{"id":335,"date":"2022-04-13T09:05:07","date_gmt":"2022-04-13T09:05:07","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=335"},"modified":"2025-08-20T01:47:21","modified_gmt":"2025-08-20T01:47:21","slug":"t-19-slopes-of-parallel-and-perpendicular-lines","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/t-19-slopes-of-parallel-and-perpendicular-lines\/","title":{"rendered":"T.19 Slopes of parallel and perpendicular lines"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color has-huge-font-size\" style=\"color:#00056d;text-transform:uppercase\"><strong>Slopes of parallel and perpendicular lines<\/strong><\/h2>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<div style=\"position: relative; width: 100%; height: 0; padding-top: 56.2500%;\n padding-bottom: 0; box-shadow: 0 2px 8px 0 rgba(63,69,81,0.16); margin-top: 1.6em; margin-bottom: 0.9em; overflow: hidden;\n border-radius: 8px; will-change: transform;\">\n  <iframe loading=\"lazy\" style=\"position: absolute; width: 100%; height: 100%; top: 0; left: 0; border: none; padding: 0;margin: 0;\"\n    src=\"https:\/\/www.canva.com\/design\/DAGwIUfFSik\/taZitJnvE_cBT0yZ29Rxnw\/watch?embed\" allowfullscreen=\"allowfullscreen\" allow=\"fullscreen\">\n  <\/iframe>\n<\/div>\n<a href=\"https:&#x2F;&#x2F;www.canva.com&#x2F;design&#x2F;DAGwIUfFSik&#x2F;taZitJnvE_cBT0yZ29Rxnw&#x2F;watch?utm_content=DAGwIUfFSik&amp;utm_campaign=designshare&amp;utm_medium=embeds&amp;utm_source=link\" target=\"_blank\" rel=\"noopener\">T.19 Slopes of parallel and perpendicular lines<\/a> by Delta publications\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-2e84b20f2f773a3685e1f898dc139368\" style=\"color:#74008b\">Key notes:<\/h2>\n\n\n\n<h3 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-40ab68202108c27cefb0a59b6e4176e5\" style=\"color:#000060\"><strong>What is slope?<\/strong><\/h3>\n\n\n\n<p>The <strong>slope<\/strong> (mmm) of a line measures how steep the line is.<br>Formula: <\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>m=y<sub>2\u2212<\/sub>y<sub>1 \/<\/sub> x<sub>2\u2212<\/sub>x<sub>1<\/sub><\/strong><\/p>\n\n\n\n<p>where (x<sub>1<\/sub>,y<sub>1<\/sub>) and (x<sub>2<\/sub>,y<sub>2<\/sub>)\u200b) are two points on the line.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-33fcc7903532e55e61bb2523c5364251\" style=\"color:#000060\"><strong> Slopes of Parallel Lines<\/strong><\/h2>\n\n\n\n<p><strong>Definition:<\/strong> Two lines are <strong>parallel<\/strong> if they never meet and are always the same distance apart.<\/p>\n\n\n\n<p><strong>Rule:<\/strong><br><strong>Parallel lines have the same slope<\/strong> (if they are not vertical). <\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>m<sub>1<\/sub>=m<sub>2<\/sub><\/strong><\/p>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Line 1: y=2x+3\u2192 slope m<sub>1<\/sub>=2<\/li>\n\n\n\n<li>Line 2: y=2x\u22125 \u2192 slope m<sub>2<\/sub>=2<br>Since m<sub>1<\/sub>=m<sub>2<\/sub>, the lines are parallel.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-55983670d065636f630ce19d03e89693\" style=\"color:#000060\"><strong> Slopes of Perpendicular Lines<\/strong><\/h2>\n\n\n\n<p><strong>Definition:<\/strong> Two lines are <strong>perpendicular<\/strong> if they meet at a right angle (90\u00b0).<\/p>\n\n\n\n<p><strong>Rule:<\/strong><br><strong>The slopes of perpendicular lines are negative reciprocals of each other<\/strong> (if neither is vertical). <\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>m<sub>1<\/sub>\u00d7m<sub>2<\/sub>=\u22121<\/strong><\/p>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Line 1: y=1\/2x+4\u2192 slope m<sub>1<\/sub>=1\/2<\/li>\n\n\n\n<li>Perpendicular line: slope m2=\u22122<br>(because 1\/2\u00d7\u22122=\u22121)<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-71a0c63a7b67d6c319869feec8411ef1\" style=\"color:#000060\"><strong>Special Cases<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Horizontal line<\/strong>: slope m=0<\/li>\n\n\n\n<li><strong>Vertical line<\/strong>: slope is <strong>undefined<\/strong><\/li>\n\n\n\n<li>A horizontal line is <strong>perpendicular<\/strong> to a vertical line.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-640afadf8e977733ec66d248b28ea942\" style=\"color:#000060\"><strong> Summary Table<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Relationship<\/th><th>Slope Condition<\/th><th>Example Slopes<\/th><\/tr><\/thead><tbody><tr><td>Parallel lines<\/td><td>m<sub>1<\/sub>=m<sub>2<\/sub><\/td><td>3 and 3<\/td><\/tr><tr><td>Perpendicular lines<\/td><td><br>m<sub>1<\/sub>\u00d7m<sub>2<\/sub>\u200b=\u22121<\/td><td>2\/3 and \u22123\/2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-text-color has-link-color has-large-font-size wp-elements-d2aa10170baee726f4efd29caba4c175\" style=\"color:#000060\"><strong> Practice Questions<\/strong><\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Are the lines y=4x+1 and y=4x\u22127 parallel or perpendicular?<\/li>\n\n\n\n<li>Find the slope of a line parallel to y=\u22123\/5x+2<\/li>\n\n\n\n<li>Find the slope of a line perpendicular to y=7x\u22124<\/li>\n\n\n\n<li>Determine whether the lines with slopes 5 and \u22121\/5 are parallel or perpendicular.<\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-normal-font-size\" style=\"background-color:#bbfbf3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background has-normal-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Line <em><strong>e<\/strong><\/em> has a slope of -9\/7. line <strong>f<\/strong> is perpendicular to e. <\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-3679442ab3d380b51a0975bdd6aa9f05\" style=\"color:#b00012\">what is the slope of line <strong>f<\/strong>?<\/p>\n<\/div><\/div>\n\n\n\n<p>Line&nbsp;f&nbsp;is perpendicular to line&nbsp;<em>e<\/em>, so its slope is the opposite reciprocal. Find the opposite reciprocal.<\/p>\n\n\n\n<p>-9\/7   &#8212;&#8211;&gt;Take the slope of line&nbsp;e<\/p>\n\n\n\n<p>-7\/9   &#8212;&#8212;&gt; Find the reciprocal<\/p>\n\n\n\n<p class=\"has-normal-font-size\">7\/9    &#8212;&#8212;-&gt; Find the opposite<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1667456968180_366\" class=\"wp-block-table\"><table><tbody><tr><td>The slope of line&nbsp;<em>f<\/em>&nbsp;is  7\/9<\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-normal-font-size\" style=\"background-color:#f8c6ed\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background has-normal-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Line u has a slope of 5\/9. Line v is parallel to line u.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-ad8a0ee217cfbfd5a22595dbb24484dd\" style=\"color:#b00012\"> What is the slope of line v?<\/p>\n<\/div><\/div>\n\n\n\n<p>Line v is parallel to u, so its slope is the same.<\/p>\n\n\n\n<p>The slope of line v is 5\/9.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-normal-font-size\" style=\"background-color:#aef6c3\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background has-normal-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Line u has a lope of 6\/5. Line v is parallel to line u. <\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-e4a98645fbf72412d41c741975d986e1\" style=\"color:#b00012\">What is the slope of line v?<\/p>\n<\/div><\/div>\n\n\n\n<p>Line v is parallel to u, so its slope is the same.<\/p>\n\n\n\n<p>The slope of line v is 6\/5.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">Let&#8217;s practice:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/78264\/174\/168\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-126.png\" alt=\"\" class=\"wp-image-8079\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-126.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-126-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-126-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/78280\/062\/305\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-135.png\" alt=\"\" class=\"wp-image-8080\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-135.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-135-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-135-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Slopes of parallel and perpendicular lines T.19 Slopes of parallel and perpendicular lines by Delta publications Key notes: What is slope? The slope (mmm) of a line measures how steep the line is.Formula: m=y2\u2212y1 \/ x2\u2212x1 where (x1,y1) and (x2,y2)\u200b) are two points on the line. Slopes of Parallel Lines Definition: Two lines are parallel<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/t-19-slopes-of-parallel-and-perpendicular-lines\/\">Continue reading <span class=\"screen-reader-text\">&#8220;T.19 Slopes of parallel and perpendicular lines&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-335","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/335","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=335"}],"version-history":[{"count":24,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/335\/revisions"}],"predecessor-version":[{"id":17564,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/335\/revisions\/17564"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}