The idea, step by step
Match the figures before calculating
Similar figures have the same shape: corresponding angles are equal and corresponding lengths share one multiplier. A rotation or reflection does not change which vertices correspond.
Write the vertex correspondence first. If ABC matches DEF, AB matches DE, BC matches EF and AC matches DF. A side ratio using unmatched sides gives a believable-looking but meaningless result.
Lengths, areas and scale drawings
Define the scale factor in a direction, for example small figure to large figure. Multiply a small length by that factor; divide a large length to recover the small length. Perimeters use the same linear factor.
Areas scale with the square of the length factor because two dimensions change. A drawing scale must use the same units on both sides before a calculation.
A common trap
Multiplying an area by the linear scale factor.
Instead: If lengths are multiplied by k, each pair of dimensions is multiplied by k, so area is multiplied by k².
See the thinking
Worked example 1
Triangles ABC and DEF are similar in that order. AB=6 cm, BC=9 cm, AC=11 cm and DE=15 cm. Find EF and DF.
- AB corresponds to DE, so the small-to-large factor is 15/6=2.5.
- BC corresponds to EF: EF=9×2.5=22.5 cm.
- AC corresponds to DF: DF=11×2.5=27.5 cm. Check both sides use the same factor.
Worked example 2
A plan uses scale 1:400. A rectangular courtyard is 7 cm by 4.5 cm on the plan. Find its real dimensions and area.
- Multiply each plan length by 400: 2800 cm and 1800 cm.
- Convert centimetres to metres: 28 m and 18 m.
- Multiply the real dimensions: 28×18=504 m². The scale is a length scale, not an area multiplier.
Check your understanding
Similar figures have length factor 3 from small to large. What is the area factor?
Show answer
9
Both dimensions scale by 3, giving 3×3=9.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- Similar triangles have corresponding sides 8 cm and 20 cm. A second side of the smaller triangle is 12 cm. Find the matching larger side.2 marks
- A model and real object have scale 1:50. A model length is 14 cm. Find the real length in metres.2 marks
- Two similar shapes have length factor 1.5 from A to B. Shape A has perimeter 32 cm. Find B’s perimeter.2 marks
- A triangle of area 18 cm² is enlarged with length factor 4. Find the new area.2 marks
- Triangles PQR and XYZ are similar in that order. PQ=10, QR=14, XY=15. Find YZ.2 marks
- A student says: “The area doubled, so every side doubled.” Explain and give the actual length factor.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Similar triangles have corresponding sides 8 cm and 20 cm. A second side of the smaller triangle is 12 cm. Find the matching larger side.2 marks
k=20/8=2.5.
12×2.5=30 cm.
30 cm - A model and real object have scale 1:50. A model length is 14 cm. Find the real length in metres.2 marks
14×50=700 cm.
700 cm=7 m.
7 m - Two similar shapes have length factor 1.5 from A to B. Shape A has perimeter 32 cm. Find B’s perimeter.2 marks
Perimeters scale by the length factor.
32×1.5=48 cm.
48 cm - A triangle of area 18 cm² is enlarged with length factor 4. Find the new area.2 marks
The area factor is 4²=16.
18×16=288 cm².
288 cm² - Triangles PQR and XYZ are similar in that order. PQ=10, QR=14, XY=15. Find YZ.2 marks
PQ matches XY, so k=15/10=1.5.
QR matches YZ, giving 14×1.5=21.
21 units - A student says: “The area doubled, so every side doubled.” Explain and give the actual length factor.3 marks
Area factor k²=2.
Thus k=√2; doubling lengths would multiply area by 4.
The length factor is √2, approximately 1.414, not 2.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗