{"id":206,"date":"2022-04-13T08:42:53","date_gmt":"2022-04-13T08:42:53","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=206"},"modified":"2025-12-04T12:19:21","modified_gmt":"2025-12-04T12:19:21","slug":"l-7-tangent-lines","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/l-7-tangent-lines\/","title":{"rendered":"L.7 Tangent lines"},"content":{"rendered":"\n<h3 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Tangent lines<\/strong><\/h3>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-8550df6181cd5d83aa7a08ef336a4ca1\" style=\"color:#74008b\">Key Notes :<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udd35 <strong>What is a Tangent Line?<\/strong><\/h2>\n\n\n\n<p>A <strong>tangent line<\/strong> is a straight line that <strong>touches a circle at exactly one point<\/strong>.<br>\ud83d\udc49 This point is called the <strong>point of tangency<\/strong>.<br>\ud83c\udfaf It does <strong>not<\/strong> cut across the circle.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udfe2 <strong>Tangent Touches Only Once<\/strong><\/h2>\n\n\n\n<p>A tangent line meets the circle <strong>in only one point<\/strong>.<br>\u2714\ufe0f If it crosses the circle in two points, it becomes a <strong>secant<\/strong>, not a tangent.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udd34 <strong>Radius and Tangent Are Perpendicular<\/strong><\/h2>\n\n\n\n<p>At the point where the tangent touches the circle:<br>\ud83d\udc49 The <strong>radius is always perpendicular (90\u00b0)<\/strong> to the tangent line.<br>\ud83d\udcd0 <strong>Radius \u27c2 Tangent<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udfe3 <strong>Tangents from an External Point<\/strong><\/h2>\n\n\n\n<p>If you draw <strong>two tangent lines from the same outside point<\/strong>,<br>\u2714\ufe0f The two tangent lengths are <strong>equal<\/strong>.<br>\ud83c\udfaf This is called <strong>tangent segments theorem<\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udfe1 <strong>Real-Life Examples of Tangents<\/strong><\/h2>\n\n\n\n<p>\u2728 Wheel touching the road<br>\u2728 Bike tire touching ground<br>\u2728 A ladder leaning gently against a round pole (touches at one point)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udd35 <strong>Tangent Formula (Basic Idea)<\/strong><\/h2>\n\n\n\n<p>If <em>P<\/em> is an external point and you draw tangents to a circle that touch at <em>A<\/em> and <em>B<\/em>:<br>\ud83d\udc49 <strong>PA = PB<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-large-font-size\">\ud83d\udfe0 <strong>Difference Between Tangent and Secant<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Line Type<\/th><th>Touches Circle<\/th><th>Result<\/th><\/tr><\/thead><tbody><tr><td><strong>Tangent<\/strong><\/td><td>1 point<\/td><td>Just touches<\/td><\/tr><tr><td><strong>Secant<\/strong><\/td><td>2 points<\/td><td>Cuts across<\/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-large-font-size\">\ud83c\udf89 Summary<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A tangent touches the circle at <strong>one point only<\/strong>.<\/li>\n\n\n\n<li>Radius to the tangent is <strong>90\u00b0<\/strong>.<\/li>\n\n\n\n<li>Tangents from the same point are <strong>equal in length<\/strong>.<\/li>\n\n\n\n<li>Used in circles, wheels, and geometry constructions.<\/li>\n<\/ul>\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-large-font-size\" style=\"background-color:#c0e4c1\"><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\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color wp-elements-b5c5c46c3cfda87a2e882a0eaaeeb721\" style=\"color:#b00012\"><strong><em>UV<\/em> is tangent to \u2a00T. What is \u2220W?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-43.png\" alt=\"\" class=\"wp-image-11902\" style=\"width:510px;height:auto\"\/><\/figure><\/div>\n\n\n<p>\u2220W  =  ________\u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;diagram.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-2024-01-03T164224.375.png\" alt=\"\" class=\"wp-image-11903\" style=\"width:516px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Since&nbsp; <em>UV<\/em> is tangent to \u2a00T, \u25b3UVW is a right triangle with right angle \u2220U. So, \u2220W and \u2220V are complementary. Write an equation setting the sum of their measures equal to 90\u00b0, and solve for \u2220W.<\/p>\n\n\n\n<p>\u2220W + \u2220V  =  90\u00b0<\/p>\n\n\n\n<p>\u2220W +  36\u00b0 =  90\u00b0     Plug in \u2220V=36\u00b0<\/p>\n\n\n\n<p>\u2220W  =  54\u00b0            Subtract&nbsp;36\u00b0&nbsp;from both&nbsp;sides<\/p>\n\n\n\n<p>\u2220W is 54\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#cdd9ea\"><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\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p><em>PQ<\/em> is tangent to \u2a00N. What is \u2220Q?<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-44.png\" alt=\"\" class=\"wp-image-11906\" style=\"width:425px;height:auto\"\/><\/figure><\/div>\n\n\n<p>\u2220Q = ________\u00b0<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the&nbsp;diagram.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-2024-01-03T165751.260.png\" alt=\"\" class=\"wp-image-11907\" style=\"width:475px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Since PQ is tangent to \u2a00N, \u25b3NPQ is a right triangle with right angle \u2220P. So, \u2220N and \u2220Q are complementary. Write an equation setting the sum of their measures equal to 90\u00b0, and solve for \u2220Q.<\/p>\n\n\n\n<p>\u2220N + \u2220Q =  90\u00b0<\/p>\n\n\n\n<p>58\u00b0 + \u2220Q =  90\u00b0    Plug in \u2220N=58\u00b0<\/p>\n\n\n\n<p>\u2220Q  = 32\u00b0       Subtract&nbsp;58\u00b0&nbsp;from both&nbsp;sides<\/p>\n\n\n\n<p>\u2220Q is 32\u00b0.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#d1f3da\"><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\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>PQ is tangent to \u2a00N. What is NQ?<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-45.png\" alt=\"\" class=\"wp-image-11909\" style=\"width:530px;height:auto\"\/><\/figure><\/div>\n\n\n<p>NQ = ________km<\/p>\n<\/div><\/div>\n\n\n\n<p>Since PQ is tangent to \u2a00N , PQ is perpendicular to NP. So, \u25b3NPQ is a right triangle with hypotenuse NQ.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-2024-01-03T170930.804.png\" alt=\"\" class=\"wp-image-11911\"\/><\/figure><\/div>\n\n\n<p>Now use Pythagoras&#8217; theorem to find NQ.<\/p>\n\n\n\n<p>NP<sup>2<\/sup> + PQ<sup>2<\/sup> =  NQ<sup>2<\/sup>     <\/p>\n\n\n\n<p>8<sup>2<\/sup> + 15<sup>2<\/sup> =  NQ<sup>2 <\/sup>        Plug in NP=8 and PQ=15<\/p>\n\n\n\n<p>64 + 225 =  NQ<sup>2<\/sup>     Square<\/p>\n\n\n\n<p>289  = NQ<sup>2<\/sup>     Add<\/p>\n\n\n\n<p>17 =  NQ        Take&nbsp;the square root of both&nbsp;sides<\/p>\n\n\n\n<p>NQ is 17 kilometres.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-c9aff12d0c2da4d508e8e8c5f02920c2\" 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\/89045\/355\/600\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-77.png\" alt=\"\" class=\"wp-image-7919\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-77.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-77-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-77-150x150.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\/89046\/200\/930\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-86.png\" alt=\"\" class=\"wp-image-7920\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-86.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-86-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-86-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Tangent lines Key Notes : \ud83d\udd35 What is a Tangent Line? A tangent line is a straight line that touches a circle at exactly one point.\ud83d\udc49 This point is called the point of tangency.\ud83c\udfaf It does not cut across the circle. \ud83d\udfe2 Tangent Touches Only Once A tangent line meets the circle in only one<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/l-7-tangent-lines\/\">Continue reading <span class=\"screen-reader-text\">&#8220;L.7 Tangent 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-206","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/206","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=206"}],"version-history":[{"count":17,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/206\/revisions"}],"predecessor-version":[{"id":18291,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/206\/revisions\/18291"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}