{"id":15384,"date":"2021-10-11T23:05:41","date_gmt":"2021-10-11T23:05:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/read-pythagorean-theorem\/"},"modified":"2021-10-16T20:53:22","modified_gmt":"2021-10-16T20:53:22","slug":"read-pythagorean-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/read-pythagorean-theorem\/","title":{"raw":"Pythagorean Theorem","rendered":"Pythagorean Theorem"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve application problems involving quadratic equations<\/li>\r\n \t<li>Recognize a right triangle from other types of triangles<\/li>\r\n \t<li>Use the Pythagorean theorem to find the lengths of a right triangle<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_4903\" align=\"aligncenter\" width=\"821\"]<img class=\" wp-image-4903\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15032614\/Screen-Shot-2016-06-14-at-8.25.06-PM-300x99.png\" alt=\"Four types of triangles, scalene, right, equaliateral, and isosceles.\" width=\"821\" height=\"271\" \/> Triangles[\/caption]\r\n\r\nThe <b>Pythagorean theorem,<\/b> or <b>Pythagoras's theorem,<\/b> is a statement about the sides of a right triangle.\u00a0One of the angles of a right triangle is always equal to [latex]90[\/latex] degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The image above\u00a0shows four common kinds of triangle, including a right triangle.\r\n\r\n[caption id=\"attachment_4904\" align=\"alignleft\" width=\"205\"]<img class=\" wp-image-4904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15033329\/Screen-Shot-2016-06-14-at-8.32.56-PM-268x300.png\" alt=\"right triangle labeled with teh longest length = a, and the other two b and c.\" width=\"205\" height=\"229\" \/> Right Triangle with Sides Labeled[\/caption]\r\n<p style=\"text-align: left;\">The Pythagorean theorem is often used to find unknown lengths of the sides of right triangles. If the longest leg of a right triangle is labeled c, and the other two a, and b as in the image on the left, \u00a0The Pythagorean Theorem states that<\/p>\r\n<p style=\"text-align: center;\">[latex]a^2+b^2=c^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Given enough information, we can solve for an unknown length. \u00a0This relationship has been used for many, many years for things such as celestial navigation and early civil engineering projects. We now have digital GPS and survey equipment that have been programmed to do the calculations for us.<\/p>\r\n<p style=\"text-align: left;\">In the next example we will combine the power of the Pythagorean theorem and what we know about solving quadratic equations to find unknown lengths of right triangles.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA\u00a0right triangle has one leg with length x, another whose length is greater by two, \u00a0and the length of the hypotenuse is greater by four. \u00a0Find the lengths of the sides of the triangle. Use the image below.\r\n\r\n<img class=\"alignnone size-medium wp-image-4907\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15034937\/Screen-Shot-2016-06-14-at-8.49.02-PM-264x300.png\" alt=\"Right triangle with one leg having length = x, one with length= x+2 and the hypotenuse = x+4\" width=\"264\" height=\"300\" \/>\r\n[reveal-answer q=\"133740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133740\"]\r\n\r\n<strong><strong>Read and understand:\u00a0<\/strong><\/strong>We know the lengths of all the sides of a triangle in terms of one side. We also know that the Pythagorean theorem will give us a relationship between the side lengths of a right triangle.\r\n\r\n<strong>Translate:\u00a0<\/strong>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\x^2+\\left(x+2\\right)^2=\\left(x+4\\right)^2\\end{array}[\/latex]<\/p>\r\n<strong>Solve:<\/strong>\u00a0 To solve this equation, we need to start by simplifying the equation and moving all the terms\u00a0to one side.\u00a0 If we can factor it, then we can use the zero product principle to solve.\r\n\r\nFirst, multiply the binomials and simplify so we can see what we are working with.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x^2+\\left(x+2\\right)^2=\\left(x+4\\right)^2\\\\x^2+x^2+4x+4=x^2+8x+16\\\\2x^2+4x+4=x^2+8x+16\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now move all the terms to one side and see if we can factor.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2x^2+4x+4=x^2+8x+16\\\\\\underline{-x^2}\\,\\,\\,\\underline{-8x}\\,\\,\\,\\underline{-16}\\,\\,\\,\\,\\,\\underline{-x^2}\\,\\,\\,\\underline{-8x}\\,\\,\\,\\underline{-16}\\\\x^2-4x-12=0\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This went from a messy looking problem to something promising. We can factor using the shortcut:<\/p>\r\n<p style=\"text-align: left;\">[latex]-6\\cdot{2}=-12,\\text{ and }-6+2=-4[\/latex]<\/p>\r\n<p style=\"text-align: left;\">So we can build our binomial factors with [latex]-6[\/latex] and [latex]2[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(x-6\\right)\\left(x+2\\right)=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Set each factor equal to zero:<\/p>\r\n<p style=\"text-align: left;\">[latex]x-6=0, x=6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]x+2=0, x=-2[\/latex]<\/p>\r\n<p style=\"text-align: left;\"><strong>Interpret:\u00a0<\/strong>Ok, it doesn't make sense to have a length equal to [latex]-2[\/latex], so we can safely throw that solution out. \u00a0The lengths of the sides are as follows:<\/p>\r\n<p style=\"text-align: left;\">[latex]x=6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]x+2=6+2=8[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]x+4=6+4=10[\/latex]<\/p>\r\n<p style=\"text-align: left;\"><strong>Check:\u00a0<\/strong>Since we know the relationship between the sides of a right triangle we can check that we are correct. Sometimes it helps to draw a picture<\/p>\r\n<p style=\"text-align: left;\">.<img class=\"alignnone size-medium wp-image-4908\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15042055\/Screen-Shot-2016-06-14-at-9.20.06-PM-300x263.png\" alt=\"Screen Shot 2016-06-14 at 9.20.06 PM\" width=\"300\" height=\"263\" \/><\/p>\r\n<p style=\"text-align: left;\">We know that [latex]a^2+b^2=c^2[\/latex], so we can substitute the values we found:<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{l}6^2+8^2=10^2\\\\36+64=100\\\\100=100\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Our solution checks out.<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\nThe lengths of the sides of the right triangle are [latex]6, 8[\/latex], and [latex]10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis video example shows another way a quadratic equation can be used to find and unknown length of a right triangle.\r\n\r\nhttps:\/\/youtu.be\/xeP5pRBqsNs\r\n\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]1382[\/ohm_question]\r\n\r\n<\/div>\r\nIf you are interested in celestial navigation and the mathematics behind it, watch this video for fun.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=XWLZKmPU17M\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1DUnwdkjR5Ida8hp57wOwuAws_6OnJsOXH2k9o5cn8O4\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve application problems involving quadratic equations<\/li>\n<li>Recognize a right triangle from other types of triangles<\/li>\n<li>Use the Pythagorean theorem to find the lengths of a right triangle<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_4903\" style=\"width: 831px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4903\" class=\"wp-image-4903\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15032614\/Screen-Shot-2016-06-14-at-8.25.06-PM-300x99.png\" alt=\"Four types of triangles, scalene, right, equaliateral, and isosceles.\" width=\"821\" height=\"271\" \/><\/p>\n<p id=\"caption-attachment-4903\" class=\"wp-caption-text\">Triangles<\/p>\n<\/div>\n<p>The <b>Pythagorean theorem,<\/b> or <b>Pythagoras&#8217;s theorem,<\/b> is a statement about the sides of a right triangle.\u00a0One of the angles of a right triangle is always equal to [latex]90[\/latex] degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side. The image above\u00a0shows four common kinds of triangle, including a right triangle.<\/p>\n<div id=\"attachment_4904\" style=\"width: 215px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4904\" class=\"wp-image-4904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15033329\/Screen-Shot-2016-06-14-at-8.32.56-PM-268x300.png\" alt=\"right triangle labeled with teh longest length = a, and the other two b and c.\" width=\"205\" height=\"229\" \/><\/p>\n<p id=\"caption-attachment-4904\" class=\"wp-caption-text\">Right Triangle with Sides Labeled<\/p>\n<\/div>\n<p style=\"text-align: left;\">The Pythagorean theorem is often used to find unknown lengths of the sides of right triangles. If the longest leg of a right triangle is labeled c, and the other two a, and b as in the image on the left, \u00a0The Pythagorean Theorem states that<\/p>\n<p style=\"text-align: center;\">[latex]a^2+b^2=c^2[\/latex]<\/p>\n<p style=\"text-align: left;\">Given enough information, we can solve for an unknown length. \u00a0This relationship has been used for many, many years for things such as celestial navigation and early civil engineering projects. We now have digital GPS and survey equipment that have been programmed to do the calculations for us.<\/p>\n<p style=\"text-align: left;\">In the next example we will combine the power of the Pythagorean theorem and what we know about solving quadratic equations to find unknown lengths of right triangles.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A\u00a0right triangle has one leg with length x, another whose length is greater by two, \u00a0and the length of the hypotenuse is greater by four. \u00a0Find the lengths of the sides of the triangle. Use the image below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4907\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15034937\/Screen-Shot-2016-06-14-at-8.49.02-PM-264x300.png\" alt=\"Right triangle with one leg having length = x, one with length= x+2 and the hypotenuse = x+4\" width=\"264\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133740\">Show Solution<\/span><\/p>\n<div id=\"q133740\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong><strong>Read and understand:\u00a0<\/strong><\/strong>We know the lengths of all the sides of a triangle in terms of one side. We also know that the Pythagorean theorem will give us a relationship between the side lengths of a right triangle.<\/p>\n<p><strong>Translate:\u00a0<\/strong><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\x^2+\\left(x+2\\right)^2=\\left(x+4\\right)^2\\end{array}[\/latex]<\/p>\n<p><strong>Solve:<\/strong>\u00a0 To solve this equation, we need to start by simplifying the equation and moving all the terms\u00a0to one side.\u00a0 If we can factor it, then we can use the zero product principle to solve.<\/p>\n<p>First, multiply the binomials and simplify so we can see what we are working with.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}x^2+\\left(x+2\\right)^2=\\left(x+4\\right)^2\\\\x^2+x^2+4x+4=x^2+8x+16\\\\2x^2+4x+4=x^2+8x+16\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now move all the terms to one side and see if we can factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}2x^2+4x+4=x^2+8x+16\\\\\\underline{-x^2}\\,\\,\\,\\underline{-8x}\\,\\,\\,\\underline{-16}\\,\\,\\,\\,\\,\\underline{-x^2}\\,\\,\\,\\underline{-8x}\\,\\,\\,\\underline{-16}\\\\x^2-4x-12=0\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">This went from a messy looking problem to something promising. We can factor using the shortcut:<\/p>\n<p style=\"text-align: left;\">[latex]-6\\cdot{2}=-12,\\text{ and }-6+2=-4[\/latex]<\/p>\n<p style=\"text-align: left;\">So we can build our binomial factors with [latex]-6[\/latex] and [latex]2[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x-6\\right)\\left(x+2\\right)=0[\/latex]<\/p>\n<p style=\"text-align: left;\">Set each factor equal to zero:<\/p>\n<p style=\"text-align: left;\">[latex]x-6=0, x=6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]x+2=0, x=-2[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Interpret:\u00a0<\/strong>Ok, it doesn&#8217;t make sense to have a length equal to [latex]-2[\/latex], so we can safely throw that solution out. \u00a0The lengths of the sides are as follows:<\/p>\n<p style=\"text-align: left;\">[latex]x=6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]x+2=6+2=8[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]x+4=6+4=10[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Check:\u00a0<\/strong>Since we know the relationship between the sides of a right triangle we can check that we are correct. Sometimes it helps to draw a picture<\/p>\n<p style=\"text-align: left;\">.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4908\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/15042055\/Screen-Shot-2016-06-14-at-9.20.06-PM-300x263.png\" alt=\"Screen Shot 2016-06-14 at 9.20.06 PM\" width=\"300\" height=\"263\" \/><\/p>\n<p style=\"text-align: left;\">We know that [latex]a^2+b^2=c^2[\/latex], so we can substitute the values we found:<\/p>\n<p style=\"text-align: left;\">[latex]\\begin{array}{l}6^2+8^2=10^2\\\\36+64=100\\\\100=100\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Our solution checks out.<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p>The lengths of the sides of the right triangle are [latex]6, 8[\/latex], and [latex]10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This video example shows another way a quadratic equation can be used to find and unknown length of a right triangle.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Factoring Application - Find the Lengths of Three Sides of a Right Triangle (Pythagorean Theorem)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xeP5pRBqsNs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1382\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1382&theme=oea&iframe_resize_id=ohm1382&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>If you are interested in celestial navigation and the mathematics behind it, watch this video for fun.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Celestial Navigation Math\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/XWLZKmPU17M?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1DUnwdkjR5Ida8hp57wOwuAws_6OnJsOXH2k9o5cn8O4\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/p>\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-15384\">\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>Pythagorean Theorem, Description and Examples. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Factoring Application - Find the Lengths of Three Sides of a Right Triangle (Pythagorean Theorem). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xeP5pRBqsNs\">https:\/\/youtu.be\/xeP5pRBqsNs<\/a>. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Celestial Navigation Math. <strong>Authored by<\/strong>: TabletClass Math. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=XWLZKmPU17M\">https:\/\/www.youtube.com\/watch?v=XWLZKmPU17M<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Pythagorean Theorem. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem\">https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/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":167848,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Celestial Navigation Math\",\"author\":\"TabletClass Math\",\"organization\":\"\",\"url\":\"https:\/\/www.youtube.com\/watch?v=XWLZKmPU17M\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"original\",\"description\":\"Pythagorean Theorem, Description and Examples\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Pythagorean Theorem\",\"author\":\"\",\"organization\":\"Wikipedia\",\"url\":\"https:\/\/en.wikipedia.org\/wiki\/Pythagorean_theorem\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Factoring Application - Find the Lengths of Three Sides of a Right Triangle (Pythagorean Theorem)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/xeP5pRBqsNs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"4f34e13f8ee047d5b04458cd40046ecf, ebf0d16aacf14ecf8993830b8c6c66dc","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15384","chapter","type-chapter","status-publish","hentry"],"part":13821,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15384","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15384\/revisions"}],"predecessor-version":[{"id":15580,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15384\/revisions\/15580"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/13821"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15384\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15384"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15384"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15384"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}