The triangle PQR shows a triangular garden. The angle at P between PQ and PR is marked 60 degrees. The side QR is not labelled.

Scale 1 : 300. On the drawing, PQ = 4 cm, PR = 3 cm and ∠QPR = 60°.
Figure data

This table is included in print while it is open.

Drawing measurements
ItemOn the drawing
Scale1 : 300
PQ4 cm
PR3 cm
∠QPR60°

Worked example 2

Read real lengths from the drawing

Find the real lengths of PQ and PR. On an accurate drawing, QR measures 3.6 cm to the nearest millimetre. Estimate the real length of QR.

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  1. Scale the given sides

    Method

    Multiply each drawing length by 3 m.

    Reason

    Each centimetre on the drawing represents 300 cm, which is 3 m.

    Working

    Answer

    PQ = 4 × 3 = 12 m,; PR = 3 × 3 = 9 m.

  2. Scale the measured side

    Method

    Multiply the measured 3.6 cm by 3 m.

    Reason

    QR was measured with a ruler, so the result is an estimate.

    Working

    Answer

    QR ≈ 3.6 × 3 = 10.8 m.

Check your understanding 2

Draw a plan at 1 : 200

A floor plan uses the scale 1 : 200. A wall is 7.4 m long. A window on the plan is 0.6 cm wide.

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The wall is 740 cm long, so on the plan it is 740÷200 = 3.7 cm. The window is 0.6 × 200 = 120 cm, which is 1.2 m.

Check your understanding 3

Find the scale of a model

A model car is 21 cm long. The real car is 4.2 m long. Write the scale of the model in the form 1 : n. The real car is 1.8 m wide. How wide is the model?

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In centimetres, the lengths are 21 and 420, so the scale is 21:420 = 1:20. The real width is 180 cm, so the model is 180÷20 = 9 cm wide.

Common mistakes

  • Adding instead of multiplying. An enlargement multiplies every length by the scale factor.
  • Changing the angles. Enlargement and reduction keep every angle the same.
  • Forgetting to convert units. On a 1 : 300 plan, 1 cm represents 300 cm, which is 3 m.
  • Using the length factor for area. Areas change by the square of the scale factor.

Questions

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