{"id":16517,"date":"2019-10-03T17:08:41","date_gmt":"2019-10-03T17:08:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-parallel-lines\/"},"modified":"2020-10-22T09:18:06","modified_gmt":"2020-10-22T09:18:06","slug":"read-or-watch-parallel-lines","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-or-watch-parallel-lines\/","title":{"raw":"9.4.c - Slopes of Parallel and Perpendicular Lines","rendered":"9.4.c &#8211; Slopes of Parallel and Perpendicular Lines"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify slopes of parallel and perpendicular lines<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"Characterize the slopes of parallel and perpendicular lines\">Characterize the slopes of parallel and perpendicular lines<\/h2>\r\nWhen you graph two or more linear equations in a coordinate plane, they generally cross at a point. However, when two lines in a coordinate plane never cross, they are called <b>parallel lines<\/b>. You will also look at the case where two lines in a coordinate plane cross at a right angle. These are called <b>perpendicular lines<\/b>. The slopes of the graphs in each of these cases have a special relationship to each other.\r\n\r\nParallel lines are two or more lines in a plane that never intersect. Examples of parallel lines are all around us, such as the opposite sides of a rectangular picture frame and the shelves of a bookcase.\r\n\r\n<img class=\"aligncenter wp-image-1402\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/07212406\/Graphing-Linear-Inequalities-Module-2.png\" alt=\"Line y=2x+3 and line y=2x-3. Caption says Equations of parallel lines will have the same slopes and different intercepts.\" width=\"315\" height=\"243\" \/>\r\n\r\nPerpendicular lines are two or more lines that intersect at a [latex]90[\/latex]-degree angle, like the two lines drawn on this graph. These [latex]90[\/latex]-degree angles are also known as right angles.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064331\/image054.jpg\" alt=\"Two lines that cross to form a 90 degree angle.\" width=\"390\" height=\"340\" \/>\r\n\r\nPerpendicular lines are also everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt.\r\n<div class=\"textbox shaded\">\r\n<h3>Parallel Lines<\/h3>\r\nTwo non-vertical lines in a plane are parallel if they have both:\r\n<ul>\r\n \t<li>the same slope<\/li>\r\n \t<li>different [latex]y[\/latex]-intercepts<\/li>\r\n<\/ul>\r\nAny two vertical lines in a plane are parallel.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the slope of a line parallel to the line [latex]y=\u22123x+4[\/latex].\r\n\r\n[reveal-answer q=\"350329\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"350329\"]\r\n\r\nIdentify the slope of the given line.\r\n\r\nThe given line is written in [latex]y=mx+b[\/latex]\u00a0form, with [latex]m=\u22123[\/latex] and [latex]b=4[\/latex]. The slope is [latex]\u22123[\/latex].\r\n\r\nA line parallel to the given line has the same slope.\r\n<h4>Answer<\/h4>\r\nThe slope of the parallel line is [latex]\u22123[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDetermine whether the lines [latex]y=6x+5[\/latex] and [latex]y=6x\u20131[\/latex]\u00a0are parallel.\r\n\r\n[reveal-answer q=\"619259\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"619259\"]\r\n\r\nIdentify the slopes of the given lines.\r\n\r\nThe given lines are written in [latex]y=mx+b[\/latex]\u00a0form, with [latex]m=6[\/latex]\u00a0for the first line and [latex]m=6[\/latex]\u00a0for the second line. The slope of both lines is [latex]6[\/latex].\r\n\r\nLook at b, the [latex]y[\/latex]-value of the [latex]y[\/latex]-intercept, to see if the lines are perhaps exactly the same line, in which case we don\u2019t say they are parallel.\r\n\r\nThe first line has a y-intercept at [latex](0,5)[\/latex], and the second line has a y-intercept at [latex](0,\u22121)[\/latex]. They are not the same line.\r\n\r\nThe slopes of the lines are the same and they have different y-intercepts, so they are not the same line and they are parallel.\r\n<h4>Answer<\/h4>\r\nThe lines are parallel.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Perpendicular Lines<\/h3>\r\nTwo non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.\r\n\r\n<\/div>\r\nIf the slope of the first equation is [latex]4[\/latex], then the slope of the second equation will need to be [latex]-\\frac{1}{4}[\/latex] for the lines to be perpendicular.\u00a0 You can also check the two slopes to see if the lines are perpendicular by multiplying the two slopes together. If they are perpendicular, the product of the slopes will be [latex]\u22121[\/latex]. For example, [latex] 4\\cdot-\\frac{1}{4}=\\frac{4}{1}\\cdot-\\frac{1}{4}=-1[\/latex].\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the slope of a line perpendicular to the line [latex]y=2x\u20136[\/latex].\r\n\r\n[reveal-answer q=\"331107\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"331107\"]The given line is written in\u00a0<span class=\"s1\">[latex]y=mx+b[\/latex]<\/span>\u00a0form, with [latex]m=2[\/latex] and [latex]b=-6[\/latex]. The slope is [latex]2[\/latex] or [latex]\\frac{2}{1}[\/latex]. The reciprocal of [latex]\\frac{2}{1}[\/latex] is [latex]\\frac{1}{2}[\/latex]. The negative or opposite of [latex]\\frac{1}{2}[\/latex] is [latex]-\\frac{1}{2}[\/latex]\r\n\r\nIdentify the slope of the given line.\r\n<h4>Answer<\/h4>\r\nThe slope of the perpendicular line is [latex]-\\tfrac{1}{2}[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nTo find the slope of a perpendicular line, find the reciprocal, [latex] \\displaystyle \\tfrac{1}{2}[\/latex], and then find the opposite of this reciprocal [latex] \\displaystyle -\\tfrac{1}{2}[\/latex].\r\n\r\nNote that the product [latex]2\\left(-\\frac{1}{2}\\right)=\\frac{2}{1}\\left(-\\frac{1}{2}\\right)=-1[\/latex], so this means the slopes are perpendicular.\r\n\r\nIn the case where one of the lines is vertical, the slope of that line is undefined and it is not possible to calculate the product with an undefined number. When one line is vertical, the line perpendicular to it will be horizontal, having a slope of zero ([latex]m=0[\/latex]).\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDetermine whether the lines [latex]y=\u22128x+5[\/latex]\u00a0and [latex] \\displaystyle y\\,\\text{=}\\,\\,\\frac{1}{8}x-1[\/latex] are parallel, perpendicular, or neither.\r\n\r\n[reveal-answer q=\"981152\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"981152\"]\r\n\r\nIdentify the slopes of the given lines.\r\n\r\nThe given lines are written in [latex]y=mx+b[\/latex] form, with [latex]m=\u22128[\/latex] for the first line and\u00a0[latex]m=\\frac{1}{8}[\/latex] for the second line.\r\n\r\nDetermine if the slopes are the same or if they are opposite reciprocals.\r\n\r\n[latex]-8\\ne\\frac{1}{8}[\/latex], so the lines are not parallel.\r\n\r\nThe opposite reciprocal of [latex]\u22128[\/latex] is [latex] \\displaystyle \\frac{1}{8}[\/latex], so the lines are perpendicular.\r\n\r\nThe slopes of the lines are opposite reciprocals, so the lines are perpendicular.\r\n<h4>Answer<\/h4>\r\nThe lines are perpendicular.[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"yt watch-title-container\">Watch the video below for an example of how to determine when two lines are parallel or perpendicular.<\/p>\r\nhttps:\/\/youtu.be\/IIy4N2lAkDs\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7737[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify slopes of parallel and perpendicular lines<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"Characterize the slopes of parallel and perpendicular lines\">Characterize the slopes of parallel and perpendicular lines<\/h2>\n<p>When you graph two or more linear equations in a coordinate plane, they generally cross at a point. However, when two lines in a coordinate plane never cross, they are called <b>parallel lines<\/b>. You will also look at the case where two lines in a coordinate plane cross at a right angle. These are called <b>perpendicular lines<\/b>. The slopes of the graphs in each of these cases have a special relationship to each other.<\/p>\n<p>Parallel lines are two or more lines in a plane that never intersect. Examples of parallel lines are all around us, such as the opposite sides of a rectangular picture frame and the shelves of a bookcase.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1402\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/07212406\/Graphing-Linear-Inequalities-Module-2.png\" alt=\"Line y=2x+3 and line y=2x-3. Caption says Equations of parallel lines will have the same slopes and different intercepts.\" width=\"315\" height=\"243\" \/><\/p>\n<p>Perpendicular lines are two or more lines that intersect at a [latex]90[\/latex]-degree angle, like the two lines drawn on this graph. These [latex]90[\/latex]-degree angles are also known as right angles.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064331\/image054.jpg\" alt=\"Two lines that cross to form a 90 degree angle.\" width=\"390\" height=\"340\" \/><\/p>\n<p>Perpendicular lines are also everywhere, not just on graph paper but also in the world around us, from the crossing pattern of roads at an intersection to the colored lines of a plaid shirt.<\/p>\n<div class=\"textbox shaded\">\n<h3>Parallel Lines<\/h3>\n<p>Two non-vertical lines in a plane are parallel if they have both:<\/p>\n<ul>\n<li>the same slope<\/li>\n<li>different [latex]y[\/latex]-intercepts<\/li>\n<\/ul>\n<p>Any two vertical lines in a plane are parallel.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the slope of a line parallel to the line [latex]y=\u22123x+4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q350329\">Show Solution<\/span><\/p>\n<div id=\"q350329\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the slope of the given line.<\/p>\n<p>The given line is written in [latex]y=mx+b[\/latex]\u00a0form, with [latex]m=\u22123[\/latex] and [latex]b=4[\/latex]. The slope is [latex]\u22123[\/latex].<\/p>\n<p>A line parallel to the given line has the same slope.<\/p>\n<h4>Answer<\/h4>\n<p>The slope of the parallel line is [latex]\u22123[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Determine whether the lines [latex]y=6x+5[\/latex] and [latex]y=6x\u20131[\/latex]\u00a0are parallel.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q619259\">Show Solution<\/span><\/p>\n<div id=\"q619259\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the slopes of the given lines.<\/p>\n<p>The given lines are written in [latex]y=mx+b[\/latex]\u00a0form, with [latex]m=6[\/latex]\u00a0for the first line and [latex]m=6[\/latex]\u00a0for the second line. The slope of both lines is [latex]6[\/latex].<\/p>\n<p>Look at b, the [latex]y[\/latex]-value of the [latex]y[\/latex]-intercept, to see if the lines are perhaps exactly the same line, in which case we don\u2019t say they are parallel.<\/p>\n<p>The first line has a y-intercept at [latex](0,5)[\/latex], and the second line has a y-intercept at [latex](0,\u22121)[\/latex]. They are not the same line.<\/p>\n<p>The slopes of the lines are the same and they have different y-intercepts, so they are not the same line and they are parallel.<\/p>\n<h4>Answer<\/h4>\n<p>The lines are parallel.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Perpendicular Lines<\/h3>\n<p>Two non-vertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.<\/p>\n<\/div>\n<p>If the slope of the first equation is [latex]4[\/latex], then the slope of the second equation will need to be [latex]-\\frac{1}{4}[\/latex] for the lines to be perpendicular.\u00a0 You can also check the two slopes to see if the lines are perpendicular by multiplying the two slopes together. If they are perpendicular, the product of the slopes will be [latex]\u22121[\/latex]. For example, [latex]4\\cdot-\\frac{1}{4}=\\frac{4}{1}\\cdot-\\frac{1}{4}=-1[\/latex].<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the slope of a line perpendicular to the line [latex]y=2x\u20136[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q331107\">Show Solution<\/span><\/p>\n<div id=\"q331107\" class=\"hidden-answer\" style=\"display: none\">The given line is written in\u00a0<span class=\"s1\">[latex]y=mx+b[\/latex]<\/span>\u00a0form, with [latex]m=2[\/latex] and [latex]b=-6[\/latex]. The slope is [latex]2[\/latex] or [latex]\\frac{2}{1}[\/latex]. The reciprocal of [latex]\\frac{2}{1}[\/latex] is [latex]\\frac{1}{2}[\/latex]. The negative or opposite of [latex]\\frac{1}{2}[\/latex] is [latex]-\\frac{1}{2}[\/latex]<\/p>\n<p>Identify the slope of the given line.<\/p>\n<h4>Answer<\/h4>\n<p>The slope of the perpendicular line is [latex]-\\tfrac{1}{2}[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>To find the slope of a perpendicular line, find the reciprocal, [latex]\\displaystyle \\tfrac{1}{2}[\/latex], and then find the opposite of this reciprocal [latex]\\displaystyle -\\tfrac{1}{2}[\/latex].<\/p>\n<p>Note that the product [latex]2\\left(-\\frac{1}{2}\\right)=\\frac{2}{1}\\left(-\\frac{1}{2}\\right)=-1[\/latex], so this means the slopes are perpendicular.<\/p>\n<p>In the case where one of the lines is vertical, the slope of that line is undefined and it is not possible to calculate the product with an undefined number. When one line is vertical, the line perpendicular to it will be horizontal, having a slope of zero ([latex]m=0[\/latex]).<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Determine whether the lines [latex]y=\u22128x+5[\/latex]\u00a0and [latex]\\displaystyle y\\,\\text{=}\\,\\,\\frac{1}{8}x-1[\/latex] are parallel, perpendicular, or neither.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q981152\">Show Solution<\/span><\/p>\n<div id=\"q981152\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify the slopes of the given lines.<\/p>\n<p>The given lines are written in [latex]y=mx+b[\/latex] form, with [latex]m=\u22128[\/latex] for the first line and\u00a0[latex]m=\\frac{1}{8}[\/latex] for the second line.<\/p>\n<p>Determine if the slopes are the same or if they are opposite reciprocals.<\/p>\n<p>[latex]-8\\ne\\frac{1}{8}[\/latex], so the lines are not parallel.<\/p>\n<p>The opposite reciprocal of [latex]\u22128[\/latex] is [latex]\\displaystyle \\frac{1}{8}[\/latex], so the lines are perpendicular.<\/p>\n<p>The slopes of the lines are opposite reciprocals, so the lines are perpendicular.<\/p>\n<h4>Answer<\/h4>\n<p>The lines are perpendicular.<\/p><\/div>\n<\/div>\n<\/div>\n<p class=\"yt watch-title-container\">Watch the video below for an example of how to determine when two lines are parallel or perpendicular.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Slopes of Parallel and Perpendicular Lines\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IIy4N2lAkDs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7737&theme=oea&iframe_resize_id=ohm7737&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-16517\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Slope of Parallel and Perpendicular Lines. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IIy4N2lAkDs\">https:\/\/youtu.be\/IIy4N2lAkDs<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Slope of Parallel 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