Justify the model, then scale
AA establishes similarity when two corresponding angles are equal. Vertical objects on level ground provide right angles; parallel sunlight gives another equal pair. Match height with height and shadow with shadow
- State the direction For a scale factor from A to B, divide a B length by its corresponding A length. B can be larger or smaller \(\displaystyle k=L_B/L_A\)
- Use matching units Convert corresponding lengths to the same unit before forming their ratio
- Apply or undo the scale From A to B, multiply by k. From B to A, divide by k. \(\displaystyle L_B=kL_A\)
Worked example
Two similar triangular panels have corresponding lengths of 9 cm and 15 cm. A length of 12 cm on the smaller panel corresponds to x on the larger panel. Find x
- \(\displaystyle k=\frac{15}{9}=\frac{5}{3}\)The scale factor is the larger corresponding length divided by the smaller one
- \(\displaystyle x=12\times\frac{5}{3}=\frac{60}{3}\)Multiply the smaller-panel length by the scale factor
- \(\displaystyle x=20\ \mathrm{cm}\)The corresponding length on the larger panel is 20 cm
Common mistakes
Difficulty levels
Questions about this topic
Does the image have to be larger?+
No. A factor between 0 and 1 gives a reduction. Always multiply from original to image, and divide in the reverse direction
Why subtract after scaling a nested triangle?+
Scaling gives a whole corresponding side. If the requested segment is only the part outside the inner triangle, subtract the inner side from the whole
When can shadows be compared?+
Both objects must be vertical on level ground and use the same sunlight angle, usually by measuring at the same time
Which unit should be used?+
Either unit is suitable, provided the corresponding lengths use the same unit before division. Convert the final answer to the requested unit
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 7: Geometry and similarity