Meaning of Similar Figures and Similar Triangles
Simple Explanation
Two figures are similar if they have exactly the same shape, though possibly different sizes — one is an enlargement or reduction of the other. For triangles specifically, this means corresponding angles are equal and corresponding sides are all in the same fixed ratio (the scale factor).
Why Do We Need It?
Similarity is how mathematics captures the everyday idea of "same shape, different size" — the foundation for scale models, maps, photo enlargements, and every theorem in this chapter.
See It
Triangle ABC and a larger triangle A-prime B-prime C-prime with the same shape, showing corresponding vertices
Formula
Similarity Ratio
AB/A'B' = BC/B'C' = CA/C'A' = k, and corresponding angles are equal
Two triangles are similar exactly when their corresponding angles are equal and their corresponding sides are all in the same fixed ratio, k (the scale factor).
- k
- — the scale factor — how many times larger (or smaller) one triangle is than the other
- AB, BC, CA / A'B', B'C', C'A'
- — corresponding side lengths of the two similar triangles
When to use it: Whenever you need to confirm two triangles are similar, or use their known similarity to find a missing side length.
Worked Example
Find a missing side using similarity
△ABC ~ △DEF (read "is similar to"), with AB=4, BC=6, and DE=6. Find EF.
Why Does This Work?
Similarity means every length in one figure is scaled by the exact same factor k to produce the other figure — so any two corresponding sides must be in that same ratio k, which is why setting corresponding side ratios equal to each other always gives a correct proportion.
Real-Life Example
Scale models and maps
An architect's scale model of a building, or a road map of a city, are both smaller similar copies of the real thing.
Every real-world distance is shrunk by the same scale factor in the model or map — exactly the mathematical definition of similarity in action.
Practice
What must be true for two triangles to be similar?
EasyCommon mistake
Confusing similar (same shape, possibly different size) with congruent (same shape AND same size) — congruent figures are always similar with scale factor 1, but similar figures are not always congruent.
Quick Review
- Similar figures have the same shape; corresponding angles are equal, corresponding sides share one common ratio.
- AB/A'B' = BC/B'C' = CA/C'A' — this common ratio is the scale factor.
- Congruent is a special case of similar, with scale factor exactly 1.