lesson
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When you pinch to zoom in on a photo on your phone, everything gets bigger, but nobody's face gets squished or stretched out of shape.
Geometry has a special name for this exact kind of perfect resizing: similarity.
Dilations and Similarity
A dilation is a geometric transformation that enlarges or reduces a figure proportionally from a fixed point called the center of dilation.
Two figures are similar (written with the symbol ∼) if one can be mapped onto the other through a sequence of rigid motions (slides, turns, or flips) and a dilation.
📊Interactive/animated diagram showing a dilation from a center point O. A small triangle ABC (with side lengths 3, 4, 5) expands outward along dashed projection rays originating at point O to form a larger triangle A'B'C' (with side lengths 6, 8, 10). Rays connect O to A to A', O to B to B', and O to C to C'. The scale factor k = 2 is displayed clearly with a slider or toggle showing 1x to 2x expansion. Color scheme: Navy text #1e2945, background #f8f9fa, original triangle ABC in teal (#0ea5e9), dilated triangle A'B'C' in royal blue (#3b82f6), projection rays in dashed warm gray.
What actually happens to the angles and the sides when a shape expands or shrinks like this?
The Two Rules of Similar Figures
First, corresponding angles are congruent (equal in measure). Expanding a triangle pushes the sides farther apart, but the rays point in the exact same directions, so the angle openings never change.
Second, corresponding sides are proportional. Every single side length of the original figure is multiplied by the same value, called the scale factor (k).
📊Visual comparison diagram of two similar triangles, triangle ABC and triangle DEF. Triangle ABC has angles 40° (green arc at A), 60° (orange arc at B), and 80° (purple arc at C), with side lengths AB = 5, BC = 7, AC = 6. Triangle DEF is scaled up by k = 3: angles are identical (40° at D, 60° at E, 80° at F), while sides are DE = 15, EF = 21, DF = 18. Callout boxes highlight: 1) Angle Match: ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F. 2) Side Ratio: DE/AB = EF/BC = DF/AC = 3. Clear, modern, colorful labels.
How do we write this mathematically to solve for an unknown length on an exam?