{"id":2791,"date":"2025-08-13T18:40:43","date_gmt":"2025-08-13T18:40:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2791"},"modified":"2025-10-22T20:18:44","modified_gmt":"2025-10-22T20:18:44","slug":"parametric-functions-and-vectors-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-functions-and-vectors-background-youll-need-3\/","title":{"raw":"Parametric Functions and Vectors: Background You'll Need 3","rendered":"Parametric Functions and Vectors: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use the Pythagorean Theorem to find the distance between two points<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<p data-start=\"3012\" data-end=\"3218\">The Pythagorean Theorem connects directly to the distance formula on the coordinate plane. To find the distance between two points, think of the segment between them as the hypotenuse of a right triangle.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>distance formula<\/h3>\r\n<p data-start=\"3234\" data-end=\"3301\">For points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]:<\/p>\r\n<p data-start=\"3234\" data-end=\"3301\"><\/p>\r\n<p style=\"text-align: center;\" data-start=\"3303\" data-end=\"3387\">[latex]\r\n\\begin{align}\r\nd &amp;= \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"3303\" data-end=\"3387\"><\/p>\r\n<p data-start=\"3389\" data-end=\"3444\">This formula comes from [latex]a^2 + b^2 = c^2[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p data-start=\"3460\" data-end=\"3534\">Find the distance between [latex](1, 2)[\/latex] and [latex](5, 8)[\/latex].<\/p>\r\n<p data-start=\"3536\" data-end=\"3689\">[latex]\r\n\\begin{align}\r\nd &amp;= \\sqrt{(5 - 1)^2 + (8 - 2)^2} \\\\\r\n&amp;= \\sqrt{4^2 + 6^2} \\\\\r\n&amp;= \\sqrt{16 + 36} \\\\\r\n&amp;= \\sqrt{52} \\\\\r\n&amp;\\approx 7.21\r\n\\end{align}\r\n[\/latex]<\/p>\r\n<p data-start=\"3691\" data-end=\"3751\">The distance between the points is approximately 7.21 units.<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]313902[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use the Pythagorean Theorem to find the distance between two points<\/span><\/li>\n<\/ul>\n<\/section>\n<p data-start=\"3012\" data-end=\"3218\">The Pythagorean Theorem connects directly to the distance formula on the coordinate plane. To find the distance between two points, think of the segment between them as the hypotenuse of a right triangle.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>distance formula<\/h3>\n<p data-start=\"3234\" data-end=\"3301\">For points [latex](x_1, y_1)[\/latex] and [latex](x_2, y_2)[\/latex]:<\/p>\n<p data-start=\"3234\" data-end=\"3301\">\n<p style=\"text-align: center;\" data-start=\"3303\" data-end=\"3387\">[latex]\\begin{align}  d &= \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}  \\end{align}[\/latex]<\/p>\n<p data-start=\"3303\" data-end=\"3387\">\n<p data-start=\"3389\" data-end=\"3444\">This formula comes from [latex]a^2 + b^2 = c^2[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p data-start=\"3460\" data-end=\"3534\">Find the distance between [latex](1, 2)[\/latex] and [latex](5, 8)[\/latex].<\/p>\n<p data-start=\"3536\" data-end=\"3689\">[latex]\\begin{align}  d &= \\sqrt{(5 - 1)^2 + (8 - 2)^2} \\\\  &= \\sqrt{4^2 + 6^2} \\\\  &= \\sqrt{16 + 36} \\\\  &= \\sqrt{52} \\\\  &\\approx 7.21  \\end{align}[\/latex]<\/p>\n<p data-start=\"3691\" data-end=\"3751\">The distance between the points is approximately 7.21 units.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm313902\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313902&theme=lumen&iframe_resize_id=ohm313902&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"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":520,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2791"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2791\/revisions"}],"predecessor-version":[{"id":4827,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2791\/revisions\/4827"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2791\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2791"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2791"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2791"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2791"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}