Similarity & Congruence
Grades 8–10 Geometry Ratios
Congruent means identical in every way — same shape, same size (copies). Similar means same shape, possibly different size (photos resized, scale models).
Similar figures keep two promises
- Matching angles are equal
- Matching sides are proportional — every side multiplied by the same scale factor
Small triangle: 3, 4, 5
Big triangle (same shape, ×2): 6, 8, 10
Missing side? Small 6 cm matches big 18 cm → scale factor 3.
Big side matching small 10 cm = 10 × 3 = 30 cm
Why it matters
Maps are similar figures. A 1:50,000 map means 1 cm on paper = 50,000 cm = 500 m in the world. Shadows, camera zoom, screen resolutions, and model trains all run on scale factors.
Watch out
Scale factor applies to lengths. Areas scale by the
factor squared, volumes by the factor cubed. A 2× photo
isn't twice the ink — it's 4×.
Practice
Triangle sides 5 cm and 15 cm match — scale factor 3. A 5 cm side matches a ___ cm side on the big one.
Sides 4 ↔ 12 are matching — the scale factor is ___
Similar figures always have equal:
A shape scaled ×2 — its area is multiplied by ___
Next: ratios & proportions · Back: Pythagorean theorem