{"id":208,"date":"2022-04-13T08:43:11","date_gmt":"2022-04-13T08:43:11","guid":{"rendered":"http:\/\/9thclass.deltapublications.in\/?page_id=208"},"modified":"2025-12-04T12:19:23","modified_gmt":"2025-12-04T12:19:23","slug":"l-8-perimeter-of-polygons-with-an-inscribed-circle","status":"publish","type":"page","link":"https:\/\/9thclass.deltapublications.in\/index.php\/l-8-perimeter-of-polygons-with-an-inscribed-circle\/","title":{"rendered":"L.8 Perimeter of polygons with an inscribed circle"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Perimeter of polygons with an inscribed circle<\/strong><\/h2>\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\udfe2 <strong>What is an Inscribed Circle (Incircle)?<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A circle drawn <strong>inside a polygon<\/strong> that <strong>touches all its sides<\/strong>.<\/li>\n\n\n\n<li>The point where the circle touches a side is called the <strong>point of tangency<\/strong>.<\/li>\n\n\n\n<li>The center of the circle is called the <strong>incenter<\/strong> (\ud83d\udd3a for triangles).<\/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-large-font-size\">\ud83d\udfe3 <strong>Key Property of Tangents from a Point<\/strong><\/h2>\n\n\n\n<p>\ud83d\udc49 If two tangents are drawn from the <strong>same vertex<\/strong> to the circle,<br>then they are <strong>equal in length<\/strong>.<\/p>\n\n\n\n<p>Example in a quadrilateral:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>From vertex <strong>A<\/strong>, the two tangent lengths are <strong>AP = AS<\/strong>.<\/li>\n\n\n\n<li>From vertex <strong>B<\/strong>, the two tangent lengths are <strong>BP = BQ<\/strong>, and so on.<\/li>\n<\/ul>\n\n\n\n<p>This property is the <strong>secret<\/strong> behind finding the perimeter! \ud83d\udd11\u2728<\/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>How Perimeter Is Found Using Tangent Lengths<\/strong><\/h2>\n\n\n\n<p>Each side of the polygon is formed by adding <strong>two tangent segments<\/strong>,<br>one from each of its endpoints.<\/p>\n\n\n\n<p>Example for a quadrilateral ABCD:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Side <strong>AB = AP + BP<\/strong><\/li>\n\n\n\n<li>Side <strong>BC = BQ + CQ<\/strong><\/li>\n\n\n\n<li>Side <strong>CD = CR + DR<\/strong><\/li>\n\n\n\n<li>Side <strong>DA = DS + AS<\/strong><\/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-large-font-size\">\ud83d\udfe0 <strong>Amazing Result: Perimeter = Sum of All Tangent Lengths \u00d7 2<\/strong><\/h2>\n\n\n\n<p>Since every tangent length repeats <strong>twice<\/strong> (once for each side),<br>the <strong>perimeter = (AP + BP + CQ + DR + \u2026) \u00d7 2<\/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\udfe2 <strong>Special Case \u2014 Triangle<\/strong><\/h2>\n\n\n\n<p>For a triangle, if tangent lengths are <strong>x, y, z<\/strong> at three vertices,<br>then:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Perimeter = x + y + z + x + y + z = 2(x + y + z)<\/strong><\/li>\n\n\n\n<li>Or simpler:<br>\u27a4 <strong>Perimeter = sum of the three sides<\/strong> (as usual!)<br>because each side is formed by adding tangent lengths.<\/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-large-font-size\">\ud83d\udd34 <strong>Formula Summary<\/strong> \u2728<\/h2>\n\n\n\n<h3 class=\"wp-block-heading has-normal-font-size\">\u2b50 <strong>Perimeter of polygon with an inscribed circle<\/strong><\/h3>\n\n\n\n<p>P = 2(x1 + x2 + x3 + \u2026)<\/p>\n\n\n\n<p>where x1 , x2 , x3&#8230; are tangent lengths from each vertex.<\/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>Visual Shortcut<\/strong><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Tangent lengths from each vertex are <strong>equal<\/strong>.<\/li>\n\n\n\n<li>Add all tangent lengths.<\/li>\n\n\n\n<li>Multiply by <strong>2<\/strong>.<\/li>\n\n\n\n<li>\ud83c\udf89 You get the <strong>perimeter<\/strong>!<\/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-large-font-size\">\ud83d\udfe1 <strong>Why Useful?<\/strong><\/h2>\n\n\n\n<p>This method is useful when:<br>\u2714 Side lengths are <strong>not given<\/strong><br>\u2714 <strong>Tangent lengths<\/strong> (from vertices to points of tangency) <strong>are provided<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-link-color has-large-font-size wp-elements-3fefd20f8ce119a518e5f53333ab5486\" 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:#d3e4b7\"><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-b44c614db0b205b1476e8100ff679b44\" style=\"color:#b00012\"><strong>What is FG?<\/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-49.png\" alt=\"\" class=\"wp-image-11920\" style=\"width:504px;height:auto\"\/><\/figure><\/div>\n\n\n<p>FG= _______<\/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\"><img decoding=\"async\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/01\/image-removebg-preview-2024-01-03T174142.504.png\" alt=\"\" class=\"wp-image-11924\"\/><\/figure><\/div>\n\n\n<p>Find&nbsp;the unknown segment&nbsp;lengths.<\/p>\n\n\n\n<p>JK and IJ are tangents to the inscribed circle from J. So , JK is congruent to IJ.<br>JK=IJ=8.<\/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-2024-01-03T174324.531.png\" alt=\"\" class=\"wp-image-11926\"\/><\/figure><\/div>\n\n\n<p>You know JK and FJ. Use the Additive Property of Length to write an equation and find FK.<\/p>\n\n\n\n<p>FK+JK= FJ            Additive Property of Length<\/p>\n\n\n\n<p>FK + 8 = 13       Plug in JK=8 and FJ=13<\/p>\n\n\n\n<p>FK =  5   Subtract 8 from both sides<\/p>\n\n\n\n<p>So, FK is 5.<\/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-2024-01-03T174556.190.png\" alt=\"\" class=\"wp-image-11930\"\/><\/figure><\/div>\n\n\n<p>FG and FK are tangents to the inscribed circle from F. So , FG is congruent to FK.   <br>FG=FK=5.<\/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-2024-01-03T174756.215.png\" alt=\"\" class=\"wp-image-11931\" style=\"width:476px;height:auto\"\/><\/figure><\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#bbe1eb\"><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\" style=\"color:#b00012\"><strong>What is TU?<\/strong><\/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-50.png\" alt=\"\" class=\"wp-image-11936\"\/><\/figure><\/div>\n\n\n<p>TU=______<\/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-2024-01-03T175420.113.png\" alt=\"\" class=\"wp-image-11937\" style=\"width:423px;height:auto\"\/><\/figure><\/div>\n\n\n<p>Find&nbsp;the unknown segment&nbsp;lengths.<\/p>\n\n\n\n<p>SX and WX are tangents to the inscribed circle from X. So , SX is congruent to WX .<\/p>\n\n\n\n<p>SX=WX=1.<\/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-2024-01-03T175619.073.png\" alt=\"\" class=\"wp-image-11938\"\/><\/figure><\/div>\n\n\n<p>You know SX and TX. Use the Additive Property of Length to write an equation and find ST.<\/p>\n\n\n\n<p>SX + ST =  TX       Additive Property of Length<\/p>\n\n\n\n<p>1 + ST =  2          Plug in SX=1 and TX=2<\/p>\n\n\n\n<p>ST= 1              Subtract 1 from both sides<\/p>\n\n\n\n<p>So, ST is 1.<\/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-2024-01-03T175817.335.png\" alt=\"\" class=\"wp-image-11939\"\/><\/figure><\/div>\n\n\n<p>TU and ST are tangents to the inscribed circle from T. So , TU is congruent to ST.<br>TU=ST=1.<\/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-2024-01-03T175949.416.png\" alt=\"\" class=\"wp-image-11940\"\/><\/figure><\/div><\/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\/88997\/970\/294\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-76.png\" alt=\"\" class=\"wp-image-7916\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-76.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-76-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-76-150x150.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\/88997\/335\/238\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-85.png\" alt=\"\" class=\"wp-image-7917\" srcset=\"https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-85.png 500w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-85-300x300.png 300w, https:\/\/9thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-85-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Perimeter of polygons with an inscribed circle Key Notes : \ud83d\udfe2 What is an Inscribed Circle (Incircle)? \ud83d\udfe3 Key Property of Tangents from a Point \ud83d\udc49 If two tangents are drawn from the same vertex to the circle,then they are equal in length. Example in a quadrilateral: This property is the secret behind finding the<a class=\"more-link\" href=\"https:\/\/9thclass.deltapublications.in\/index.php\/l-8-perimeter-of-polygons-with-an-inscribed-circle\/\">Continue reading <span class=\"screen-reader-text\">&#8220;L.8 Perimeter of polygons with an inscribed circle&#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-208","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/208","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=208"}],"version-history":[{"count":16,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/208\/revisions"}],"predecessor-version":[{"id":18292,"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/208\/revisions\/18292"}],"wp:attachment":[{"href":"https:\/\/9thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}