{"id":42,"date":"2017-08-23T07:09:06","date_gmt":"2017-08-23T07:09:06","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=42"},"modified":"2017-09-18T14:14:54","modified_gmt":"2017-09-18T14:14:54","slug":"parallel-perpendicular-lines","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/parallel-perpendicular-lines\/","title":{"rendered":"Parallel &#038; Perpendicular Lines"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><!--<a href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/math_02_00.html\" class=\"button button-primary\">\u2b05 Previous Lesson<\/a>--><br \/>\n<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/absolute-values-2\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Parallel &amp; Perpendicular Lines<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will study the slopes of parallel and perpendicular lines.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <strong><em>slope\u00a0<\/em><\/strong>of a line equals\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p1_clip_image002.gif\" width=\"132\" height=\"38\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li>The <strong>equation of a line<\/strong> can\u00a0be in either of these forms: <em>ax\u00a0+ by = c<\/em>, which is called\u00a0standard form, or <em>y = mx + b<\/em> which is called slope-intercept form, where <em>a, b, c,<\/em> and <em>m\u00a0<\/em>are real numbers.<\/li>\n<\/ul>\n<section>\n<h3>What are parallel lines?<\/h3>\n<p><abbr title=\"lines in the same plane that do not intersect\">Parallel\u00a0lines<\/abbr> are lines in the same <abbr title=\"a surface of such nature that a straight line joining two of its points lies wholly in the surface or a collections of points forming a flat surface.\">plane\u00a0<\/abbr>that do not <abbr title=\"to meet or cross at a point\">intersect<\/abbr>.<\/p>\n<p>We use the symbol <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/parallel.gif\" width=\"9\" height=\"21\" align=\"absmiddle\" \/> to stand for the words &#8220;is\u00a0parallel to.&#8221;<\/p>\n<p>So, given two lines, AB and CD, as shown below<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/Math%20Mod%202.4%20Art%20001.JPG\" alt=\"Two Parallel Lines\" width=\"309\" height=\"124\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/center>then AB <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/parallel.gif\" width=\"9\" height=\"21\" align=\"absmiddle\" \/>CD.<\/p>\n<p>Because parallel lines maintain the same distance from each\u00a0other, their <abbr title=\"upward or downward slant or inclination or degree of slant\">slopes<\/abbr> must be equal. In their equations, the only difference between the\u00a0two lines is in the <em>y<\/em>-intercept.\u00a0Therefore, because the slope-intercept form of the equation of a\u00a0line is <em>y\u00a0= mx + b,<\/em> we might\u00a0have the equation <em>y\u00a0= 3x + 4.<\/em><\/p>\n<p>Another line that is\u00a0parallel to the first might have the equation <em>y\u00a0= 3x \u2013 10.<\/em><\/p>\n<p>In both cases, the slope of\u00a0the line (<em>m<\/em>)\u00a0is the same.<\/p>\n<p>A line that differs from\u00a0another by its <em>y<\/em>-intercept\u00a0only is called a <abbr title=\"to move an object without rotating or reflecting it. A translation can be vertical or horizontal.\">vertical\u00a0translation\u00a0<\/abbr>of the line.<\/p>\n<p>For example, find the slope\u00a0of a line that is parallel to <em>3y \u2013 2x = 12.<\/em><\/p>\n<p>Because the equation is in\u00a0standard form, we must first put it into slope-intercept\u00a0form. To do so, solve for <em>y<\/em>.<\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p2_clip_image002.gif\" width=\"110\" height=\"81\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/center>Because the two lines are parallel, both have a slope of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p2_clip_image004.gif\" width=\"9\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Now, write equations for\u00a0two lines that have a slope of 2, given that one line\u2019s <em>y<\/em>-intercept\u00a0is at the point (0, 5) and the other line&#8217;s <em>y<\/em>-intercept\u00a0is at the point (0, \u20138).<\/p>\n<p>The equations must be in\u00a0the form <em>y = mx + b<\/em>. Simply plug in the slope and <em>y<\/em>-intercepts\u00a0of each line into the equation.<\/p>\n<p>The first equation becomes\u00a0<em>y = 2x + 5<\/em> and\u00a0the second line becomes <em>y = 2x \u2013 8.<\/em><\/p>\n<p>Two lines are <abbr title=\" lines that meet at 4 right angles.\">perpendicular<\/abbr> to each other if their intersection forms four right angles. For\u00a0example, a vertical line and a horizontal line are perpendicular.\u00a0The slopes of perpendicular lines are negative reciprocals of each\u00a0other. This means that the product of the two slopes is equal to\u00a0\u20131. Thus, a line with a slope of 2 is perpendicular to a\u00a0line with a slope of\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p3_clip_image002.gif\" width=\"22\" height=\"34\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Given the line <em>y = 3x + 4<\/em>, determine\u00a0the equation of a line that is perpendicular and has a <em>y<\/em>-intercept of 8.<\/p>\n<p>Because the slope of a\u00a0perpendicular line will be the negative reciprocal of 3, the slope\u00a0of the new line is\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p3_clip_image004.gif\" width=\"20\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Because the <em>y<\/em>-intercept\u00a0is 8, the equation of the new line is <em>y = <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images\/s1_p3_clip_image005.gif\" width=\"20\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>x\u00a0<\/em>+ 8.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><!--<a href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/math_02_00.html\" class=\"button button-primary\">\u2b05 Previous Lesson<\/a>--><br \/>\n<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/absolute-values-2\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Workshop Index\u00a0Next Lesson \u27a1 Parallel &amp; Perpendicular Lines Objective In this lesson, you will study the slopes of parallel and perpendicular lines. Previously Covered: The slope\u00a0of a line equals\u00a0 The equation of a line can\u00a0be in either of these forms: ax\u00a0+ by = c, which is called\u00a0standard form, or y = mx + b which [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-42","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/42","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=42"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/42\/revisions"}],"predecessor-version":[{"id":722,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/42\/revisions\/722"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=42"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}