Setting Up Proportions in Similar Figures

Similar figures are scaled copies of each other: same angles, sides that grow or shrink by one consistent factor. That factor is the whole idea. If you can find it using one known pair of corresponding sides, it unlocks every other side. The hard part is never the algebra — it's making sure side A in the first figure lines up with the side in the second figure that plays the same role, not just the one that looks closest to it on the page.

The safe method: name the figures so matching letters correspond, like triangle ABC similar to triangle PQR, meaning A matches P, B matches Q, C matches R. Then AB corresponds to PQ, BC to QR, and so on. Write your proportion as (small figure's side)/(big figure's side) = (small figure's other side)/(big figure's other side), keeping the same figure on top in both fractions. Cross-multiply and solve. Shadows work the same way: the sun's rays create two similar right triangles at the same instant, so height over shadow length is one fixed ratio no matter how tall the object is.

This setup feeds directly into the next ideas in the unit. The triangle proportionality theorem — a line parallel to one side of a triangle cuts the other two sides into proportional pieces — is just similar figures in disguise, since that parallel line creates a smaller triangle similar to the original. Later, when right triangles and trig ratios enter, you're still comparing proportional sides; similarity is the reason sine, cosine, and tangent give the same value for any right triangle with a given angle, regardless of its size.