{"id":228321,"date":"2025-06-07T08:01:31","date_gmt":"2025-06-07T08:01:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=228321"},"modified":"2025-06-07T08:01:33","modified_gmt":"2025-06-07T08:01:33","slug":"the-coordinates-of-the-endpoints-of-ab-and-cd-are-ax1-ay1-bx2-by2-cx3-cy3-and-dx4-dy4","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/07\/the-coordinates-of-the-endpoints-of-ab-and-cd-are-ax1-ay1-bx2-by2-cx3-cy3-and-dx4-dy4\/","title":{"rendered":"The coordinates of the endpoints of AB and CD are (Ax1, Ay1), (Bx2, By2), (Cx3, Cy3), and (Dx4, Dy4)."},"content":{"rendered":"\n

The coordinates of the endpoints of AB and CD are (Ax1, Ay1), (Bx2, By2), (Cx3, Cy3), and (Dx4, Dy4). Which condition proves that AB || CD? A. (Ax1 – Bx2) \/ (Ay1 – By2) = (Cx3 – Dx4) \/ (Cy3 – Dy4) B. (Ax1 – Cx3) \/ (Ay1 – Cy3) = (Bx2 – Dx4) \/ (By2 – Dy4) C. (Ax1 – Dx4) \/ (Ay1 – Dy4) = (Bx2 – Cx3) \/ (By2 – Cy3) D. (Ay1 – By2) \/ (Ax1 – Bx2) = (Cy3 – Dy4) \/ (Cx3 – Dx4)<\/p>\n\n\n\n

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

The correct answer is: D. (Ay1 – By2) \/ (Ax1 – Bx2) = (Cy3 – Dy4) \/ (Cx3 – Dx4)<\/strong><\/p>\n\n\n\n

\n\n\n\n

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

To determine whether two line segments, AB<\/strong> and CD<\/strong>, are parallel<\/strong>, we compare their slopes<\/strong>. Two lines are parallel if and only if they have the same slope<\/strong> (assuming they are not vertical lines).<\/p>\n\n\n\n

The slope<\/strong> of a line segment connecting two points (x1,y1)(x_1, y_1)(x1\u200b,y1\u200b) and (x2,y2)(x_2, y_2)(x2\u200b,y2\u200b) is calculated by:slope=y2\u2212y1x2\u2212x1\\text{slope} = \\frac{y_2 – y_1}{x_2 – x_1}slope=x2\u200b\u2212x1\u200by2\u200b\u2212y1\u200b\u200b<\/p>\n\n\n\n

Now, let\u2019s apply this to the segments:<\/p>\n\n\n\n

    \n
  • Segment AB<\/strong> has endpoints (Ax1,Ay1)(Ax1, Ay1)(Ax1,Ay1) and (Bx2,By2)(Bx2, By2)(Bx2,By2).
    Its slope is: mAB=By2\u2212Ay1Bx2\u2212Ax1m_{AB} = \\frac{By2 – Ay1}{Bx2 – Ax1}mAB\u200b=Bx2\u2212Ax1By2\u2212Ay1\u200b<\/li>\n\n\n\n
  • Segment CD<\/strong> has endpoints (Cx3,Cy3)(Cx3, Cy3)(Cx3,Cy3) and (Dx4,Dy4)(Dx4, Dy4)(Dx4,Dy4).
    Its slope is: mCD=Dy4\u2212Cy3Dx4\u2212Cx3m_{CD} = \\frac{Dy4 – Cy3}{Dx4 – Cx3}mCD\u200b=Dx4\u2212Cx3Dy4\u2212Cy3\u200b<\/li>\n<\/ul>\n\n\n\n

    For AB and CD to be parallel:mAB=mCD\u21d2By2\u2212Ay1Bx2\u2212Ax1=Dy4\u2212Cy3Dx4\u2212Cx3m_{AB} = m_{CD} \\Rightarrow \\frac{By2 – Ay1}{Bx2 – Ax1} = \\frac{Dy4 – Cy3}{Dx4 – Cx3}mAB\u200b=mCD\u200b\u21d2Bx2\u2212Ax1By2\u2212Ay1\u200b=Dx4\u2212Cx3Dy4\u2212Cy3\u200b<\/p>\n\n\n\n

    However, option D<\/strong> is just a rearrangement using equivalent expressions (the slopes written from the first point to the second, and signs flipped):Ay1\u2212By2Ax1\u2212Bx2=Cy3\u2212Dy4Cx3\u2212Dx4\\frac{Ay1 – By2}{Ax1 – Bx2} = \\frac{Cy3 – Dy4}{Cx3 – Dx4}Ax1\u2212Bx2Ay1\u2212By2\u200b=Cx3\u2212Dx4Cy3\u2212Dy4\u200b<\/p>\n\n\n\n

    Since:Ay1\u2212By2Ax1\u2212Bx2=\u2212(By2\u2212Ay1)\u2212(Bx2\u2212Ax1)=By2\u2212Ay1Bx2\u2212Ax1\\frac{Ay1 – By2}{Ax1 – Bx2} = \\frac{-(By2 – Ay1)}{-(Bx2 – Ax1)} = \\frac{By2 – Ay1}{Bx2 – Ax1}Ax1\u2212Bx2Ay1\u2212By2\u200b=\u2212(Bx2\u2212Ax1)\u2212(By2\u2212Ay1)\u200b=Bx2\u2212Ax1By2\u2212Ay1\u200b<\/p>\n\n\n\n

    the slopes remain equal. Therefore, option D correctly expresses the condition for AB and CD to be parallel<\/strong>.<\/p>\n\n\n\n

    Options A, B, and C do not compare the slopes of AB and CD directly, and thus do not prove parallelism between the two segments.<\/p>\n\n\n\n

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

    The coordinates of the endpoints of AB and CD are (Ax1, Ay1), (Bx2, By2), (Cx3, Cy3), and (Dx4, Dy4). Which condition proves that AB || CD? A. (Ax1 – Bx2) \/ (Ay1 – By2) = (Cx3 – Dx4) \/ (Cy3 – Dy4) B. (Ax1 – Cx3) \/ (Ay1 – Cy3) = (Bx2 – Dx4) \/ […]<\/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-228321","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/228321","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=228321"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/228321\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=228321"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=228321"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=228321"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}