{"id":234,"date":"2022-04-13T10:24:38","date_gmt":"2022-04-13T10:24:38","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=234"},"modified":"2025-02-08T08:57:18","modified_gmt":"2025-02-08T08:57:18","slug":"l-6-perpendicular-bisector-theorem","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/l-6-perpendicular-bisector-theorem\/","title":{"rendered":"L.6 Perpendicular Bisector Theorem"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Perpendicular Bisector 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<p class=\"has-large-font-size\">A&nbsp;point is equidistant from other points if it is the same distance from&nbsp;them.<\/p>\n\n\n\n<p class=\"has-large-font-size\">The&nbsp;<strong>Perpendicular&nbsp;Bisector&nbsp;Theorem<\/strong>&nbsp;states that a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the&nbsp;segment.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-23.png\" alt=\"\" class=\"wp-image-10965\" style=\"width:228px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-23.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-23-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled-design-23-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p class=\"has-large-font-size\">X&nbsp;is on the perpendicular bisector of WY if and only if WX \u2245 XY .<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">Definition:<\/h3>\n\n\n\n<p class=\"has-large-font-size\">The Perpendicular Bisector Theorem is a geometric principle that relates to a line segment and its perpendicular bisector.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">Statement:<\/h3>\n\n\n\n<p class=\"has-large-font-size\">If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">Key Concepts:<\/h3>\n\n\n\n<ol class=\"wp-block-list has-large-font-size\">\n<li><strong>Perpendicular Bisector:<\/strong>\n<ul class=\"wp-block-list\">\n<li class=\"has-large-font-size\">A line, segment, or ray that intersects another line segment at its midpoint and forms a right angle (90 degrees) with the segment.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li class=\"has-large-font-size\"><strong>Equidistant:<\/strong>\n<ul class=\"wp-block-list\">\n<li>The distance from a point to each endpoint of a segment is equal.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">Theorem Statement:<\/h3>\n\n\n\n<p class=\"has-large-font-size\">If a point <em>P<\/em> lies on the perpendicular bisector of segment <em>AB<\/em>, then <em>PA<\/em>=<em>PB<\/em>, where <em>PA<\/em> and <em>PB<\/em> are the distances from point <em>P<\/em> to the endpoints <em>A<\/em> and <em>B<\/em> respectively.<\/p>\n\n\n\n<h3 class=\"wp-block-heading has-large-font-size\">Example:<\/h3>\n\n\n\n<p class=\"has-large-font-size\">Consider line segment <em>AB<\/em> with midpoint <em>M<\/em>. If <em>P<\/em> is a point on the perpendicular bisector of <em>AB<\/em>, then <em>PM<\/em>=<em>PA<\/em>=<em>PB<\/em>.<\/p>\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:#c7eeba\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group\"><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-text-color has-link-color wp-elements-0c468b3d76268ecfb4065b58b7a648f9\" style=\"color:#b00012\"><strong>What&nbsp;is&nbsp;GI?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"334\" height=\"231\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-38.png\" alt=\"\" class=\"wp-image-10020\" style=\"width:305px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-38.png 334w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-38-300x207.png 300w\" sizes=\"auto, (max-width: 334px) 100vw, 334px\" \/><\/figure><\/div>\n\n\n<p><em>GI<\/em>= ______________<\/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 loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__24_-removebg-preview.png\" alt=\"\" class=\"wp-image-10966\" style=\"width:348px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__24_-removebg-preview.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__24_-removebg-preview-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__24_-removebg-preview-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>since GH = HI = 84 , H &nbsp;is equidistant from the endpoints of GI . Therefore H&nbsp;lies on the perpendicular bisector of GI by the Perpendicular Bisector&nbsp;Theorem.<\/p>\n\n\n\n<p>Since&nbsp;H&nbsp;is on the perpendicular bisector of&nbsp;GI and HJI  is a right angle ,HJ &nbsp;is the perpendicular bisector of&nbsp;GI .<\/p>\n\n\n\n<p>So, GJ = IJ = 69 and G . I = 2.69 = 138.<\/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:#c8e7f1\"><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-text-color has-link-color wp-elements-6b5dc268050e3bec4296a999253b1068\" style=\"color:#b00012\"><strong> What&nbsp;is&nbsp;QR?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"261\" height=\"263\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-43.png\" alt=\"\" class=\"wp-image-10025\" style=\"width:280px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-43.png 261w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-43-150x150.png 150w\" sizes=\"auto, (max-width: 261px) 100vw, 261px\" \/><\/figure><\/div>\n\n\n<p><em>QR<\/em>=_____________<\/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 loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__25_-removebg-preview.png\" alt=\"\" class=\"wp-image-10975\" style=\"width:354px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__25_-removebg-preview.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__25_-removebg-preview-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__25_-removebg-preview-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>Since \u2220QPR is a right angle and PQ = PS = 9 , PR<bdo dir=\"ltr\" lang=\"PR\"> <\/bdo>is the perpendicular bisector of QS .<\/p>\n\n\n\n<p>Therefore , QR \u2245 RS  by the Perpendicular Bisector&nbsp;Theorem.<\/p>\n\n\n\n<p>So , QR = RS = 11.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#e7ffd3\"><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-text-color has-link-color wp-elements-3a316f284974a5cdb20a695beb19875d\" style=\"color:#b00012\"><strong>If&nbsp;<em>QR<\/em>=<em>RS<\/em>=66&nbsp;and&nbsp;<em>ST<\/em>=42,&nbsp;what is&nbsp;QS?<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"229\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-45.png\" alt=\"\" class=\"wp-image-10027\" style=\"width:330px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-45.png 323w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/11\/image-45-300x213.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><\/figure><\/div>\n\n\n<p><em>QS<\/em>=__________<\/p>\n<\/div><\/div>\n\n\n\n<p>Label&nbsp;the diagram with the information given in the&nbsp;question.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__26_-removebg-preview.png\" alt=\"\" class=\"wp-image-10981\" style=\"width:338px;height:auto\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__26_-removebg-preview.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__26_-removebg-preview-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/12\/Untitled_design__26_-removebg-preview-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/figure><\/div>\n\n\n<p>Since QR = RS = 66 , R &nbsp;is equidistant from the endpoints of QS . Therefore R&nbsp;lies on the perpendicular bisector of QS by the Perpendicular Bisector&nbsp;Theorem.<\/p>\n\n\n\n<p>Since&nbsp;R&nbsp;is on the perpendicular bisector of&nbsp;QS and \u2220QTR  is a right angle ,RT &nbsp;is the perpendicular bisector of&nbsp;QS .<\/p>\n\n\n\n<p>So, QT = ST = 42 and QS = 2. 42 = 84.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\"><strong>let&#8217;s practice!<\/strong><\/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\/86342\/410\/509\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-70.png\" alt=\"\" class=\"wp-image-7040\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-70.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-70-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-70-150x150.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\/63830\/722\/515\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-88.png\" alt=\"\" class=\"wp-image-7041\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-88.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-88-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-88-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Perpendicular Bisector Theorem Key Notes : A&nbsp;point is equidistant from other points if it is the same distance from&nbsp;them. The&nbsp;Perpendicular&nbsp;Bisector&nbsp;Theorem&nbsp;states that a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the&nbsp;segment. X&nbsp;is on the perpendicular bisector of WY if and only if WX<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/l-6-perpendicular-bisector-theorem\/\">Continue reading <span class=\"screen-reader-text\">&#8220;L.6 Perpendicular Bisector 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-234","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/234","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=234"}],"version-history":[{"count":16,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/234\/revisions"}],"predecessor-version":[{"id":17512,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/234\/revisions\/17512"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}