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3 Responses
Speed run: list each triangle’s side lengths in ascending order and match shortest-to-shortest, middle-to-middle, longest-to-longest (equivalently, match largest angle to largest), then confirm all three side ratios are equal-the common ratio is your scale factor, rotations and mirror shenanigans included. Want to test it on a quick example together, or peek at this tidy walkthrough from Khan Academy: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-similarity/a/corresponding-parts-of-similar-figures?
Here’s my go-to: ignore the order and orientation, rank the sides in each triangle as shortest/middle/longest (or match vertices by “longest side ⇔ largest angle”), then pair short↔short, mid↔mid, long↔long to get the correspondence. The scale factor is the common ratio of those pairs (k = long₂/long₁ = mid₂/mid₁ = short₂/short₁)-like matching socks by size, not by which way the toes point.
Love this! When a triangle’s been spun, flipped, and its labels scrambled, my no-nonsense trick is the small–medium–large method: in each triangle, rank the sides (or angles) by size, then relabel the vertices so A is opposite the smallest angle/side, B the middle, C the largest-do the same on the other triangle, and boom: A ↔ A′, B ↔ B′, C ↔ C′ are your corresponding parts. This survives any rotation or mirror because similarity preserves the order of sizes. Then grab any matched pair to get the scale factor k = (length in image)/(length in original), and quickly confirm with a second pair to avoid a sneaky mismatch. If you only have side lengths, compute side ratios across the two triangles; the two equal ratios point to the corresponding sides, and that common value is k. I like thinking of it like matching T-shirt sizes or nesting measuring cups: smallest with smallest, middle with middle, largest with largest-orientation doesn’t matter. I used to trip over mirrored triangles in a practice set until a teacher had me literally jot S, M, L on the sides; once I did that, the vertex order fell into place and the scale factor popped right out. For a tidy walkthrough with examples, this Khan Academy page is great: https://www.khanacademy.org/math/geometry/hs-geo-similarity/hs-geo-similarity/e/determine-scale-factor-similar-figures.