{"id":2535,"date":"2024-08-05T22:01:55","date_gmt":"2024-08-05T22:01:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2535"},"modified":"2025-08-15T16:35:13","modified_gmt":"2025-08-15T16:35:13","slug":"circles-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/circles-learn-it-1\/","title":{"raw":"Circles: Learn It 1","rendered":"Circles: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write the equations for circles using the standard form<\/li>\r\n \t<li>Graph a circle<\/li>\r\n \t<li>Solve system of equations involving circles<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h2>\r\nA circle is one of the most familiar shapes in geometry. It's defined as the set of all points in a plane that are the same distance from a given point in the plane. The given point is called the center, [latex](h,k)[\/latex], and the fixed distance is called the radius, [latex]r[\/latex], of the circle.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>equation of a circle<\/h3>\r\nThe standard form of the equation of a circle with center at <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex](h,k)[\/latex] and radius [latex]r[\/latex] is:<\/span>\r\n<p style=\"text-align: center;\">[latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/p>\r\n\r\n<div class=\"page\" title=\"Page 849\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_2539\" align=\"aligncenter\" width=\"592\"]<img class=\"wp-image-2539 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM.png\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" width=\"592\" height=\"340\" \/> Circle with key features labeled[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Write the standard form of the equation of the circle with\r\n<ul>\r\n \t<li>radius [latex]3[\/latex] and center [latex](0,0)[\/latex].<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"397116\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"397116\"]\r\n\r\nSubstitute in the values [latex](h, k)=(0,0)[\/latex], where [latex]h=0[\/latex], [latex]k=0[\/latex], and simplify.\r\n<p style=\"text-align: center;\">[latex](x-(0))^2+(y-(0))^2 = 3^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x^2+y^2 = 9[\/latex]<\/p>\r\n[\/hidden-answer]\r\n<ul>\r\n \t<li>radius [latex]2[\/latex] and center [latex](-1,3)[\/latex].<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"307324\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"307324\"]\r\n\r\nSubstitute in the values [latex](h, k)=(1, -3)[\/latex], where [latex]h=1[\/latex], [latex]k=-3[\/latex], and simplify.\r\n<p style=\"text-align: center;\">[latex](x-(-1))^2+(y-(3))^2 = 2^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](x+1)^2+(y-3)^2 = 4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24875[\/ohm2_question]<\/section>\r\n<h2 data-type=\"title\">Graph a Circle<\/h2>\r\nGraphing a circle involves a few key steps that build upon our understanding of the standard form equation [latex](x-h)^2+(y-k)^2 = r^2[\/latex]. Let's break down the process.\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Graph a Circle\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify the Center and Radius<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">The center is given by the point [latex](h,k)[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The radius is the square root of the right side of the equation, [latex]r[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Plot the Center<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Locate and mark the point [latex](h,k)[\/latex] on the coordinate plane.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Draw the Radius<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">From the center, measure [latex]r[\/latex] units in all directions.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">You can do this by measuring [latex]r[\/latex] units right, left, up, and down from the center.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Complete the Circle<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Connect the points to form a smooth, round shape.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Check Key Points<\/strong>\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Verify the four points where the circle intersects the axes:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](h+r,k)[\/latex], [latex](h-r,k)[\/latex], [latex](h,k+r)[\/latex], and [latex](h,k-r)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the center and radius and then graph the circle:\r\n<p style=\"text-align: center;\">[latex](x+3)^2 + (y+4)^2 = 4[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"324817\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"324817\"]<\/p>\r\nUse the standard form of the equation of a circle. Identify the center, [latex](h,k)[\/latex] and radius, [latex]r[\/latex].\r\n<p style=\"text-align: center;\">[latex](x+3)^2 + (y+4)^2 = 4[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](x-(-3))^2 + (y-(-4))^2 = 2^2[\/latex]<\/p>\r\n<strong>Thus, the center is [latex](-3, -4)[\/latex] and the radius is [latex]2[\/latex].<\/strong>\r\n\r\nGraph the circle:\r\n\r\n[caption id=\"attachment_2537\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-2537\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"292\" \/> Circle on a coordinate plane with center and radius labeled[\/caption]\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24876[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write the equations for circles using the standard form<\/li>\n<li>Graph a circle<\/li>\n<li>Solve system of equations involving circles<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h2>\n<p>A circle is one of the most familiar shapes in geometry. It&#8217;s defined as the set of all points in a plane that are the same distance from a given point in the plane. The given point is called the center, [latex](h,k)[\/latex], and the fixed distance is called the radius, [latex]r[\/latex], of the circle.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>equation of a circle<\/h3>\n<p>The standard form of the equation of a circle with center at <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex](h,k)[\/latex] and radius [latex]r[\/latex] is:<\/span><\/p>\n<p style=\"text-align: center;\">[latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/p>\n<div class=\"page\" title=\"Page 849\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>&nbsp;<\/p>\n<figure id=\"attachment_2539\" aria-describedby=\"caption-attachment-2539\" style=\"width: 592px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2539 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM.png\" alt=\"Figure shows circle with center at (h, k) and a radius of r. A point on the circle is labeled x, y. The formula is open parentheses x minus h close parentheses squared plus open parentheses y minus k close parentheses squared equals r squared.\" width=\"592\" height=\"340\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM.png 592w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM-300x172.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM-65x37.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM-225x129.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05220817\/Screenshot-2024-08-05-at-3.08.12%E2%80%AFPM-350x201.png 350w\" sizes=\"(max-width: 592px) 100vw, 592px\" \/><figcaption id=\"caption-attachment-2539\" class=\"wp-caption-text\">Circle with key features labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Write the standard form of the equation of the circle with<\/p>\n<ul>\n<li>radius [latex]3[\/latex] and center [latex](0,0)[\/latex].<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q397116\">Show Answer<\/button><\/p>\n<div id=\"q397116\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute in the values [latex](h, k)=(0,0)[\/latex], where [latex]h=0[\/latex], [latex]k=0[\/latex], and simplify.<\/p>\n<p style=\"text-align: center;\">[latex](x-(0))^2+(y-(0))^2 = 3^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x^2+y^2 = 9[\/latex]<\/p>\n<\/div>\n<\/div>\n<ul>\n<li>radius [latex]2[\/latex] and center [latex](-1,3)[\/latex].<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q307324\">Show Answer<\/button><\/p>\n<div id=\"q307324\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute in the values [latex](h, k)=(1, -3)[\/latex], where [latex]h=1[\/latex], [latex]k=-3[\/latex], and simplify.<\/p>\n<p style=\"text-align: center;\">[latex](x-(-1))^2+(y-(3))^2 = 2^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex](x+1)^2+(y-3)^2 = 4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24875\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24875&theme=lumen&iframe_resize_id=ohm24875&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2 data-type=\"title\">Graph a Circle<\/h2>\n<p>Graphing a circle involves a few key steps that build upon our understanding of the standard form equation [latex](x-h)^2+(y-k)^2 = r^2[\/latex]. Let&#8217;s break down the process.<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Graph a Circle<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Identify the Center and Radius<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The center is given by the point [latex](h,k)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">The radius is the square root of the right side of the equation, [latex]r[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Plot the Center<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Locate and mark the point [latex](h,k)[\/latex] on the coordinate plane.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Draw the Radius<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">From the center, measure [latex]r[\/latex] units in all directions.<\/li>\n<li class=\"whitespace-normal break-words\">You can do this by measuring [latex]r[\/latex] units right, left, up, and down from the center.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Complete the Circle<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Connect the points to form a smooth, round shape.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Check Key Points<\/strong>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Verify the four points where the circle intersects the axes:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](h+r,k)[\/latex], [latex](h-r,k)[\/latex], [latex](h,k+r)[\/latex], and [latex](h,k-r)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the center and radius and then graph the circle:<\/p>\n<p style=\"text-align: center;\">[latex](x+3)^2 + (y+4)^2 = 4[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q324817\">Show Answer<\/button><\/p>\n<div id=\"q324817\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the standard form of the equation of a circle. Identify the center, [latex](h,k)[\/latex] and radius, [latex]r[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex](x+3)^2 + (y+4)^2 = 4[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex](x-(-3))^2 + (y-(-4))^2 = 2^2[\/latex]<\/p>\n<p><strong>Thus, the center is [latex](-3, -4)[\/latex] and the radius is [latex]2[\/latex].<\/strong><\/p>\n<p>Graph the circle:<\/p>\n<figure id=\"attachment_2537\" aria-describedby=\"caption-attachment-2537\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2537\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM.png\" alt=\"\" width=\"400\" height=\"292\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM.png 592w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM-300x219.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM-65x47.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM-225x164.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/05215625\/Screenshot-2024-08-05-at-2.56.21%E2%80%AFPM-350x255.png 350w\" sizes=\"(max-width: 400px) 100vw, 400px\" \/><figcaption id=\"caption-attachment-2537\" class=\"wp-caption-text\">Circle on a coordinate plane with center and radius labeled<\/figcaption><\/figure>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24876\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24876&theme=lumen&iframe_resize_id=ohm24876&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":345,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2535"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2535\/revisions"}],"predecessor-version":[{"id":7879,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2535\/revisions\/7879"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/345"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2535\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2535"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2535"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2535"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}