{"id":32267,"date":"2023-07-24T06:19:35","date_gmt":"2023-07-24T06:19:35","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=32267"},"modified":"2023-07-24T07:17:39","modified_gmt":"2023-07-24T07:17:39","slug":"concurrent-lines-definition-formula-examples-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines","title":{"rendered":"Concurrent Lines &#8211; Definition, Formula, Examples, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-25f403b6-882e-41f9-b829-db6d1d3b3da7\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#0-what-are-concurrent-lines-in-geometry>What Are Concurrent Lines in Geometry?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#2-what-is-the-point-of-concurrency>What Is the Point of Concurrency?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#7-difference-between-concurrent-lines-and-intersecting-lines>Difference between Concurrent Lines and Intersecting Lines<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#9-solved-examples-on-concurrent-lines>Solved Examples on Concurrent Lines<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#10-practice-problems-on-concurrent-lines>Practice Problems on Concurrent Lines<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines#11-frequently-asked-questions-on-concurrent-lines>Frequently Asked Questions on Concurrent Lines<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-are-concurrent-lines-in-geometry\">What Are Concurrent Lines in Geometry?<\/h2>\n\n\n\n<p><strong>Concurrent lines are three or more lines meeting exactly at a single point.<\/strong><\/p>\n\n\n\n<p>We know that two <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/line\">lines<\/a> can intersect at one point only. What if there are three or more lines meeting at a single point? Take a look at the image below. The lines l, m, and n intersect each other at a single point, O. Thus, lines l, m, and n are concurrent.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"424\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines.png\" alt=\"Concurrent lines l, m, and n intersecting at the point O\" class=\"wp-image-32274\" title=\"Concurrent lines l, m, and n intersecting at the point O\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-300x205.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/add-like-fractions-using-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/fractions_add_like_frac_sum_upto_1_nl_pt.png\" alt=\"Add Like Fractions using Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Add Like Fractions using Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/divide-using-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_facts_divide_nl_pt.png\" alt=\"Divide using Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Divide using Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-fractions-on-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/fractions_identify_fraction_nl_pt.png\" alt=\"Identify Fractions on Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Fractions on Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-lines-line-segments-rays-angles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_identify_pt_line_ray_angle_pt.png\" alt=\"Identify Lines, Line Segments, Rays, Angles Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Lines, Line Segments, Rays, Angles Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-parallel-and-perpendicular-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_identify_types_of_lines_pt.png\" alt=\"Identify Parallel and Perpendicular Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify Parallel and Perpendicular Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-the-lines-of-symmetry-in-irregular-shapes\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_symm_irregular_shape_pt.png\" alt=\"Identify the LInes of Symmetry in Irregular Shapes Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify the LInes of Symmetry in Irregular Shapes Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/identify-the-lines-of-symmetry-in-quadrilaterals\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geometry_symm_regular_shape_pt.png\" alt=\"Identify the LInes of Symmetry in Quadrilaterals Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Identify the LInes of Symmetry in Quadrilaterals Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/mark-fractions-on-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/fractions_place_fraction_nl_pt.png\" alt=\"Mark Fractions on Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Mark Fractions on Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/mark-unit-fractions-on-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/fractions_place_unit_fraction_nl_pt.png\" alt=\"Mark Unit Fractions on Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Mark Unit Fractions on Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/missing-denominators-on-number-lines\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/fractions_missing_deno_nl_pt.png\" alt=\"Missing Denominators on Number Lines Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Missing Denominators on Number Lines Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-concurrent-lines-definition\">Concurrent Lines: Definition<\/h2>\n\n\n\n<p><strong>Three or more lines that intersect at one common point are said to be concurrent lines.&nbsp;<\/strong><\/p>\n\n\n\n<p>When two lines meet each other, they are said to be intersecting. When a third line passes through the point of intersection made by the first two lines, then these three lines are known as the concurrent lines.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-what-is-the-point-of-concurrency\">What Is the Point of Concurrency?<\/h2>\n\n\n\n<p>The point of intersection of three or more lines is known as the \u201cpoint of concurrency.\u201d It is the point where three or more lines meet.<\/p>\n\n\n\n<p>Concurrent lines examples:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"367\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-and-point-of-concurrency.png\" alt=\"Concurrent lines and their point of concurrency\" class=\"wp-image-32276\" title=\"Concurrent lines and their point of concurrency\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-and-point-of-concurrency.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-and-point-of-concurrency-300x178.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-condition-for-concurrent-lines-concurrent-lines-formula\">Condition for Concurrent Lines (Concurrent Lines Formula)<\/h2>\n\n\n\n<p>If three lines are concurrent, then the point of intersection of two lines satisfies the equation of the third line (i.e., it lies on the third line).&nbsp;<\/p>\n\n\n\n<p>Consider three lines whose equations are:<\/p>\n\n\n\n<p>$a_{1} x + b_{1} y + c_{1}\u00a0 = 0$ \u00a0 \u2026\u2026\u2026\u2026\u2026. (1)\u00a0\u00a0<\/p>\n\n\n\n<p>$a_{2} x + b_{2} y + c_{2} = 0$ \u00a0 \u2026\u2026\u2026\u2026\u2026. (2)\u00a0<\/p>\n\n\n\n<p>$a_{3} x + b_{3} y + c_{3} = 0$ \u00a0 \u2026\u2026\u2026\u2026\u2026. (3)<\/p>\n\n\n\n<p>If the determinant of the coefficients is 0, then the lines are concurrent. Thus, the condition for the lines to be concurrent is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"358\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/condition-for-concurrent-lines.png\" alt=\"Condition for concurrent lines\" class=\"wp-image-32277\" title=\"Condition for concurrent lines\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/condition-for-concurrent-lines.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/condition-for-concurrent-lines-300x173.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-how-to-know-if-lines-are-concurrent\">How to Know if Lines Are concurrent<\/h2>\n\n\n\n<p>How can we know if the given lines are concurrent? There are two methods to find if three lines are concurrent or not:<\/p>\n\n\n\n<p><strong>Method 1: Determinant Method<\/strong><\/p>\n\n\n\n<p>If the three lines are concurrent, then the determinant of coefficients is 0. Thus, the lines should follow the condition for concurrent lines.<\/p>\n\n\n\n<p>Suppose, the equation of three lines are:<\/p>\n\n\n\n<p>$2x \\;\u2013\\; 3y + 5 = 0$<\/p>\n\n\n\n<p>$3x + 4y \\;\u2013\\; 7 = 0$<\/p>\n\n\n\n<p>$9x \\;\u2013\\; 5y + 8 = 0$<\/p>\n\n\n\n<p>Determinant $D\u00a0 = 2(32 \\;\u2013\\; 35) \\;-\\; (\\;-\\;3)(24 + 63) + 5(\\;-\\;15 \\;\u2013\\; 36)$<\/p>\n\n\n\n<p>\u00a0\u00a0\u00a0\u00a0$= 2(\\;-\\;3) + 3(87) + 5(\\;-\\;51)$<\/p>\n\n\n\n<p>\u00a0\u00a0\u00a0$= \\;\u2013\\; 6 + 261 \\;-\\;255$<\/p>\n\n\n\n<p>\u00a0\u00a0\u00a0\u00a0$= 0$<\/p>\n\n\n\n<p>Hence, the lines are concurrent.&nbsp;<\/p>\n\n\n\n<p><strong>Method 2: Solving the Equations<\/strong><\/p>\n\n\n\n<p>Here, we first find the point of intersection of two lines and then check if the point lies on the third line. It ensures that all three lines are concurrent.&nbsp;<\/p>\n\n\n\n<p>Consider equations of three lines.<\/p>\n\n\n\n<p>$2x \\;-\\; 3y + 5 = 0$ &#8212;&#8211;(1)<\/p>\n\n\n\n<p>$3x + 4y \\;-\\; 7 = 0$ &#8212;&#8211;(2)<\/p>\n\n\n\n<p>$9x \\;-\\; 5y + 8 = 0$ &#8212;&#8211;(3)\u00a0 \u00a0<\/p>\n\n\n\n<p>Solving (1) and (2) by making the coefficient of x equal.<\/p>\n\n\n\n<p>$6x \\;-\\; 9y + 15 = 0$<\/p>\n\n\n\n<p>$6x + 8y \\;-\\; 14 = 0$<\/p>\n\n\n\n<p>Subtracting, we get<\/p>\n\n\n\n<p>$17y = 29$<\/p>\n\n\n\n<p>$y = \\frac{29}{17}$<\/p>\n\n\n\n<p>Putting the value of $y = \\frac{29}{17}$ in the equation $6x \\;-\\; 9y + 15 = 0$, we get the value of \u201cx.\u201d\u00a0<\/p>\n\n\n\n<p>$6x \\;-\\; 9(\\frac{29}{17}) + 15 = 0 =$\u00a0<\/p>\n\n\n\n<p>$6x = \\frac{261}{17} \\;-\\; 15 = \\frac{6}{17}$<\/p>\n\n\n\n<p>Thus, $x = \\frac{1}{17}$<\/p>\n\n\n\n<p>Therefore, line 1 and line 2 intersect at a point $(\\frac{1}{17},\\; \\frac{29}{17})$.\u00a0<\/p>\n\n\n\n<p>Substitute the point of intersection of the first two lines in the equation of the third line.&nbsp;<\/p>\n\n\n\n<p>Equation of the third line is $9x \\;-\\; 5y + 8 = 0$.\u00a0\u00a0\u00a0\u00a0<\/p>\n\n\n\n<p>Substituting the values of (4,6) in the equation (3), we get&nbsp;<\/p>\n\n\n\n<p>$9x \\;-\\; 5y + 8 = 9(\\frac{1}{17})\\;-\\; 5(\\frac{29}{17}) + 8$<\/p>\n\n\n\n<p>\u00a0$= \\frac{9}{17} \\;-\\; \\frac{145}{17} + \\frac{136}{17}$<\/p>\n\n\n\n<p>\u00a0$= 0$<\/p>\n\n\n\n<p>The point satisfies the equation of the third line.<\/p>\n\n\n\n<p>Hence, these are concurrent lines.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-concurrent-lines-in-triangles\">Concurrent Lines in Triangles<\/h2>\n\n\n\n<p>Concurrent lines can be found in triangles when special types of <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/line-segment\">line segments<\/a> are drawn inside a triangle. In a triangle, the four important types of concurrent lines are altitudes, <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/angle-bisector\">angle bisectors<\/a>, medians, and perpendicular bisectors. Thus, there are four different points of concurrency.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Incenter:<\/strong> The point of intersection of three angle bisectors of a triangle is known as the incenter of a triangle.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Circumcenter:<\/strong> The point of intersection of three perpendicular bisectors of a triangle is known as the circumcenter of a triangle.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Centroid:<\/strong>&nbsp; The point of intersection of three medians of a triangle is known as the centroid of a triangle&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Orthocenter:<\/strong> The point of intersection of three altitudes of a triangle is known as the orthocenter of a triangle.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"482\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-triangles-1.png\" alt=\"Concurrent lines inside a triangle\" class=\"wp-image-32279\" title=\"Concurrent lines inside a triangle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-triangles-1.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-triangles-1-300x233.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-concurrent-lines-in-geometry\">Concurrent Lines in Geometry<\/h2>\n\n\n\n<p>Let\u2019s see some examples of concurrent lines in different <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometric-shapes\">geometric shapes<\/a>.<\/p>\n\n\n\n<p><strong>Circles<\/strong><\/p>\n\n\n\n<p>The diameters of a circle are concurrent such that the point of concurrency is the center of the circle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"432\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-circle.png\" alt=\"Diameters of a circle are concurrent at the center\" class=\"wp-image-32280\" title=\"Diameters of a circle are concurrent at the center\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-circle.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-in-circle-300x209.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Quadrilaterals<\/strong><\/p>\n\n\n\n<p>In quadrilaterals, the line segments joining the midpoints of opposite sides, and the two diagonals are concurrent lines.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-difference-between-concurrent-lines-and-intersecting-lines\">Difference between Concurrent Lines and Intersecting Lines<\/h2>\n\n\n\n<figure class=\"wp-block-table wj-custom-table\"><table class=\"wj-table-class\"><thead><tr><th><strong>Concurrent Lines<\/strong><\/th><th><strong>Intersecting Lines<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Three or more lines that pass through a single point in a plane are called concurrent lines.<\/td><td>Two lines meeting at a single point are called intersecting lines.<\/td><\/tr><tr><td>The point where the concurrent lines meet is called the point of concurrency.<\/td><td>The point where two intersecting lines meet is called the point of intersection.<\/td><\/tr><tr><td><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-example.png\" alt=\"Three concurrent lines\"><\/td><td><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/intersecting-lines.png\" alt=\"Two intersecting lines\"><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-facts-about-concurrent-lines\">Facts about Concurrent Lines<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/parallel-lines\">Parallel lines<\/a> are not concurrent at any point on a plane.<\/li>\n\n\n\n<li>In <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/geometry\">geometry<\/a>, the concept of concurrent lines is essential, and they play a significant role in various proofs and constructions involving angles, triangles, and other geometric shapes.<\/li>\n\n\n\n<li>The point of concurrency of three perpendicular bisectors in a triangle is the circumcenter.<\/li>\n\n\n\n<li>The point of concurrency of three angle bisectors in a triangle is the incenter.<\/li>\n\n\n\n<li>The point of concurrency of three medians in a triangle is the centroid.<\/li>\n\n\n\n<li>The point of concurrency of three altitudes in a triangle is the orthocenter.<\/li>\n\n\n\n<li>In a <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/triangle\">triangle<\/a>, the three medians, altitudes, angle bisectors, and perpendicular bisectors are examples of concurrent lines.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-solved-examples-on-concurrent-lines\">Solved Examples on Concurrent Lines<\/h2>\n\n\n\n<p><strong>1. Which lines are concurrent in the given figure? Also, tell the point of concurrency.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"285\" height=\"166\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/Solved-Examples-1.png\" alt=\"\" class=\"wp-image-32288\" title=\"Lines p, q, l , m, and n\"\/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Lines p, q, and m are concurrent at point A.<\/p>\n\n\n\n<p>Lines l, p, and n are concurrent at point B.<\/p>\n\n\n\n<p><strong>2. Name all the pairs of concurrent lines and the point of concurrency in the following figure.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"292\" height=\"132\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/identify-concurrent-lines.png\" alt=\"Lines a, b, c, m, p, q, r\" class=\"wp-image-32286\" title=\"Lines a, b, c, m, p, q, r\"\/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Lines a, b, c, and r are concurrent at point A.&nbsp;<\/p>\n\n\n\n<p>Lines p, q, and r are concurrent at point D.<\/p>\n\n\n\n<p><strong>3. Check whether the lines are concurrent or not using the determinant method.<\/strong><\/p>\n\n\n\n<p>$3x \\;-\\; 4y \\;-\\; 13\u00a0 = 0$<\/p>\n\n\n\n<p>$8x \\;-\\; 11 y \\;-\\; 33 = 0$<\/p>\n\n\n\n<p>$2x \\;-\\; 3y \\;-\\; 7 = 0$<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Condition for concurrent lines is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"197\" height=\"130\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/solved-example-on-concurrent-lines.png\" alt=\"Determinant condition for concurrent lines\" class=\"wp-image-32290\" title=\"Determinant condition for concurrent lines\"\/><\/figure>\n\n\n\n<p>Substitute the values.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"226\" height=\"130\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/concurrent-lines-problem.png\" alt=\"Determinant condition for concurrent lines - Example\" class=\"wp-image-32291\" title=\"Determinant condition for concurrent lines - Example\"\/><\/figure>\n\n\n\n<p>Solve the determinant.<\/p>\n\n\n\n<p>$D = 3(77\\;-\\;99)\\;-\\;(\\;-\\;4)(\\;-\\;56 + 66)\\;-\\;13(\\;-\\;24 + 22)$<\/p>\n\n\n\n<p>$D = 3 \\times (\\;-\\;22) + 4 \\times 10\\;-\\;13 \\times (\\;-\\;2)$<\/p>\n\n\n\n<p>$D = \\;-\\;66 + 40 + 26 = 0$<\/p>\n\n\n\n<p>Hence, the lines are concurrent.&nbsp;<\/p>\n\n\n\n<p><strong>5. Check whether the lines are concurrent or not?&nbsp;<\/strong><\/p>\n\n\n\n<p>$x + 2y \\;-\\; 4\u00a0 = 0,\\; x\\;-\\; y \\;-\\; 1 = 0,\\; 4x + 5y \\;-\\; 13 = 0$<\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Let us first consider the first two equations.<\/p>\n\n\n\n<p>$x + 2y \\;-\\; 4\u00a0 = 0$<strong>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8211;(1)<\/strong><\/p>\n\n\n\n<p>$x\\;-\\; y \\;-\\; 1 = 0$<strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &#8211;(2)<\/strong><\/p>\n\n\n\n<p>Subtracting (1) and (2)<\/p>\n\n\n\n<p>$x + 2y\\;-\\;4\\;-\\;(x\\;-\\;y\\;-\\;1) = 0$<\/p>\n\n\n\n<p>$x + 2y\\;-\\;4\\;-\\;x + y + 1 = 0$<\/p>\n\n\n\n<p>$2y + y\\;-\\;4 + 1 = 0$<\/p>\n\n\n\n<p>$3y\\;-\\;3 = 0$<\/p>\n\n\n\n<p>$y = 1$<\/p>\n\n\n\n<p>Substituting value of $y = 1$ in (2)<\/p>\n\n\n\n<p>$x\\;-\\;y\\;-\\;1 = 0\\Rightarrow x\\;-\\;1\\;-\\;1 = 0$<\/p>\n\n\n\n<p>$x\\;-\\;2 = 0\\Rightarrow x = 2$<\/p>\n\n\n\n<p>The two lines intersect at the point (2,1).<\/p>\n\n\n\n<p>Substituting values of x and y in $4x + 5y \\;-\\; 13 = 0$<strong>,<\/strong> we get<\/p>\n\n\n\n<p>$(4 \\times 2) + (5 \\times 1)\\;-\\; 13 = 8 + 5 \\;-\\; 13 = 0$<\/p>\n\n\n\n<p>Hence, the lines are concurrent.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-practice-problems-on-concurrent-lines\">Practice Problems on Concurrent Lines<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Concurrent Lines - Definition, Formula, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">The lines p, q, and s are ______.<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/07\/find-concurrent-lines.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">parallel<\/div><div class=\"spq_answer_block\" data-value=\"1\">concurrent<\/div><div class=\"spq_answer_block\" data-value=\"2\">perpendicular<\/div><div class=\"spq_answer_block\" data-value=\"3\">none of the above<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: concurrent<br\/>The lines p, q, and s are concurrent lines.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">If the determinant of the three lines is 0, then the lines are _________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">concurrent<\/div><div class=\"spq_answer_block\" data-value=\"1\">intersecting<\/div><div class=\"spq_answer_block\" data-value=\"2\">parallel<\/div><div class=\"spq_answer_block\" data-value=\"3\">can not be determined<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: concurrent<br\/>If the determinant of three lines is zero, then the lines are concurrent.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">If the lines $2x + y \\;\u2013\\; 3 = 0,\\; 5x + ky \\;\u2013\\; 3 = 0$ and $3x \\;\u2013\\; y \\;\u2013\\; 2 = 0$ are concurrent, find the value of k. <\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\;\u2013\\; 2$<\/div><div class=\"spq_answer_block\" data-value=\"1\">0<\/div><div class=\"spq_answer_block\" data-value=\"2\">2<\/div><div class=\"spq_answer_block\" data-value=\"3\">$\\;\u2013\\; 1$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\\;\u2013\\; 2$<br\/>$2x + y \\;\u2013\\; 3 = 0$                    \u2013 (1)<br>\r\n$5x + ky \\;\u2013\\; 3 = 0$                  \u2013 (2)<br>\r\n$3x \\;\u2013\\; y \\;\u2013\\; 2 = 0$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$2x + y \\;\u2013\\; 3 + 3x \\;\u2013\\; y \\;\u2013\\; 2 = 0$<br>\r\n$5x = 5$<br>\r\n$x = 1$<br>\r\n\r\nSubstituting the value of $x = 1$ in (1)<br>\r\n$2(1) + y\\;-\\;3 = 0$<br>\r\n$y = 1$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$5(1) + k(1) \\;\u2013\\; 3 = 0$<br>\r\n$k = \\;-\\;2$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Which of the following are concurrent lines in triangles?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">Medians<\/div><div class=\"spq_answer_block\" data-value=\"1\">Perpendicular bisectors<\/div><div class=\"spq_answer_block\" data-value=\"2\">Angle bisectors<\/div><div class=\"spq_answer_block\" data-value=\"3\">All of the above<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: All of the above<br\/>In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        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\"name\": \"If the lines $$2x + y \\\\;\u2013\\\\; 3 = 0,\\\\; 5x + ky \\\\;\u2013\\\\; 3 = 0$$ and $$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$ are concurrent, find the value of k. \",\n                    \"text\": \"If the lines $$2x + y \\\\;\u2013\\\\; 3 = 0,\\\\; 5x + ky \\\\;\u2013\\\\; 3 = 0$$ and $$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$ are concurrent, find the value of k. \",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"2\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\;\u2013\\\\; 1$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\;\u2013\\\\; 2$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$2x + y \\\\;\u2013\\\\; 3 = 0$$                    \u2013 (1)<br>\r\n$$5x + ky \\\\;\u2013\\\\; 3 = 0$$                  \u2013 (2)<br>\r\n$$3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$                         \u2013 (3)<br>\r\nAdding (1) and (3)<br>\r\n$$2x + y \\\\;\u2013\\\\; 3 + 3x \\\\;\u2013\\\\; y \\\\;\u2013\\\\; 2 = 0$$<br>\r\n$$5x = 5$$<br>\r\n$$x = 1$$<br>\r\n\r\nSubstituting the value of $$x = 1$$ in (1)<br>\r\n$$2(1) + y\\\\;-\\\\;3 = 0$$<br>\r\n$$y = 1$$<br>\r\n\r\nSubstituting the values of x and y in (2)<br>\r\n$$5(1) + k(1) \\\\;\u2013\\\\; 3 = 0$$<br>\r\n$$k = \\\\;-\\\\;2$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following are concurrent lines in triangles?\",\n                    \"text\": \"Which of the following are concurrent lines in triangles?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Medians\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Perpendicular bisectors\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Angle bisectors\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"All of the above\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"In a triangle, medians, perpendicular bisectors and angle bisectors are concurrent lines.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"11-frequently-asked-questions-on-concurrent-lines\">Frequently Asked Questions on Concurrent Lines<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-336e4ba9-f428-4cbe-aba2-d23b19411bac\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-336e4ba9-f428-4cbe-aba2-d23b19411bac\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-336e4ba9-f428-4cbe-aba2-d23b19411bac\"><strong>Are parallel lines concurrent?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-336e4ba9-f428-4cbe-aba2-d23b19411bac\">\n\n<p>No, parallel lines are not concurrent lines, because they do not intersect each other.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-336e4ba9-f428-4cbe-aba2-d23b19411bac\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-336e4ba9-f428-4cbe-aba2-d23b19411bac\"><strong>What is the difference between intersecting lines and concurrent lines?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-336e4ba9-f428-4cbe-aba2-d23b19411bac\">\n\n<p>Two lines are said to be intersecting lines if they meet at a common point whereas three or more lines are said to be concurrent lines if they meet at a common point.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-336e4ba9-f428-4cbe-aba2-d23b19411bac\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-336e4ba9-f428-4cbe-aba2-d23b19411bac\"><strong><strong>What are collinear points?<\/strong><\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-336e4ba9-f428-4cbe-aba2-d23b19411bac\">\n\n<p>Collinear points are the points that lie on the same line.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-336e4ba9-f428-4cbe-aba2-d23b19411bac\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-336e4ba9-f428-4cbe-aba2-d23b19411bac\"><strong>Are concurrent lines coplanar?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-336e4ba9-f428-4cbe-aba2-d23b19411bac\">\n\n<p>Yes, concurrent lines are <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/coplanar\">coplanar<\/a> as they intersect at the same plane.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are Concurrent Lines in Geometry? Concurrent lines are three or more lines meeting exactly at a single point. We know that two lines can intersect at one point only. What if there are three or more lines meeting at a single point? Take a look at the image below. The lines l, m, and &#8230; <a title=\"Concurrent Lines &#8211; Definition, Formula, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/concurrent-lines\" aria-label=\"More on Concurrent Lines &#8211; Definition, Formula, Examples, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-32267","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32267","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=32267"}],"version-history":[{"count":9,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32267\/revisions"}],"predecessor-version":[{"id":32298,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32267\/revisions\/32298"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=32267"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=32267"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=32267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}