{"id":293,"date":"2022-04-13T10:35:20","date_gmt":"2022-04-13T10:35:20","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=293"},"modified":"2024-10-23T14:17:56","modified_gmt":"2024-10-23T14:17:56","slug":"o-5-identify-medians-altitudes-angle-bisectors-and-perpendicular-bisectors","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/o-5-identify-medians-altitudes-angle-bisectors-and-perpendicular-bisectors\/","title":{"rendered":"O.5 Identify medians, altitudes, angle bisectors and perpendicular bisectors"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Identify medians, altitudes, angle bisectors and perpendicular bisectors<\/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<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h4 class=\"wp-block-heading\">1. <strong>Median of a Triangle<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition<\/strong>: A median is a line segment drawn from a vertex of the triangle to the midpoint of the opposite side.<\/li>\n\n\n\n<li><strong>Key Properties<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Every triangle has three medians, one from each vertex.<\/li>\n\n\n\n<li>The medians intersect at a point called the <strong>centroid<\/strong>. The centroid divides each median into two parts, with the longer part being twice the shorter part.<\/li>\n\n\n\n<li>The centroid is the <strong>center of mass<\/strong> or <strong>balance point<\/strong> 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=\"339\" height=\"149\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-14.png\" alt=\"\" class=\"wp-image-14299\" style=\"width:451px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-14.png 339w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-14-300x132.png 300w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/figure><\/div>\n\n\n<h4 class=\"wp-block-heading\">2. <strong>Altitude of a Triangle<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition<\/strong>: An altitude is a perpendicular line segment from a vertex to the opposite side (or the line containing the opposite side).<\/li>\n\n\n\n<li><strong>Key Properties<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Every triangle has three altitudes, one from each vertex.<\/li>\n\n\n\n<li>The altitudes meet at a point called the <strong>orthocenter<\/strong>.<\/li>\n\n\n\n<li>In a <strong>right-angled triangle<\/strong>, the two legs of the triangle are the altitudes.<\/li>\n\n\n\n<li>In an <strong>obtuse triangle<\/strong>, one of the altitudes will lie outside 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=\"161\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-15.png\" alt=\"\" class=\"wp-image-14300\" style=\"width:442px;height:auto\"\/><\/figure><\/div>\n\n\n<h4 class=\"wp-block-heading\">3. <strong>Angle Bisector of a Triangle<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition<\/strong>: An angle bisector is a line segment that divides an angle of the triangle into two equal angles.<\/li>\n\n\n\n<li><strong>Key Properties<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Every triangle has three angle bisectors.<\/li>\n\n\n\n<li>The angle bisectors meet at a point called the <strong>incenter<\/strong>, which is equidistant from the sides of the triangle.<\/li>\n\n\n\n<li>The incenter is the center of the <strong>inscribed circle<\/strong> (the largest circle that can fit inside the triangle and touch all three sides).<\/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=\"283\" height=\"178\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-16.png\" alt=\"\" class=\"wp-image-14301\" style=\"width:369px;height:auto\"\/><\/figure><\/div>\n\n\n<h4 class=\"wp-block-heading\">4. <strong>Perpendicular Bisector of a Triangle<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition<\/strong>: A perpendicular bisector is a line that is perpendicular to a side of the triangle and divides that side into two equal parts.<\/li>\n\n\n\n<li><strong>Key Properties<\/strong>:\n<ul class=\"wp-block-list\">\n<li>Every triangle has three perpendicular bisectors.<\/li>\n\n\n\n<li>The perpendicular bisectors meet at a point called the <strong>circumcenter<\/strong>, which is equidistant from the three vertices of the triangle.<\/li>\n\n\n\n<li>The circumcenter is the center of the <strong>circumscribed circle<\/strong> (the circle that passes through all three vertices of the triangle).<\/li>\n\n\n\n<li>In a <strong>right-angled triangle<\/strong>, the circumcenter lies at the midpoint of the hypotenuse.<\/li>\n\n\n\n<li>In an <strong>acute triangle<\/strong>, the circumcenter is inside the triangle, while in an <strong>obtuse triangle<\/strong>, it lies outside<\/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=\"237\" height=\"212\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2024\/10\/image-17.png\" alt=\"\" class=\"wp-image-14302\" style=\"width:319px;height:auto\"\/><\/figure><\/div>\n\n\n<h4 class=\"wp-block-heading\">5. <strong>Comparison of Properties<\/strong>:<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Centroid<\/strong>: Intersection of medians; divides medians in a 2:1 ratio.<\/li>\n\n\n\n<li><strong>Orthocenter<\/strong>: Intersection of altitudes; can be inside, on, or outside the triangle.<\/li>\n\n\n\n<li><strong>Incenter<\/strong>: Intersection of angle bisectors; center of the inscribed circle.<\/li>\n\n\n\n<li><strong>Circumcenter<\/strong>: Intersection of perpendicular bisectors; center of the circumscribed circle.<\/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:#e9adad\"><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>\u2220QRT \u2245 \u2220SRT.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"454\" height=\"385\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-50.png\" alt=\"\" class=\"wp-image-11418\" style=\"width:308px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-50.png 454w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-50-300x254.png 300w\" sizes=\"auto, (max-width: 454px) 100vw, 454px\" \/><\/figure><\/div>\n\n\n<p class=\"has-text-color has-link-color wp-elements-59bba2c6e7777b9dd024c6f03e150138\" style=\"color:#b00012\">Which&nbsp;term describes&nbsp;RT ?<\/p>\n\n\n\n<p>a ) median<\/p>\n\n\n\n<p>b ) angle bisector<\/p>\n\n\n\n<p>c ) perpendicular bisector<\/p>\n\n\n\n<p>d ) altitude <\/p>\n<\/div><\/div>\n\n\n\n<p>\u2220QRT \u2245 \u2220SRT ,&nbsp;so&nbsp;RT bisects  \u2220QRS.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"478\" height=\"406\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154207.043.png\" alt=\"\" class=\"wp-image-11419\" style=\"width:421px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154207.043.png 478w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154207.043-300x255.png 300w\" sizes=\"auto, (max-width: 478px) 100vw, 478px\" \/><\/figure><\/div>\n\n\n<p>Therefore,&nbsp;RT is an angle&nbsp;bisector.<\/p>\n\n\n\n<p>Also,&nbsp;since there is not enough information to know whether&nbsp;RT&nbsp;bisects&nbsp;QS&nbsp;or whether&nbsp;RT&nbsp;is perpendicular to&nbsp;QS,&nbsp;you cannot conclude that&nbsp;RT&nbsp;is a median, perpendicular bisector, or&nbsp;altitude.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#95b6ec\"><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>\u2220FEG \u2245 \u2220GEH.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"610\" height=\"459\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-51.png\" alt=\"\" class=\"wp-image-11420\" style=\"width:419px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-51.png 610w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-51-300x226.png 300w\" sizes=\"auto, (max-width: 610px) 100vw, 610px\" \/><\/figure><\/div>\n\n\n<p class=\"has-text-color has-link-color wp-elements-e21e98bfc1c7211793fa7aa862eb7a32\" style=\"color:#b00012\">Which&nbsp;term describes&nbsp;EG ?<\/p>\n\n\n\n<p>a ) angle bisector<\/p>\n\n\n\n<p>b ) altitude<\/p>\n\n\n\n<p>c ) perpendicular bisector<\/p>\n\n\n\n<p>d )  median<\/p>\n<\/div><\/div>\n\n\n\n<p>\u2220FEG \u2245 \u2220GEH,&nbsp;so EG &nbsp;bisects&nbsp;\u2220FEH.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"595\" height=\"419\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154808.194.png\" alt=\"\" class=\"wp-image-11421\" style=\"width:558px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154808.194.png 595w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T154808.194-300x211.png 300w\" sizes=\"auto, (max-width: 595px) 100vw, 595px\" \/><\/figure><\/div>\n\n\n<p>Therefore,&nbsp;EG is an angle&nbsp;bisector.<\/p>\n\n\n\n<p>Also,&nbsp;since there is not enough information to know whether&nbsp;EG&nbsp;bisects&nbsp;FH&nbsp;or whether&nbsp;EG&nbsp;is perpendicular to&nbsp;FH,&nbsp;you cannot conclude that&nbsp;EG&nbsp;is a median, perpendicular bisector, or&nbsp;altitude.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f6c0c0\"><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>\u2220GJH \u2245 \u2220HJI.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"545\" height=\"235\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-52.png\" alt=\"\" class=\"wp-image-11422\" style=\"width:429px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-52.png 545w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-52-300x129.png 300w\" sizes=\"auto, (max-width: 545px) 100vw, 545px\" \/><\/figure><\/div>\n\n\n<p class=\"has-text-color has-link-color wp-elements-c7bbe2e3891f18aa8ee68c6d5a5599a2\" style=\"color:#b00012\">Which&nbsp;term describes&nbsp; JH ?<\/p>\n\n\n\n<p>a ) median<\/p>\n\n\n\n<p>b ) altitude <\/p>\n\n\n\n<p>c ) perpendicular bisector<\/p>\n\n\n\n<p>d )  angle bisector<\/p>\n<\/div><\/div>\n\n\n\n<p>\u2220GJH \u2245 \u2220HJI, &nbsp;so&nbsp; JH&nbsp; bisects&nbsp; \u2220GJI.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"720\" height=\"277\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T155715.446.png\" alt=\"\" class=\"wp-image-11423\" style=\"width:460px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T155715.446.png 720w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/image-removebg-preview-2023-12-30T155715.446-300x115.png 300w\" sizes=\"auto, (max-width: 720px) 100vw, 720px\" \/><\/figure><\/div>\n\n\n<p>Therefore,&nbsp;JH is an angle&nbsp;bisector.<\/p>\n\n\n\n<p>Also,&nbsp;since there is not enough information to know whether&nbsp;JH&nbsp;bisects&nbsp;GI&nbsp;or whether&nbsp;JH&nbsp;is perpendicular to&nbsp;GI ,&nbsp;you cannot conclude that&nbsp;JH&nbsp;is a median, perpendicular bisector, or&nbsp;altitude.<\/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\/80470\/513\/327\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-96.png\" alt=\"\" class=\"wp-image-7148\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-96.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-96-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-96-150x150.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\/80217\/087\/481\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-114.png\" alt=\"\" class=\"wp-image-7149\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-114.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-114-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-114-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Identify medians, altitudes, angle bisectors and perpendicular bisectors Key Notes : 1. Median of a Triangle: 2. Altitude of a Triangle: 3. Angle Bisector of a Triangle: 4. Perpendicular Bisector of a Triangle: 5. Comparison of Properties: Learn with an example \u2220QRT \u2245 \u2220SRT. Which&nbsp;term describes&nbsp;RT ? a ) median b ) angle bisector c<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/o-5-identify-medians-altitudes-angle-bisectors-and-perpendicular-bisectors\/\">Continue reading <span class=\"screen-reader-text\">&#8220;O.5 Identify medians, altitudes, angle bisectors and perpendicular bisectors&#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-293","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/293","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=293"}],"version-history":[{"count":11,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/293\/revisions"}],"predecessor-version":[{"id":14303,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/293\/revisions\/14303"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}