Similarity of circles

Learning Objectives Define similarity in the context of circles. Prove that any two circles are similar using transformations. Determine the scale factor and translation that maps one circle onto another. Calculate the scale factor of two circles given their radii, diameters, circumferences, or areas. Relate the ratio of radii to the ratio of circumferences and the ratio of areas of two similar circles. Apply the properties of similar circles to solve geometric problems. Ever used the pinch-to-zoom feature on a map to make a circular city boundary bigger or smaller? 🗺️ You were using the principle of circle similarity! In this tutorial, we will explore a fundamental property of circles: they are all similar to each other. We will prove this concept using geometric transform...

TermDefinitionExample SimilarityTwo geometric figures are similar if there is a sequence of rigid motions (translation, rotation, reflection) followed by a dilation that maps one figure exactly onto the other. Similar figures have the same shape but can be different sizes.A 5x7 photo and an 8x11.2 photo of the same image are similar. One is an enlargement of the other. DilationA transformation that enlarges or reduces a figure by a specific scale factor about a fixed point called the center of dilation. It preserves angles but not side lengths.Applying a dilation with a scale factor of 2 to a circle of radius 3 cm results in a circle of radius 6 cm. Scale Factor (k)The ratio of the lengths of corresponding sides of two similar figures. For circles, it is the ratio of their radii.If Circle...

Proof of Circle Similarity Any two circles can be proven similar through a two-step transformation: 1. Translation, 2. Dilation. First, translate the center of the first circle to coincide with the center of the second circle. Second, dilate the translated circle by a scale factor equal to the ratio of the second circle's radius to the first circle's radius. Scale Factor of Circles k = r_2 / r_1 = d_2 / d_1 The scale factor (k) that maps Circle 1 onto Circle 2 is the ratio of their radii (r_2 / r_1) or the ratio of their diameters (d_2 / d_1). Ratio of Circumferences and Areas Ratio of Circumferences: C_2 / C_1 = k. Ratio of Areas: A_2 / A_1 = k^2. The ratio of the circumferences of two similar circles is equal to their scale factor (k). The ratio of their...