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
\[A_B=k^2A_A\]
- 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 the square of the length factor when finding an original area
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
\[A_A=9\times6\text{ cm}^2\]
- \(\displaystyle k=\frac{12}{9}=\frac{4}{3}\)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
Using one ratio as proof of similarity
✕\(\displaystyle \frac{12}{9}=\frac{4}{3}\;\Rightarrow\;\text{similar}\)✓\(\displaystyle \frac{12}{9}=\frac{8}{6}=\frac{4}{3}\)
For these rectangles the right angles already match; both width and height ratios must agree. For general polygons, equal side ratios alone do not establish equal corresponding anglesInverting the scale factor
✕\(\displaystyle k=\frac{9}{12}=\frac{3}{4}\)✓\(\displaystyle k=\frac{12}{9}=\frac{4}{3}\)
The direction from Figure A to Figure B is image length divided by original lengthScaling area by k instead of k squared
✕\(\displaystyle A_B=\frac{3}{2}A_A\)✓\(\displaystyle A_B=\left(\frac{3}{2}\right)^2A_A=\frac{9}{4}A_A\)
Area has two dimensions, so the length factor must be used twiceDifficulty levels
introUse gridded similar rectangles with an integer length factor
standardCheck described similar figures, find a fractional factor, scale an area, or find an unknown length
challengeCheck whether ratios agree, recover a length factor from an area factor and work backwards for a separate marked segment
examChoose a scale factor or unknown length from options containing inverted, additive, and non-corresponding values
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 the length factor multiplies the area by the square of that factor
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 I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity
In this chapter
Round to decimal places and significant figuresScientific notationMultiply and divide in scientific notationFind a side of a right-angled trianglePythagoras in three dimensions10C Perimeter and area: Convert measurement unitsPerimeter and area of standard shapesArea of a triangle from three sidesCircumference and area of a circlePerimeter and area of composite shapesArc length and sector area10E Volume and capacity: Convert measurement unitsVolume of prisms and cylinders10E Volume and capacity: Volume of cones, pyramids and spheresCapacity of a container10F Volume of pyramids: Volume of cones, pyramids and spheresSurface area of solidsSimilar figures and scale factorsTests for similar trianglesSimilar solids and scaling volume