{"id":26156,"date":"2023-03-12T15:00:33","date_gmt":"2023-03-12T15:00:33","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=26156"},"modified":"2024-02-05T15:41:14","modified_gmt":"2024-02-05T15:41:14","slug":"triangle-proportionality-theorem-statement-proof-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem","title":{"rendered":"Triangle Proportionality Theorem: Statement, Proof, Examples"},"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-44ce178d-ac39-4522-b217-bfbd10f515e9\" 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\/triangle-proportionality-theorem#0-what-is-the-triangle-proportionality-theorem>What Is the Triangle Proportionality Theorem?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem#2-proof-of-the-triangle-proportionality-theorem>Proof of the Triangle Proportionality Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem#3-converse-of-the-triangle-proportionality-theorem>Converse of the Triangle Proportionality Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem#7-solved-examples-on-triangle-proportionality-theorem>Solved Examples on Triangle Proportionality Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem#8-practice-problems-on-triangle-proportionality-theorem>Practice Problems on Triangle Proportionality Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem#9-frequently-asked-questions-on-triangle-proportionality-theorem>Frequently Asked Questions on Triangle Proportionality Theorem<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-the-triangle-proportionality-theorem\">What Is the Triangle Proportionality Theorem?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Triangle proportionality theorem is also known as \u201cbasic proportionality theorem\u201d or \u201cThales theorem,\u201d or \u201cside-splitter theorem.\u201d It was proposed by a famous Greek mathematician Thales.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The theorem is helpful in understanding the concept of similar triangles.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The triangle proportionality theorem uses a line drawn parallel to one side of a triangle to determine the proportions of the other two sides.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">If you draw a line parallel to any one side of a triangle that intersects the other two sides of the triangle in two distinct points, then the line divides the other two sides in the same ratio. <\/p>\n\n\n\n<p class=\"eplus-wrapper\">y.<\/p>\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\/answer-questions-related-to-triangles\" 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_classify_tri_on_side_pt.png\" alt=\"Answer Questions Related to Triangles 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\">Answer Questions Related to Triangles 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\/classify-triangles-and-rectangles-as-closed-shape\" 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_triangle_rectangle_2_pt.png\" alt=\"Classify Triangles and Rectangles as Closed Shape 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\">Classify Triangles and Rectangles as Closed Shape 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\/classify-triangles\" 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_classify_triangles_pt.png\" alt=\"Classify Triangles 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\">Classify Triangles 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-type-of-triangles-based-on-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_classify_tri_on_angle_pt.png\" alt=\"Identify the Type of Triangles Based on 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 the Type of Triangles Based on 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-triangles-and-rectangles\" 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_amusement_park_18_20_gm.png\" alt=\"Identify Triangles and Rectangles 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 Triangles and Rectangles 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-triangles-and-squares\" 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_triangles_squares_pt.png\" alt=\"Identify Triangles and Squares 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 Triangles and Squares 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-types-of-triangles\" 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_types_of_triangles_pt.png\" alt=\"Identify Types of Triangles 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 Types of Triangles 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\/match-triangles-and-squares\" 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_amusement_park_4_5_gm.png\" alt=\"Match Triangles and Squares 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\">Match Triangles and Squares 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\/triangles-and-rectangles\" 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_triangle_rectangle_1_pt.png\" alt=\"Triangles and Rectangles 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\">Triangles and Rectangles 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 eplus-wrapper\" id=\"1-triangle-proportionality-theorem-statement\">Triangle Proportionality Theorem: Statement<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The triangle proportionality theorem states that if a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the remaining two sides proportionally.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"411\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-.png\" alt=\"Triangle Proportionality Theorem\" class=\"wp-image-37503\" title=\"Triangle proportionality theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem--300x199.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Here, the line DE is parallel to the side BC. It intersects sides AB and AC at two distinct points, D and E.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the triangle proportionality theorem, a ratio of AD to DB is equal to the ratio of CE to AE.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, $\\frac{AD}{BD} = \\frac{AE}{CE}$<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/rectangles-and-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/rectangles-and-triangles.jpeg\" alt=\"Rectangles and Triangles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/separate-out-the-triangles\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/separate-out-the-triangles.jpeg\" alt=\"Separate Out the Triangles Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-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 eplus-wrapper\" id=\"2-proof-of-the-triangle-proportionality-theorem\">Proof of the Triangle Proportionality Theorem<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Given:<\/strong> In $\\Delta ABC$, the line segment DE is drawn parallel to the side BC.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>To prove:<\/strong> $\\frac{AD}{BD} = \\frac{AE}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Construction:<\/strong> Join the vertex B to the point E. Join the vertex C to the point D.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Draw $EG\\; \\bot \\;AB$ and $DF\\; \\bot \\;AC$ as shown in the given figure.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"483\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem.png\" alt=\"Triangle proportionality theorem construction\" class=\"wp-image-37504\" title=\"Triangle proportionality theorem construction\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-300x234.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">In $\\Delta ADE$ and $\\Delta BDE$,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{(Area\\; of\\;&nbsp; \\Delta ADE)}{(Area\\; of\\;&nbsp; \\Delta BDE)} = \\frac{\\frac{1}{2} \\times AD \\times EG}{\\frac{1}{2} \\times BD \\times EG}$ (Both the triangles have the same height EG.)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{(Area\\; of\\;&nbsp;\\Delta ADE)}{(Area\\; of\\;&nbsp;\\Delta BDE)} = \\frac{AD}{BD}$ \u2026(1)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Now In $\\Delta CDE$ and&nbsp; $\\Delta ADE$,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{(Area\\; of\\; \\Delta ADE)}{(Area\\; of\\;&nbsp;\\Delta CDE)} = \\frac{\\frac{1}{2} \\times AE \\times DF}{\\frac{1}{2} \\times CE \\times DF}$ (Both the triangles have the same height DF.)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{(Area\\; of\\;&nbsp;\\Delta ADE)}{(Area\\; of\\;&nbsp;\\Delta CDE)} = \\frac{AE}{CE}$ \u2026(2)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Delta BDE$ and $\\Delta CDE$ are lying between the same parallel lines DE and BC.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Triangles having the same base and lying between the same parallels are equal in area.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\therefore\\; Area\\; of\\; \\Delta BDE = Area\\; of\\; \\Delta CDE$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Using this result in equation (1) and (2), we get left sides equal.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{(Area\\; of\\; ADE)} {(Area\\; of\\; BDE)} = \\frac{(Area\\; of\\; ADE)} {(Area\\; of\\; CDE)}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, from equation (1) and (2), we get<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AD}{BD} = \\frac{AE}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, proved.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-converse-of-the-triangle-proportionality-theorem\">Converse of the Triangle Proportionality Theorem<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">If a line segment divides any two sides of a triangle proportionally (in the same ratio), then the line is parallel to the third side of the triangle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"353\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem.png\" alt=\"Converse of triangle proportionality theorem\" class=\"wp-image-37505\" title=\"Converse of triangle proportionality theorem\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem-300x171.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Here, if the line segment DE divides sides AB and AC in equal proportion, i.e., $\\frac{AD}{BD} = \\frac{AE}{CE}$, then&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">by the converse of the triangle proportionality theorem, we can say that the line DE must be parallel to BC or $DE\\; || \\;BC$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-proof-of-the-converse-of-the-triangle-proportionality-theorem\">Proof of the Converse of the Triangle Proportionality Theorem<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Given: <\/strong>In<strong> <\/strong>\u0394ABC, the line segment DE divides the sides AB and AC in equal proportion. ADBD=AECE.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>To prove:<\/strong> DE || BC<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Construction: <\/strong>Let us assume DE is not parallel to BC. So draw another line DF, which is parallel to BC.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"361\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem-proof-.png\" alt=\"Converse of triangle proportionality theorem proof\" class=\"wp-image-37506\" title=\"Converse of triangle proportionality theorem proof\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem-proof-.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem-proof--300x175.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Proof:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Now, we have to show that points E and F are identical or that DE and DF represent the same lines.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the Basic Proportionality theorem,&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AD}{BD} = \\frac{AF}{CF}$ \u2026 (1)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Also, $\\frac{AD}{BD} = \\frac{AE}{CE}$ (given) \u2026 (2)<\/p>\n\n\n\n<p class=\"eplus-wrapper\">From equation (1) and (2), we get<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AF}{CF} = \\frac{AE}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Adding 1 on both sides, we get<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AF}{CF} + 1 = \\frac{AE}{CE} + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AF + CF}{CF} = \\frac{AE + CE}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AC}{CF} = \\frac{AC}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\therefore\\; CF = CE$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Therefore, from the above statement, we can conclude that the points E and F are the same points. So, DE and DF represent the same lines.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, $DE\\; || \\;BC$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, it proves the converse of the basic proportionality theorem.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-facts-about-triangle-proportionality-theorem\">Facts about Triangle Proportionality Theorem<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The well-known Greek mathematician Thales proposed the basic proportionality theorem, which is why it is also known as the Thales theorem.<\/li>\n<\/ul>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">In daily life, the triangle proportionality theorem has applications in the construction of houses, bridges, towers, etc.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we have learned about the triangle proportionality and converse of triangle proportionality theorem with statement, proof, applications, facts. Let\u2019s solve a few examples.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-solved-examples-on-triangle-proportionality-theorem\">Solved Examples on Triangle Proportionality Theorem<\/h2>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>In the figure given below ST || QR, find the ratio PT : TR.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"187\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-example.png\" alt=\"Triangle proportionality theorem example\" class=\"wp-image-37507\" title=\"Triangle proportionality theorem example\"\/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PS = 8,\\; SQ = 12$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$ST\\; || \\;QR$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the triangle proportionality theorem, the ratio of PS to SQ is equal to the ratio of PT to TR.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{PS}{SQ} = \\frac{PT}{TR}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{8}{12} = \\frac{PT}{TR}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow \\frac{2}{3} = \\frac{PT}{TR}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow PT\\;:\\;TR =2\\;:\\;3$<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>Check whether the line PQ is parallel to the BC or not.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"278\" height=\"186\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/converse-of-triangle-proportionality-theorem-example-.png\" alt=\"Converse of triangle proportionality theorem example\" class=\"wp-image-37508\" title=\"Converse of triangle proportionality theorem example\"\/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AP = 6,\\; BP = 9,\\; AQ = 5,$ and $CQ = 7.5$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AP}{BP} = \\frac{6}{9} = \\frac{2}{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AQ}{CQ} = \\frac{5}{7.5} = \\frac{2}{3}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Since, $\\frac{AP}{BP} = \\frac{AQ}{CQ}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, by the converse of triangle proportionality theorem, the line PQ is parallel to the BC.<\/p>\n\n\n\n<ol start=\"3\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>In the figure given below <\/strong>$DE\\; || \\;BC$<strong>, find the value of \u201cx.\u201d<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"247\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-example-.png\" alt=\"Triangle proportionality theorem example\" class=\"wp-image-37509\" title=\"Triangle proportionality theorem example\"\/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Given, $DE\\; || \\;BC,\\; AD = 2$ in, $BD = 5$ in, $AE = 1$ in, and $CE = x + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the triangle proportionality theorem,&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AD}{BD} = \\frac{AE}{CE}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow \\frac{2}{5} = \\frac{1}{x + 1}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow 2x + 2 = 5$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow 2x = 3$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow x = \\frac{3}{2} = 1.5$ in<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the value of x is 1.5 inches.<\/p>\n\n\n\n<ol start=\"4\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>In the figure given below, <\/strong>$PQ\\; || \\;BC$<strong>, find the value of x.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"284\" height=\"125\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2024\/01\/triangle-proportionality-theorem-example-1-1.png\" alt=\"Triangle proportionality theorem example\" class=\"wp-image-37510\" title=\"Triangle proportionality theorem example\"\/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Given, $PQ\\; || \\;BC,\\; AP = 2,\\; BP = x \\;\u2212\\; 1,\\; AQ = 3,$ and $CQ = x + 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the triangle proportionality theorem,&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\frac{AP}{BP} = \\frac{AQ}{CQ}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow \\frac{2}{x \\;-\\;1} = \\frac{3}{x\\;+\\;1}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow 2(x\\;+\\;1) = 3(x\\;-\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\Rightarrow 2x\\;+\\;2 = 3x\\;-\\;3$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$3x\\;-\\;2x = 3+2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = 5$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the value of x is 5.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-practice-problems-on-triangle-proportionality-theorem\">Practice Problems on Triangle Proportionality Theorem<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Triangle Proportionality Theorem: Statement, Proof, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">In the figure below $DE\\; || \\;BC$. By the triangle proportionality theorem, we can write $\\frac{AE}{EC}=$ ?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\\frac{2}{5}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\frac{3}{5}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\frac{2}{3}$<\/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: $\\frac{2}{3}$<br\/>From the given figure, $\\frac{AD}{BD} = \\frac{2}{3} = 2\\; : \\;3$.<br>\r\nFor $DE\\; || \\;BC$, the values of AE and CE must be in the same ratio as $AD\\;: \\;BD$<br>\r\n$AE\\; : \\;CE = 4\\; : \\;6 = 2\\; : \\;3$..<\/span><div class=\"spq_explanation_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-1-1.png\"><\/div><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Which of the following is correct for the figure below?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$AE\\; : \\;CE = 1\\; : \\;2$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$DE\\; || \\;BC$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$AD\\; : \\;BD = 1\\; : \\;2$<\/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\/>From the given figure, $AE\\; : \\;CE = 2\\;: \\;4 = 1\\;: \\;2,\\; AD\\;: \\;BD = 1\\;:\\;2$<br>\r\nSince, $\\frac{AD}{BD} = \\frac{AE}{CE}$, hence, according to the converse of triangle proportionality theorem $DE\\; || \\;BC$.<br>\r\nTherefore, all of the above i.e. option (d) is correct.<\/span><div class=\"spq_explanation_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-2-1.png\"><\/div><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Which of the following is the triangle proportionality theorem\u2019s condition such that the equation $\\frac{AP}{PB} = \\frac{AQ}{QC}$ becomes true?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-3.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$PQ = BC$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$PQ\\; || \\;BC$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$AB = BC = AC$<\/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: $PQ\\; || \\;BC$<br\/>If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $PQ\\; || \\;BC$, we have $\\frac{AP}{PB} = \\frac{AQ}{QC}$.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Triangle Proportionality Theorem: Statement, Proof, Examples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Triangle Proportionality Theorem\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"In the figure below $$DE\\\\; || \\\\;BC$$. By the triangle proportionality theorem, we can write $$\\\\frac{AE}{EC}=$$ ?\",\n                    \"text\": \"In the figure below $$DE\\\\; || \\\\;BC$$. By the triangle proportionality theorem, we can write $$\\\\frac{AE}{EC}=$$ ?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$..\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{2}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$..\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\frac{3}{5}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$..\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"None of the above\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$..\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\\\\frac{2}{3}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$.. <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-1-1.png\\\"\/>\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"From the given figure, $$\\\\frac{AD}{BD} = \\\\frac{2}{3} = 2\\\\; : \\\\;3$$.<br>\r\nFor $$DE\\\\; || \\\\;BC$$, the values of AE and CE must be in the same ratio as $$AD\\\\;: \\\\;BD$$<br>\r\n$$AE\\\\; : \\\\;CE = 4\\\\; : \\\\;6 = 2\\\\; : \\\\;3$$.. <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-1-1.png\\\"\/>\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is correct for the figure below?\",\n                    \"text\": \"Which of the following is correct for the figure below?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$AE\\\\; : \\\\;CE = 1\\\\; : \\\\;2$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$DE\\\\; || \\\\;BC$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$AD\\\\; : \\\\;BD = 1\\\\; : \\\\;2$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct.\"\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\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct. <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-2-1.png\\\"\/>\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"From the given figure, $$AE\\\\; : \\\\;CE = 2\\\\;: \\\\;4 = 1\\\\;: \\\\;2,\\\\; AD\\\\;: \\\\;BD = 1\\\\;:\\\\;2$$<br>\r\nSince, $$\\\\frac{AD}{BD} = \\\\frac{AE}{CE}$$, hence, according to the converse of triangle proportionality theorem $$DE\\\\; || \\\\;BC$$.<br>\r\nTherefore, all of the above i.e. option (d) is correct. <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-2-1.png\\\"\/>\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is the triangle proportionality theorem\u2019s condition such that the equation $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$ becomes true?\",\n                    \"text\": \"Which of the following is the triangle proportionality theorem\u2019s condition such that the equation $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$ becomes true? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/Practice-Problems-3.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$PQ = BC$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$AB = BC = AC$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"None of the above\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$PQ\\\\; || \\\\;BC$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"If a line parallel to one side of a triangle intersects the other two sides at different points, then it divides the two sides proportionally.<br>\r\nThus, if $$PQ\\\\; || \\\\;BC$$, we have $$\\\\frac{AP}{PB} = \\\\frac{AQ}{QC}$$.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-frequently-asked-questions-on-triangle-proportionality-theorem\">Frequently Asked Questions on Triangle Proportionality Theorem<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-df784316-1cd2-4542-a4cb-16aa559407b0\" 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-df784316-1cd2-4542-a4cb-16aa559407b0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-df784316-1cd2-4542-a4cb-16aa559407b0\"><strong>What are the uses of the triangle proportionality theorem?<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\">\n\n<p class=\"eplus-wrapper\">In daily life, it is used in the construction of houses, bridges, towers, etc., where precision and proportions are important.<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-df784316-1cd2-4542-a4cb-16aa559407b0\"><strong>What is the other name of the triangle proportionality theorem?<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\">\n\n<p class=\"eplus-wrapper\">The triangle proportionality theorem is also known as the Thales theorem or basic proportionality theorem.<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-df784316-1cd2-4542-a4cb-16aa559407b0\"><strong>What is the formula of the triangle proportionality theorem?<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\">\n\n<p class=\"eplus-wrapper\">In any $\\Delta ABC$ where line segment DE is drawn parallel to the side BC of $\\Delta ABC,\\;&nbsp;\\frac{AD}{BD} = \\frac{AE}{CE}$.<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-df784316-1cd2-4542-a4cb-16aa559407b0\"><strong>What is the converse of the triangle proportionality theorem?<\/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-df784316-1cd2-4542-a4cb-16aa559407b0\">\n\n<p class=\"eplus-wrapper\">If a line segment divides any two sides of a triangle in equal proportion, then the line is parallel to the third side of the triangle.<\/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-4-df784316-1cd2-4542-a4cb-16aa559407b0\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-df784316-1cd2-4542-a4cb-16aa559407b0\"><strong>What is the basic proportionality theorem?<\/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-4-df784316-1cd2-4542-a4cb-16aa559407b0\">\n\n<p class=\"eplus-wrapper\">The basic proportionality theorem (B.P.T.) states that if a line is drawn parallel to one side of a triangle and is cutting the other two sides, then it divides the other two sides in equal proportion.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">It is also known as the Thales theorem.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Triangle Proportionality Theorem? Triangle proportionality theorem is also known as \u201cbasic proportionality theorem\u201d or \u201cThales theorem,\u201d or \u201cside-splitter theorem.\u201d It was proposed by a famous Greek mathematician Thales.&nbsp; The theorem is helpful in understanding the concept of similar triangles. The triangle proportionality theorem uses a line drawn parallel to one side of &#8230; <a title=\"Triangle Proportionality Theorem: Statement, Proof, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/triangle-proportionality-theorem\" aria-label=\"More on Triangle Proportionality Theorem: Statement, Proof, Examples\">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-26156","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\/26156","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=26156"}],"version-history":[{"count":29,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26156\/revisions"}],"predecessor-version":[{"id":39949,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26156\/revisions\/39949"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=26156"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=26156"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=26156"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}