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AMC 10 & 12

Triangle Basics & Similarity

Congruence and similarity criteria, and the special right-triangle ratios that let you skip the Pythagorean theorem entirely.

Congruence vs. similarity

Two triangles are congruent (identical in size and shape) if they satisfy SSS, SAS, ASA, AAS, or (for right triangles) HL. They're similar (same shape, possibly different size) if they satisfy AA, SAS similarity, or SSS similarity — same angle/side conditions, but only requiring proportional sides rather than equal ones.

AA similarity is the one you'll use constantly: if two angles of one triangle match two angles of another, the triangles are similar — full stop, no side information needed.

What similarity gives you

If two triangles are similar with ratio kk (every side of the second is kk times the corresponding side of the first):

corresponding sides scale by k,areas scale by k2\text{corresponding sides scale by } k, \qquad \text{areas scale by } k^2

That area-scales-by-k2k^2 fact is easy to forget under time pressure and is a favorite AMC trap.

Special right triangles

Two ratios are worth memorizing cold, since they turn a Pythagorean-theorem computation into instant recall:

  • 45-45-90: legs equal, hypotenuse is 2\sqrt2 times a leg — side ratio 1:1:21:1:\sqrt2.
  • 30-60-90: side ratio 1:3:21:\sqrt3:2 (short leg : long leg : hypotenuse).
  • 3-4-5 (and its multiples, like 6-8-10 or 5-12-13): common integer-sided right triangles worth recognizing on sight.

Worked example

Problem: In △ABC\triangle ABC, DE∥BCDE \parallel BC with DD onABAB and EE on ACAC. If AD=4AD=4, DB=6DB=6, andBC=15BC=15, find DEDE.

Solution: Since DE∥BCDE \parallel BC, corresponding angles are equal, so△ADE∼△ABC\triangle ADE \sim \triangle ABC by AA. The similarity ratio isADAB=44+6=410=25\tfrac{AD}{AB} = \tfrac{4}{4+6} = \tfrac{4}{10} = \tfrac25. So:

DE=25⋅BC=25⋅15=6DE = \dfrac25 \cdot BC = \dfrac25 \cdot 15 = 6

Practice problems

1.A 45-45-90 triangle has hypotenuse 10. Find the length of each leg.

2.A 30-60-90 triangle has short leg 5. Find the hypotenuse and long leg.

3.Two similar triangles have a side ratio of 3:53:5. If the smaller triangle has area 18, find the larger triangle's area.

4.A right triangle has legs 9 and 12. Find the hypotenuse.

Special Segments in Triangles