Similar figures and scale factors
Similar figures have equal corresponding angles and one common ratio for all corresponding lengths. The scale factor from Figure A to Figure B is an image length divided by its corresponding original length
- Pair corresponding lengths Match dimensions with the same role and keep the order consistent \(\displaystyle \frac{12}{8}=\frac{7.5}{5}=1.5\)
- Find the length factor Divide a length on Figure B by its corresponding length on Figure A. \(\displaystyle k=\frac{12}{8}=\frac{3}{2}=1.5\)
- Square the factor for area Area has two dimensions, so its scale factor is k squared \(\displaystyle k^2=\left(\frac{3}{2}\right)^2=\frac{9}{4}=2.25\)
- Use the direction of the factor Multiply from Figure A to Figure B and divide when finding an original length from an image length \(\displaystyle x=10\times\frac{3}{2}=15\text{ cm}\)
- Perimeters and angles Perimeters scale by k because they are sums of lengths. Corresponding angles stay equal
- Recover a length factor Take the positive square root of the area ratio. Divide by k² when finding an original area
- Use consistent units Convert both corresponding lengths to the same unit before forming a ratio. A scale factor is dimensionless
- Combine and test scale models Multiply successive length factors. Check that the image is uniformly scaled; unequal stretching invalidates a single-factor model
Worked example
Figure A is a rectangle with width 9 cm and height 6 cm. Figure B is a similar rectangle with width 12 cm and height 8 cm. Find the area of Figure B.
- \(\displaystyle k=\frac{12}{9}=\frac{4}{3}\)Find the image-to-original length factor from the corresponding widths
- \(\displaystyle k^2=\left(\frac{4}{3}\right)^2=\frac{16}{9}\)Square the length factor because area has two dimensions
- \(\displaystyle A_A=9\times6=54\text{ cm}^2\)Calculate the original rectangle's area
- \(\displaystyle A_B=54\times\frac{16}{9}=96\text{ cm}^2\)Multiply the original area by the area scale factor
Common mistakes
Difficulty levels
Questions about this topic
What makes two figures similar?+
Their corresponding angles are equal and all corresponding lengths have one common ratio
Why is the area factor k squared?+
A figure's area has two length dimensions, so multiplying each dimension by k multiplies the area by k × k = k²
When should a length be divided by k?+
Divide by k when the known length is on Figure B and the unknown corresponding length is on Figure A.
Can equal side ratios prove any two polygons are similar?+
No. A square and a non-square rhombus can have equal side ratios but different angles. Triangle SSS is a special test
Do successive scale factors add?+
No. Multiply them, then square the resulting length factor if an area factor is needed
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity