Similarity & Ratios

Angles give you lengths.

Similarity is the bridge from angle information to length information, and AA is all you need to cross it. A parallel line is the most common trigger: $DE \parallel BC$ instantly gives $\triangle ADE \sim \triangle ABC$.

Keep the companions close: the angle bisector theorem ($BD/DC = AB/AC$), the midsegment (parallel and half), and the altitude to a hypotenuse, which produces the geometric mean $AH^2 = BH \cdot HC$.

One trap causes more lost marks than any other here: areas scale as the square of the length ratio, never the ratio itself.

More techniques

• Pigeonhole PrincipleMore pigeons than holes means a shared hole.
• Bounding & SqueezingTrap the answer between two walls.
• All 29 techniques →