{"id":156108,"date":"2022-03-26T01:23:24","date_gmt":"2022-03-25T19:53:24","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/concentric-circles\/"},"modified":"2025-06-23T16:59:00","modified_gmt":"2025-06-23T11:29:00","slug":"concentric-circles","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/","title":{"rendered":"Concentric Circles"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/#Concentric_Circles_Equation\" title=\"Concentric Circles Equation\">Concentric Circles Equation<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/#Concentric_Circles_Meaning\" title=\"Concentric Circles Meaning\">Concentric Circles Meaning<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/#Concentric_Circle_Equations\" title=\"Concentric Circle Equations\">Concentric Circle Equations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/#Concentric_Circles_-_Theorem\" title=\"Concentric Circles \u2013 Theorem\">Concentric Circles \u2013 Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/concentric-circles\/#Proof\" title=\"Proof\">Proof<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Concentric_Circles_Equation\"><\/span>Concentric Circles Equation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A concentric circle equation is a mathematical equation that defines a circle that has a center point and radii that are all the same length. The equation can be used to define the location of the center point and the size of the circle.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156107 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/concentric-circles.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/concentric-circles.jpg?v=1648238002 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/concentric-circles-300x212.jpg?v=1648238002 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Concentric_Circles_Meaning\"><\/span>Concentric Circles Meaning<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"aligncenter\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/09\/concentric-circles.jpg\" alt=\"Concentric Circles\" width=\"249\" height=\"250\" \/><\/p>\n<p>The circles with a common centre are known as concentric circles and have different radii. In other words, it is defined as two or more circles that have the same centre point. The region between two concentric circles are of different radii is known as an annulus.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Concentric_Circle_Equations\"><\/span>Concentric Circle Equations<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Let the equation of the circle with centre (-g, -f) and radius \u221a[g<sup>2<\/sup>+f<sup>2<\/sup>-c] be<\/p>\n<p>x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c =0<\/p>\n<p>Therefore, the equation of the circle concentric with the other circle be<\/p>\n<p>x<sup>2<\/sup> + y<sup>2<\/sup> + 2gx + 2fy + c\u2019 =0<\/p>\n<p>It is observed that both the equations have the same centre (-g, -f), but they have different radii, where c\u2260 c\u2019<\/p>\n<p>Similarly, a circle with centre (h, k), and the radius is equal to r, then the equation becomes<\/p>\n<p>( x \u2013 h )<sup>2<\/sup> + ( y \u2013 k )<sup>2 <\/sup>= r<sup>2<\/sup><\/p>\n<p>Therefore, the equation of a circle concentric with the circle is<\/p>\n<p><strong>( x \u2013 h )<sup>2<\/sup> + ( y \u2013 k )<sup>2 <\/sup>= r<sub>1<\/sub><sup>2<\/sup><\/strong><\/p>\n<p>Where r \u2260 r<sub>1<\/sub><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Concentric_Circles_-_Theorem\"><\/span><b>Concentric Circles \u2013 Theorem<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Proof\"><\/span><b>Proof<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Given:<\/strong><\/p>\n<p>Consider two concentric circles C<sub>1<\/sub> and C<sub>2<\/sub>, with centre O and a chord AB of the larger circle C<sub>1<\/sub>, touching the smaller circle C<sub>2<\/sub> at the point P as shown in the figure below.<\/p>\n<p>Construction:<\/p>\n<p>Join OP.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2021\/11\/Concentric-circles-theorem.png\" alt=\"Concentric circles 2\" width=\"560\" height=\"514\" \/><\/p>\n<p><strong>To prove:<\/strong> AP = BP<\/p>\n<p><strong>Proof:<\/strong><\/p>\n<p>Since AB is the chord of larger circle C<sub>1<\/sub>, it becomes the tangent to C<sub>2<\/sub> at P.<\/p>\n<p>OP is the radius of circle C<sub>2<\/sub>.<\/p>\n<p>We know that the radius is perpendicular to the tangent at the point of contact.<\/p>\n<p>So, OP \u22a5 AB<\/p>\n<p>Now AB is a chord of the circle C1 and OP \u22a5 AB.<\/p>\n<p>Therefore, OP is the bisector of the chord AB.<\/p>\n<p>Thus, the perpendicular from the centre bisects the chord, i.e., AP = BP.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Concentric Circles Equation A concentric circle equation is a mathematical equation that defines a circle that has a center point [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Concentric Circles","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"A concentric circle equation is a mathematical equation that defines a circle that has a center point and radii that are all the same length.","custom_permalink":"maths\/concentric-circles\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - 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