We claim that all circles are similar. Proof. Consider two circles - PowerPoint PPT Presentation

D AY 121 – S IMILARITY IN CIRCLES

I NTRODUCTION When we talk of similarity, we always think of geometrical figures with sides and vertices, that is, triangles, quadrilaterals, pentagon among others. However, we may also be interesting to compare others shapes and see what we can say with respect to similarity. These shapes may be ellipse, circles, sectors among others. In this lesson, we are going to prove that all circles are similar.

V OCABULARY Circle A plane figure where all points on it are at equal distance from a fixed point called the center. Similarity A term used to imply that two more figures are proportional in shape and size. Likewise, two images are similar of there is a dilation followed by a translation

Prove similarity in circles We claim that all circles are similar. Proof. Consider two circles of radius 𝑠 and 𝑆 not equal where 𝑆 > 𝑠. Let the radius be picked arbitrarily. 𝑠 𝑆 𝑃 1 𝑃 2

If we translate the small circle to the larger circle such that their centers have at a common point, we get two concentric circle. 𝑆 𝑠 𝑃 2 Since the points on a circle are equidistant from the center, when we multiply each of the points on smaller circle by a suitable scale factor,

𝑆 Let the scale factor be 𝑙 = 𝑠 , we get the bigger circle. Since the radii were picked arbitrarily, the circles represent any possible two sets of a circles that we can think off. Hence given any two circles, we get a translation and a dilation that can map one circle onto another showing that any two circles are similar.

Example A circle has a radius of 4 in which another has a radius of 5. Show that the circles are similar hence find the scale factor of similarity. Solution Circles are equidistance from the center hence when the centers are moved made to be common, the circles are concentric. The translation is involved. Further, when we multiply by a scale factor all 5 points on the smaller circle by 4 , we get all points 5 on the larger circle. The scale factor is 4 .

HOMEWORK Find the scale factor that maps circle of radius 3 in to that of radius 9 in if they are concentric.

A NSWERS TO HOMEWORK 3

THE END