Similar Figures via Dilation

Pre-Algebra · Axiom Academy

LESSON Similar Figures via Dilation Two figures are similar exactly when one is a dilation of the other — same angles, every side scaled by the same factor. A dilation pushes every point of a figure directly toward or away from a fixed center , multiplying its distance from that center by a scale factor k . Below, a triangle is dilated from the red center by k = 2 : each vertex slides out along its own ray until it is twice as far, so every side comes out twice as long — yet the angles never change. Each point moves to k times its distance from the center So every length is multiplied by k What a dilation keeps and what it changes Angles stay the same · the shape stays the same · every length is multiplied by k · the area is multiplied by k^2 . 2. When Are Two Figures Similar? Two figures are similar when one is a dilation of the other (possibly also rotated, reflected, or moved). That single condition forces two things you can check directly: every pair of corresponding angles is equal , and every pair of corresponding sides has the same ratio . Below, triangle B is a dilation of triangle A — watch each side pair light up and report the same ratio. Corresponding angles match exactly — a dilation never bends the corners. Each side of B is the matching side of A times the same k . — all corresponding ratios agree. You may slide, turn, or flip a figure first — only shape counts. 3. Why All Circles — and All Squares — Are Similar

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