{"id":297,"date":"2022-04-13T10:35:54","date_gmt":"2022-04-13T10:35:54","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=297"},"modified":"2024-10-23T14:21:29","modified_gmt":"2024-10-23T14:21:29","slug":"o-7-triangle-inequality-theorem","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/o-7-triangle-inequality-theorem\/","title":{"rendered":"O.7 Triangle Inequality Theorem"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Triangle Inequality Theorem<\/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<h4 class=\"wp-block-heading\">1. <strong>Definition<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>Triangle Inequality Theorem<\/strong> states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.<\/li>\n\n\n\n<li>This theorem can be written as:\n<ul class=\"wp-block-list\">\n<li>( a + b > c )<\/li>\n\n\n\n<li>( b + c > a )<\/li>\n\n\n\n<li>( a + c > b )<\/li>\n\n\n\n<li>where (a), (b), and (c) are the lengths of the sides of the triangle.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"168\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-18.png\" alt=\"\" class=\"wp-image-14305\" style=\"width:444px;height:auto\"\/><\/figure><\/div>\n\n\n<h4 class=\"wp-block-heading\">2. <strong>Explanation<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The theorem ensures that a valid triangle can only be formed if the sum of any two sides is greater than the third side.<\/li>\n\n\n\n<li>This is because if one side is too long compared to the sum of the other two sides, it would prevent the sides from meeting to form a closed shape (a triangle).<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">3. <strong>Examples<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Example 1<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Consider a triangle with side lengths 3 cm, 4 cm, and 5 cm.<\/li>\n\n\n\n<li>Apply the Triangle Inequality:<\/li>\n\n\n\n<li>( 3 + 4 > 5 ) (7 > 5, true)<\/li>\n\n\n\n<li>( 3 + 5 > 4 ) (8 > 4, true)<\/li>\n\n\n\n<li>( 4 + 5 > 3 ) (9 > 3, true)<\/li>\n\n\n\n<li>Since all inequalities hold, the sides can form a triangle.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Example 2<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Consider side lengths 2 cm, 5 cm, and 8 cm.<\/li>\n\n\n\n<li>Apply the Triangle Inequality:<\/li>\n\n\n\n<li>( 2 + 5 > 8 ) (7 > 8, false)<\/li>\n\n\n\n<li>( 2 + 8 > 5 ) (10 > 5, true)<\/li>\n\n\n\n<li>( 5 + 8 > 2 ) (13 > 2, true)<\/li>\n\n\n\n<li>The first inequality fails, so these sides <strong>cannot form a triangle<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">4. <strong>Converse of the Triangle Inequality Theorem<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The converse is also true: If the sum of the lengths of any two sides of a triangle is greater than the third side, then a triangle exists.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">5. <strong>Importance of the Theorem<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Geometric Validity<\/strong>: Ensures that three given lengths can form a valid triangle.<\/li>\n\n\n\n<li><strong>Applications in Problem Solving<\/strong>: This theorem helps determine whether a triangle is possible from given side lengths and is essential in triangle construction problems.<\/li>\n\n\n\n<li><strong>Coordinate Geometry<\/strong>: The theorem can be applied to check if three points in the coordinate plane can form a triangle by calculating the distances between the points.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\"><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#ffee69\"><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\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-background\">If&nbsp;0\u2264a\u2264b\u2264c,&nbsp;then&nbsp;a,&nbsp;b,&nbsp;and&nbsp;c&nbsp;are the side lengths of a triangle if and only if&nbsp;<\/p>\n<\/div><\/div>\n<\/div><\/div>\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-secondary-background-color has-background has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-56d0df0a093e11785950d824cf39337a\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;1,&nbsp;6,&nbsp;and&nbsp;9?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=1,&nbsp;b=6,&nbsp;and&nbsp;c=9. <\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;1+6=7,&nbsp;it is false that&nbsp;1+6&gt;9.&nbsp;So, these are not the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f7bcf0\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-bee604325d41cb9ffe7b9af1f0b673c9\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;1,&nbsp;2,&nbsp;and&nbsp;3?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=1,&nbsp;b=2,&nbsp;and&nbsp;c=3.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;1+2=3,&nbsp;it is false that&nbsp;1+2&gt;3.&nbsp;So, these are not the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f6c4d1\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-dc918c6b267844b9cb27b3feba95b46c\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;3,&nbsp;6,&nbsp;and&nbsp;8?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=3,&nbsp;b=6,&nbsp;and&nbsp;c=8.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;3+6=9,&nbsp;it is true that&nbsp;3+6&gt;8.&nbsp;So, these are the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-secondary-background-color has-background has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-cecdb09b5b635c92668d6a73da340d50\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;3,&nbsp;6,&nbsp;and&nbsp;10?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=3,&nbsp;b=6,&nbsp;and&nbsp;c=10.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;3+6=9,&nbsp;it is false that&nbsp;3+6&gt;10.&nbsp;So, these are not the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#dff3c2\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-524466a5ef5df1bc64b79297e10c9998\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;2,&nbsp;3,&nbsp;and&nbsp;5?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=2,&nbsp;b=3,&nbsp;and&nbsp;c=5.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;2+3=5,&nbsp;it is false that&nbsp;2+3&gt;5.&nbsp;So, these are not the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-secondary-background-color has-background has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-9047d4d76756d4c15a423e25d0266be0\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;2,&nbsp;7,&nbsp;and&nbsp;10?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=2,&nbsp;b=7,&nbsp;and&nbsp;c=10.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;2+7=9,&nbsp;it is false that&nbsp;2+7&gt;10.&nbsp;So, these are not the side lengths of a&nbsp;triangle.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f4caff\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-background-background-color has-text-color has-background has-link-color wp-elements-c79dfddb152b08dd4723646a6ba67421\" style=\"color:#b00012\">Can&nbsp;the sides of a triangle have lengths&nbsp;1,&nbsp;10,&nbsp;and&nbsp;10?<\/p>\n\n\n\n<p>First,&nbsp;put the three numbers in order from smallest to largest:&nbsp;a=1,&nbsp;b=10,&nbsp;and&nbsp;c=10.<\/p>\n\n\n\n<p>Now&nbsp;check whether&nbsp;a+b&gt;c.&nbsp;Since&nbsp;1+10=11,&nbsp;it is true that&nbsp;1+10&gt;10.&nbsp;So, these are the side lengths of a&nbsp;triangle.<\/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\/80472\/960\/849\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-98.png\" alt=\"\" class=\"wp-image-7158\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-98.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-98-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-98-150x150.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\/80221\/706\/720\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-116.png\" alt=\"\" class=\"wp-image-7159\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-116.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-116-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-116-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Triangle Inequality Theorem Key Notes : 1. Definition: 2. Explanation: 3. Examples: 4. Converse of the Triangle Inequality Theorem: 5. Importance of the Theorem: If&nbsp;0\u2264a\u2264b\u2264c,&nbsp;then&nbsp;a,&nbsp;b,&nbsp;and&nbsp;c&nbsp;are the side lengths of a triangle if and only if&nbsp; Learn with an example Can&nbsp;the sides of a triangle have lengths&nbsp;1,&nbsp;6,&nbsp;and&nbsp;9? First,&nbsp;put the three numbers in order from smallest to<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/o-7-triangle-inequality-theorem\/\">Continue reading <span class=\"screen-reader-text\">&#8220;O.7 Triangle Inequality Theorem&#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-297","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=297"}],"version-history":[{"count":10,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/297\/revisions"}],"predecessor-version":[{"id":14306,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/297\/revisions\/14306"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}