{"id":905,"date":"2023-07-10T19:28:17","date_gmt":"2023-07-10T19:28:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=905"},"modified":"2023-08-10T00:19:52","modified_gmt":"2023-08-10T00:19:52","slug":"symmetry-of-a-point","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/symmetry-of-a-point\/","title":{"raw":"\u25aa Symmetry of a Point","rendered":"\u25aa Symmetry of a Point"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OUTCOMES<\/h3>\r\n<ul>\r\n \t<li>Find points that are symmetric to a point about the x-axis, the y-axis, and the origin.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Symmetry of a Point<\/h2>\r\nLet's consider a point [latex](2, 5)[\/latex] on a coordinate plane. What will happen if reflect the point across the [latex]x[\/latex]-axis, the [latex]y[\/latex]-axis, and the origin?\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"none\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\r\n<div class=\"textbox shaded\"><img class=\"aligncenter wp-image-911 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis.png\" alt=\"\" width=\"492\" height=\"489\" \/>(a) When (2, 5) is reflected across the x-axis, it becomes (2, -5). Those two points have the same distance from the x-axis.<\/div><\/td>\r\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\r\n<div class=\"textbox shaded\"><img class=\"aligncenter wp-image-912 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis.png\" alt=\"\" width=\"492\" height=\"492\" \/>(b)\u00a0When (2, 5) is reflected across the y-axis, it becomes (-2, 5). Those two points have the same distance from the y-axis.<\/div><\/td>\r\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\r\n<div class=\"textbox shaded\"><img class=\"aligncenter wp-image-910 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin.png\" alt=\"(a, b) becomes (-a, -b) when a point reflected across the origin\" width=\"491\" height=\"491\" \/>(c) When (2, 5) is reflected across the origin, which is reflected across the x-axis and the y-axis, it becomes (-2, -5). Those two points have the same distance from the origin.<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>\r\n<div class=\"textbox shaded\">Figure 1. (a) The point that is symmetric to [latex](2, 5)[\/latex] about the [latex]x[\/latex]-axis.\u00a0 \u00a0(b) The point that is symmetric to [latex](2, 5)[\/latex] about the [latex]y[\/latex]-axis.\u00a0 \u00a0(c)\u00a0The point that is symmetric to [latex](2, 5)[\/latex] about the origin.<\/div>\r\nUse the following DESMOS exercise to investigate more: [DESMOS]\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/w2el4zv9d7\" target=\"_blank\" rel=\"noopener\">Symmetry of a Point<\/a>\u00a0or [DESMOS Classroom]\u00a0<a href=\"https:\/\/student.desmos.com\/activitybuilder\/student-greeting\/64b18ee3a12ac9096fe3192a\" target=\"_blank\" rel=\"noopener\">Symmetry of a Point<\/a>\r\n<div class=\"textbox\">\r\n<h3>General Note: Symmetry of a Point<\/h3>\r\n(a) When we reflect a point <strong>across the <\/strong><strong>x-axis<\/strong>, its <strong>y<\/strong><strong>-coordinate<\/strong> will become its <strong>opposite<\/strong> number while its x-coordinate stays as is. So, <strong>the point that is symmetric to <\/strong><strong>(a, b)<\/strong><strong>\u00a0about the <\/strong><strong>x<\/strong><strong>-axis is <\/strong><strong>(a, -b)<\/strong>.\r\n\r\n(b) When we reflect a point <strong>across the y-axis<\/strong>, its <strong>x-coordinate<\/strong> will become its <strong>opposite<\/strong> number while its y-coordinate stays as is. So, <strong>the point that is symmetric to (a, b) about the y-axis is (-a, b)<\/strong>.\r\n\r\n(c)\u00a0When we reflect a point <strong>across the origin<\/strong>, <strong>both x- and y-coordinates<\/strong> will become its <strong>opposite<\/strong> number, respectively.\u00a0So, <strong>the point that is symmetric to (a, b) about the origin is (-a, -b).<\/strong>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE: Symmetry of a Point<\/h3>\r\nFind the points that are symmetric to it about (a) the [latex]x[\/latex]-axis, (b) the [latex]y[\/latex]-axis, and (c) the origin.\r\n<ol>\r\n \t<li>[latex](-7, 9)[\/latex]<\/li>\r\n \t<li>[latex](3, -1)[\/latex]<\/li>\r\n \t<li>[latex](-11, -8)[\/latex]<\/li>\r\n \t<li>[latex](6, 0)[\/latex]<\/li>\r\n \t<li>[latex](0, -5)[\/latex]<\/li>\r\n \t<li>[latex](0, 0)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"153207\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"153207\"]1. (a) [latex](-7, -9)[\/latex]; (b) [latex](7, 9)[\/latex]; (c) [latex](7, -9)[\/latex]\r\n\r\n2. (a) [latex](3, 1)[\/latex]; (b) [latex](-3, -1)[\/latex]; (c) [latex](-3, 1)[\/latex]\r\n\r\n3. (a) [latex](-11, 8)[\/latex]; (b) [latex](11, -8)[\/latex]; (c) [latex](11, 8)[\/latex]\r\n\r\n4. (a) [latex](6, 0)[\/latex]; (b) [latex](-6, 0)[\/latex]; (c) [latex](-6, 0)[\/latex]\r\n\r\n5. (a) [latex](0, 5)[\/latex]; (b) [latex](0, -5)[\/latex]; (c) [latex](0, 5)[\/latex]\r\n\r\n6. (a) [latex](0, 0)[\/latex]; (b) [latex](0, 0)[\/latex]; (c) [latex](0, 0)[\/latex][\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OUTCOMES<\/h3>\n<ul>\n<li>Find points that are symmetric to a point about the x-axis, the y-axis, and the origin.<\/li>\n<\/ul>\n<\/div>\n<h2>Symmetry of a Point<\/h2>\n<p>Let&#8217;s consider a point [latex](2, 5)[\/latex] on a coordinate plane. What will happen if reflect the point across the [latex]x[\/latex]-axis, the [latex]y[\/latex]-axis, and the origin?<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-911 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis.png\" alt=\"\" width=\"492\" height=\"489\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis.png 492w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis-150x150.png 150w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis-300x298.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis-65x65.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis-225x224.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_xaxis-350x348.png 350w\" sizes=\"auto, (max-width: 492px) 100vw, 492px\" \/>(a) When (2, 5) is reflected across the x-axis, it becomes (2, -5). Those two points have the same distance from the x-axis.<\/div>\n<\/td>\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-912 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis.png\" alt=\"\" width=\"492\" height=\"492\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis.png 492w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis-150x150.png 150w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis-300x300.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis-65x65.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis-225x225.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_yaxis-350x350.png 350w\" sizes=\"auto, (max-width: 492px) 100vw, 492px\" \/>(b)\u00a0When (2, 5) is reflected across the y-axis, it becomes (-2, 5). Those two points have the same distance from the y-axis.<\/div>\n<\/td>\n<td style=\"width: 33.3333%; text-align: center; vertical-align: middle;\" scope=\"rowgroup\">\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-910 size-full\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin.png\" alt=\"(a, b) becomes (-a, -b) when a point reflected across the origin\" width=\"491\" height=\"491\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin.png 491w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin-150x150.png 150w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin-300x300.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin-65x65.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin-225x225.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/1.5-Symmtry-of-a-Point_origin-350x350.png 350w\" sizes=\"auto, (max-width: 491px) 100vw, 491px\" \/>(c) When (2, 5) is reflected across the origin, which is reflected across the x-axis and the y-axis, it becomes (-2, -5). Those two points have the same distance from the origin.<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>\n<div class=\"textbox shaded\">Figure 1. (a) The point that is symmetric to [latex](2, 5)[\/latex] about the [latex]x[\/latex]-axis.\u00a0 \u00a0(b) The point that is symmetric to [latex](2, 5)[\/latex] about the [latex]y[\/latex]-axis.\u00a0 \u00a0(c)\u00a0The point that is symmetric to [latex](2, 5)[\/latex] about the origin.<\/div>\n<p>Use the following DESMOS exercise to investigate more: [DESMOS]\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\/w2el4zv9d7\" target=\"_blank\" rel=\"noopener\">Symmetry of a Point<\/a>\u00a0or [DESMOS Classroom]\u00a0<a href=\"https:\/\/student.desmos.com\/activitybuilder\/student-greeting\/64b18ee3a12ac9096fe3192a\" target=\"_blank\" rel=\"noopener\">Symmetry of a Point<\/a><\/p>\n<div class=\"textbox\">\n<h3>General Note: Symmetry of a Point<\/h3>\n<p>(a) When we reflect a point <strong>across the <\/strong><strong>x-axis<\/strong>, its <strong>y<\/strong><strong>-coordinate<\/strong> will become its <strong>opposite<\/strong> number while its x-coordinate stays as is. So, <strong>the point that is symmetric to <\/strong><strong>(a, b)<\/strong><strong>\u00a0about the <\/strong><strong>x<\/strong><strong>-axis is <\/strong><strong>(a, -b)<\/strong>.<\/p>\n<p>(b) When we reflect a point <strong>across the y-axis<\/strong>, its <strong>x-coordinate<\/strong> will become its <strong>opposite<\/strong> number while its y-coordinate stays as is. So, <strong>the point that is symmetric to (a, b) about the y-axis is (-a, b)<\/strong>.<\/p>\n<p>(c)\u00a0When we reflect a point <strong>across the origin<\/strong>, <strong>both x- and y-coordinates<\/strong> will become its <strong>opposite<\/strong> number, respectively.\u00a0So, <strong>the point that is symmetric to (a, b) about the origin is (-a, -b).<\/strong><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE: Symmetry of a Point<\/h3>\n<p>Find the points that are symmetric to it about (a) the [latex]x[\/latex]-axis, (b) the [latex]y[\/latex]-axis, and (c) the origin.<\/p>\n<ol>\n<li>[latex](-7, 9)[\/latex]<\/li>\n<li>[latex](3, -1)[\/latex]<\/li>\n<li>[latex](-11, -8)[\/latex]<\/li>\n<li>[latex](6, 0)[\/latex]<\/li>\n<li>[latex](0, -5)[\/latex]<\/li>\n<li>[latex](0, 0)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q153207\">Show Answer<\/span><\/p>\n<div id=\"q153207\" class=\"hidden-answer\" style=\"display: none\">1. (a) [latex](-7, -9)[\/latex]; (b) [latex](7, 9)[\/latex]; (c) [latex](7, -9)[\/latex]<\/p>\n<p>2. (a) [latex](3, 1)[\/latex]; (b) [latex](-3, -1)[\/latex]; (c) [latex](-3, 1)[\/latex]<\/p>\n<p>3. (a) [latex](-11, 8)[\/latex]; (b) [latex](11, -8)[\/latex]; (c) [latex](11, 8)[\/latex]<\/p>\n<p>4. (a) [latex](6, 0)[\/latex]; (b) [latex](-6, 0)[\/latex]; (c) [latex](-6, 0)[\/latex]<\/p>\n<p>5. (a) [latex](0, 5)[\/latex]; (b) [latex](0, -5)[\/latex]; (c) [latex](0, 5)[\/latex]<\/p>\n<p>6. (a) [latex](0, 0)[\/latex]; (b) [latex](0, 0)[\/latex]; (c) [latex](0, 0)[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-905\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Symmetry of a Point. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University . <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":705214,"menu_order":33,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Symmetry of a Point\",\"author\":\"Michelle Eunhee Chung\",\"organization\":\"Georgia State University \",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mchung12"],"pb_section_license":""},"chapter-type":[],"contributor":[62],"license":[],"class_list":["post-905","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/905","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/705214"}],"version-history":[{"count":47,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/905\/revisions"}],"predecessor-version":[{"id":1008,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/905\/revisions\/1008"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/905\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=905"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=905"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=905"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}