{"id":229455,"date":"2025-06-08T07:53:01","date_gmt":"2025-06-08T07:53:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=229455"},"modified":"2025-06-08T07:53:04","modified_gmt":"2025-06-08T07:53:04","slug":"the-given-line-segment-has-a-midpoint-at-31","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/08\/the-given-line-segment-has-a-midpoint-at-31\/","title":{"rendered":"The given line segment has a midpoint at (3,1)."},"content":{"rendered":"\n

The given line segment has a midpoint at (3,1). What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment? Writing Linear Equation: corelearn.edgenuity.com\/Player\/ ess 1206310 Geometry SlandS2 2016 Unit Test Active TIME REMAINING 01:25:52 The given line segment has a midpoint at (3,1). What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment? Y = y = ix-2 y = 3x y = 3x – 8 Save and Exit Next Submit Mark this and return Sign out DeLL<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find the equation of the perpendicular bisector<\/strong> of the given line segment, we need to follow these steps:<\/p>\n\n\n\n

\n\n\n\n

Step 1: Identify the endpoints of the line segment<\/strong><\/h3>\n\n\n\n

From the image, the endpoints are:<\/p>\n\n\n\n

    \n
  • A=(2,4)A = (2, 4)A=(2,4)<\/li>\n\n\n\n
  • B=(4,\u22122)B = (4, -2)B=(4,\u22122)<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 2: Find the midpoint<\/strong><\/h3>\n\n\n\n

    The midpoint formula is:Midpoint=(x1+x22,y1+y22)=(2+42,4+(\u22122)2)=(3,1)\\text{Midpoint} = \\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right) = \\left( \\frac{2 + 4}{2}, \\frac{4 + (-2)}{2} \\right) = (3, 1)Midpoint=(2×1\u200b+x2\u200b\u200b,2y1\u200b+y2\u200b\u200b)=(22+4\u200b,24+(\u22122)\u200b)=(3,1)<\/p>\n\n\n\n

    \u2705 This matches the given midpoint.<\/p>\n\n\n\n

    \n\n\n\n

    Step 3: Find the slope of the line segment AB<\/strong><\/h3>\n\n\n\n

    Slope formula:m=y2\u2212y1x2\u2212x1=\u22122\u221244\u22122=\u221262=\u22123m = \\frac{y_2 – y_1}{x_2 – x_1} = \\frac{-2 – 4}{4 – 2} = \\frac{-6}{2} = -3m=x2\u200b\u2212x1\u200by2\u200b\u2212y1\u200b\u200b=4\u22122\u22122\u22124\u200b=2\u22126\u200b=\u22123<\/p>\n\n\n\n

    \n\n\n\n

    Step 4: Find the slope of the perpendicular bisector<\/strong><\/h3>\n\n\n\n

    The perpendicular slope is the negative reciprocal<\/strong> of \u22123-3\u22123, which is:13\\frac{1}{3}31\u200b<\/p>\n\n\n\n

    \n\n\n\n

    Step 5: Use point-slope form to write the equation<\/strong><\/h3>\n\n\n\n

    We know the perpendicular bisector passes through the midpoint (3,1)(3, 1)(3,1) and has slope 13\\frac{1}{3}31\u200b.<\/p>\n\n\n\n

    Point-slope form:y\u2212y1=m(x\u2212x1)y – y_1 = m(x – x_1)y\u2212y1\u200b=m(x\u2212x1\u200b)<\/p>\n\n\n\n

    Substitute:y\u22121=13(x\u22123)y – 1 = \\frac{1}{3}(x – 3)y\u22121=31\u200b(x\u22123)<\/p>\n\n\n\n

    \n\n\n\n

    Step 6: Convert to slope-intercept form (y = mx + b)<\/strong><\/h3>\n\n\n\n

    y\u22121=13x\u22121y=13x\u22121+1y=13xy – 1 = \\frac{1}{3}x – 1 \\\\ y = \\frac{1}{3}x – 1 + 1 \\\\ y = \\frac{1}{3}xy\u22121=31\u200bx\u22121y=31\u200bx\u22121+1y=31\u200bx<\/p>\n\n\n\n

    \n\n\n\n

    \u2705 Final Answer:<\/h3>\n\n\n\n

    y=13x\\boxed{y = \\frac{1}{3}x}y=31\u200bx\u200b<\/p>\n\n\n\n

    \n\n\n\n

    Correct Option:<\/strong><\/h3>\n\n\n\n
      \n
    • \u2705 A.<\/strong> y=13xy = \\frac{1}{3}xy=31\u200bx<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Explanation<\/strong><\/h3>\n\n\n\n

      To determine the equation of the perpendicular bisector of a line segment, we need to understand two geometric concepts: the midpoint and perpendicular slope. The perpendicular bisector is a line that cuts a segment exactly in half at a 90\u00b0 angle. First, we identify the endpoints of the segment from the graph: (2, 4) and (4, -2). Using the midpoint formula, we calculate the midpoint as (3,1)(3, 1)(3,1), which matches the one given.<\/p>\n\n\n\n

      Next, we calculate the slope of the original line segment using the slope formula: rise over run. The slope from (2, 4) to (4, -2) is \u221262=\u22123\\frac{-6}{2} = -32\u22126\u200b=\u22123. A line perpendicular to this must have a slope that is the negative reciprocal of -3, which is 13\\frac{1}{3}31\u200b. This means the perpendicular bisector has a slope of 13\\frac{1}{3}31\u200b.<\/p>\n\n\n\n

      Now that we have the slope and a point the line goes through (3,1)(3, 1)(3,1), we can use the point-slope form of a line: y\u2212y1=m(x\u2212x1)y – y_1 = m(x – x_1)y\u2212y1\u200b=m(x\u2212x1\u200b). Plugging in our values gives us:y\u22121=13(x\u22123)y – 1 = \\frac{1}{3}(x – 3)y\u22121=31\u200b(x\u22123)<\/p>\n\n\n\n

      Distributing and simplifying converts this to slope-intercept form: y=13xy = \\frac{1}{3}xy=31\u200bx.<\/p>\n\n\n\n

      This equation describes a line with slope 13\\frac{1}{3}31\u200b that passes through the midpoint and is perpendicular to the original segment. Thus, the correct answer is y=13xy = \\frac{1}{3}xy=31\u200bx<\/strong>.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      The given line segment has a midpoint at (3,1). What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment? Writing Linear Equation: corelearn.edgenuity.com\/Player\/ ess 1206310 Geometry SlandS2 2016 Unit Test Active TIME REMAINING 01:25:52 The given line segment has a midpoint at (3,1). What is the equation, in slope-intercept […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-229455","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229455","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=229455"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229455\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229455"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229455"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}