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. \(\displaystyle L_B=kL_A\)
- Use the whole corresponding side For nested triangles, scaling gives a whole side. Subtract the inner part only when the remaining segment is requested
- Prove an overlapping configuration Use vertically opposite angles at the crossing and alternate angles from parallel sides. Write vertices in matching order before forming ratios
- Interpret map scale A scale of 1 : d compares equal units. Multiply map length by d for ground length; divide for the reverse direction, then convert units
- Evaluate measurement errors A shadow estimate grows as the target shadow length increases and falls as the reference shadow length increases. Test plausible extreme measurements and report the resulting uncertainty
- Report and improve the model State the measured quantities, assumptions, method, approximate result and limitations. Repeated observations do not repair a biased setup or invalid geometry
Worked example
A vertical reference of exactly 1.8 m has a 3.0 m shadow; a vertical tree has a 10.0 m shadow. They are measured simultaneously on level ground. Each shadow length may differ by up to 0.1 m. Estimate the height and its plausible range
- \(\displaystyle H=1.8\times\frac{10}{3}=6\text{ m}\)AA similarity gives equal height-to-shadow ratios
- \(\displaystyle H_{\min}=1.8\times\frac{9.9}{3.1}\)Use the smallest numerator and largest denominator
- \(\displaystyle H_{\min}\approx5.75\text{ m}\)The lower endpoint is approximately 5.75 m.
- \(\displaystyle H_{\max}=1.8\times\frac{10.1}{2.9}\)Use the largest numerator and smallest denominator
- \(\displaystyle H_{\max}\approx6.27\text{ m}\)The upper endpoint is approximately 6.27 m.
- \(\displaystyle H\approx6\text{ m}\)Report an estimate of about 6 m and the approximate range 5.75–6.27 m under the stated assumptions. Reference-height uncertainty, if present, must also be included
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
Do triangles have to be nested to use similarity?+
No. Separate, rotated or overlapping triangles can be similar. Prove the angle relationships and match corresponding vertices
Why does measurement uncertainty matter?+
The ratio uses measured lengths. Plausible changes in numerator and denominator change the inferred height, so many calculator digits do not justify a precise reported height
Does repeating measurements fix every error?+
No. Repetition helps with random variation. Consistently sloping ground, non-vertical objects or a biased measuring method require a better setup or a different model
Does a map distance always give a travel distance?+
No. A straight mapped separation does not include bends, detours or changes in terrain. Use the appropriate route or terrain model for that purpose
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 7: Geometry and similarity