← Pattern Anatomy

Pattern Anatomy

Each pattern, deconstructed.

similar-triangle-length-ratio

★ Rank 9/22 2.6% of all AMC problems Grade 6–8

A length or area inside a synthetic plane figure is recovered by spotting a pair of similar triangles — created by parallel lines, tangent circles, a shared angle, or a right-angle altitude — and solving a single proportion (for a length) or applying the squared linear ratio (for an area). The decisive move is identifying the matched triangles and the correspondence of sides; there are no coordinates and no axis-aligned Pythagorean leg decomposition.

Try a sample now
Sample problem
no figure
amc8-2018-20
Try it →
27-year frequency
1999–2026
2026 forecast slots
—

How to solve

Primary tools see-drawmath-toolssimplify
What to look for
  • Swap a parallel-line cut for a tangent-circle configuration — the similarity argument changes from AA to SAS but the single-proportion finish is identical
  • Ask for an area instead of a length to force the squared-ratio step (area scales as the square of the linear ratio)
  • Change the given lengths so the scale_ratio stays clean (small rationals) while the surface figure looks different

Sub-archetype mix (2)

Click a row to see member problems.

More data (year-over-year, tool fingerprint, grade distribution, all members)
46
members
1999–2026
Active years
Year-over-year

Tool fingerprint (1–17)

Grade distribution

  • Gr 6
    1
  • Gr 7
    2
  • Gr 8
    43