{"id":32831,"date":"2023-08-09T16:58:33","date_gmt":"2023-08-09T16:58:33","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=32831"},"modified":"2024-02-05T16:14:41","modified_gmt":"2024-02-05T16:14:41","slug":"perpendicular-bisector-of-a-chord-definition-properties-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord","title":{"rendered":"Perpendicular Bisector of a Chord: Definition, Properties, 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-ecd271ea-31b6-4237-b7f1-be5bd9e42ab0\" 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\/perpendicular-bisector-of-a-chord#0-what-is-the-perpendicular-bisector-of-a-chord>What Is the Perpendicular Bisector of a Chord?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord#1-properties-of-perpendicular-bisector-of-a-chord>Properties of Perpendicular Bisector of a Chord<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord#3-perpendicular-bisector-of-a-chord-theorem-proof>Perpendicular Bisector of a Chord Theorem Proof<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord#6-solved-examples-on-perpendicular-bisector-of-a-chord>Solved Examples on Perpendicular Bisector of a Chord<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord#7-practice-problems-on-perpendicular-bisector-of-a-chord>Practice Problems on Perpendicular Bisector of a Chord<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord#8-frequently-asked-questions-about-the-perpendicular-bisector-of-a-chord>Frequently Asked Questions about the Perpendicular Bisector of a Chord<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-the-perpendicular-bisector-of-a-chord\">What Is the Perpendicular Bisector of a Chord?<\/h2>\n\n\n\n<p><strong>The perpendicular bisector of a chord is <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/line\"><strong>a line<\/strong><\/a><strong> passing through the center of the circle such that it divides the chord into two equal parts and meets the chord at a <\/strong><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/right-angle\"><strong>right angle<\/strong><\/a><strong>.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"521\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord.png\" alt=\"Perpendicular bisector of a chord\" class=\"wp-image-32834\" title=\"Perpendicular bisector of a chord\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-300x252.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>A <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/circle\">circle<\/a> is a 2D figure without corners or edges such that all the points on the boundary are equidistant from the center. The line segment connecting the center with any point on the circumference is called the radius.&nbsp;<\/p>\n\n\n\n<p>A chord is a <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/line-segment\">line segment<\/a> that connects any two points on the <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/circumference-of-a-circle\">circumference of the circle<\/a>. The diameter of a circle is considered to be the longest chord of a circle.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"622\" height=\"488\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/circle-center-chord.png\" alt=\"Circle representing a chord\" class=\"wp-image-32835\" title=\"Circle representing a chord\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/circle-center-chord.png 622w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/circle-center-chord-300x235.png 300w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><\/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\/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><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-properties-of-perpendicular-bisector-of-a-chord\">Properties of Perpendicular Bisector of a Chord<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>A perpendicular bisector of a chord meets the chord at a right angle.<\/li>\n\n\n\n<li>A perpendicular bisector of a chord <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/bisect\">bisects<\/a> the chord.<\/li>\n\n\n\n<li>A perpendicular bisector of a chord passes through the center of the circle.<\/li>\n<\/ol>\n\n\n\n<p>If a line segment passing through the chord has any of these two properties, it also possesses the third.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A line that is <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/perpendicular\">perpendicular<\/a> to a chord and bisects it must pass through the center of the circle.<\/li>\n\n\n\n<li>A line that is perpendicular to a chord and passes through the center of the circle must bisect the chord.<\/li>\n\n\n\n<li>A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-perpendicular-bisector-of-a-chord-theorem\">Perpendicular Bisector of a Chord Theorem<\/h2>\n\n\n\n<p>We can state the perpendicular chord bisector theorem in three ways.&nbsp;<\/p>\n\n\n\n<p><strong>Theorem 1:<\/strong> The line bisecting the chord while passing through the circle&#8217;s center is perpendicular to the chord.&nbsp;<\/p>\n\n\n\n<p><strong>Theorem 2:<\/strong> The line that is perpendicular to the chord and passes through the center of a circle bisects the chord.&nbsp;<\/p>\n\n\n\n<p><strong>Theorem 3: <\/strong>The perpendicular bisector of the chord will pass through the center of the circle. This is also called the perpendicular bisector of a chord conjecture.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-perpendicular-bisector-of-a-chord-theorem-proof\">Perpendicular Bisector of a Chord Theorem Proof<\/h2>\n\n\n\n<p><strong>Theorem 1<\/strong><\/p>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-df2b8998-9266-4d1d-97cb-f03b9cf370d0\">\n\n\n<p><strong>Statement:<\/strong> The line that bisects the chord and passes through the center of a circle is perpendicular to the chord.<\/p>\n\n\n<\/div>\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"524\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-i.png\" alt=\"A line segment OC passing through the center O and bisecting a chord AB\" class=\"wp-image-32837\" title=\"A line segment OC passing through the center O and bisecting a chord AB\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-i.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-i-300x254.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Proof:<\/strong><\/p>\n\n\n\n<p>Let the line segment OC bisects the chord AB and passes through the center O.<\/p>\n\n\n\n<p>We have to prove that OC is perpendicular to AB.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th><strong>Statement<\/strong><\/th><th><strong>Reason<\/strong><\/th><\/tr><\/thead><tbody><tr><td>OA = OB<br>AC = BC<br>OC = OC<br>$\\Delta$OAC \u2245 $\\Delta$OBC<br>\u2220OCA = \u2220OCB&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \u2026(i)<br>\u2220OCA + \u2220OCB $= 180^{\\circ}$ &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \u2026(ii)<br>\u2220OCA = \u2220OCB $= 90^{\\circ}$<br>OC$\\bot$AB.<br><br>Hence, proved.<\/td><td>OA and OB are radii of the circle.<br>OC bisects the chord.<br>Common side<br>SSS congruence criterion<br>CPCTC<br>Angles on a straight line<br>From (i) and (ii)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Theorem 2<\/strong><\/p>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-9ac863cd-a79a-4b6e-8796-9d9ed9b8d7ad\">\n\n\n<p><strong>Statement:<\/strong> The line that is perpendicular to the chord and passes through the center of a circle bisects the chord.&nbsp;<\/p>\n\n\n<\/div>\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"524\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-ii.png\" alt=\"A line segment OC passing through the center O and perpendicular to the chord AB\" class=\"wp-image-32838\" title=\"A line segment OC passing through the center O and perpendicular to the chord AB\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-ii.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-ii-300x254.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Proof:<\/strong><\/p>\n\n\n\n<p>OC passes through the center O. Let OC $\\bot$ AB. We will prove that OC bisects AB. We need to prove that $AC = BC$.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th><strong>Statement<\/strong><\/th><th><strong>Reason<\/strong><\/th><\/tr><\/thead><tbody><tr><td>\u2220OCA = \u2220OCB $= 90^{\\circ}$<br>OA = OB<br>OC = OC<br>$\\Delta$OAC \u2245 $\\Delta$OBC<br>AC = BC<br>Hence, proved.<\/td><td>OC $\\bot$ AB<br>Radii of the same circle<br>Common side<br>RHS criterion<br>CPCTC<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Theorem 3<\/strong><\/p>\n\n\n<div class=\"ub-styled-box ub-notification-box\" id=\"ub-styled-box-562826d4-e34e-4cbf-b78a-83190423c612\">\n\n\n<p><strong>Statement: <\/strong>The perpendicular bisector of any chord of a circle passes through the center of the circle.<\/p>\n\n\n<\/div>\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"524\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-iii.png\" alt=\"Perpendicular bisector of any chord of a circle passes through the center of the circle.\" class=\"wp-image-32839\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-iii.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-iii-300x254.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Proof:<\/strong><\/p>\n\n\n\n<p>Given that CD is a perpendicular bisector of chord AB.&nbsp;<\/p>\n\n\n\n<p>C is the midpoint of AB.<\/p>\n\n\n\n<p>\u2220DCA = \u2220DCB $= 90^{\\circ}$ &nbsp; &nbsp; &nbsp; &nbsp; \u2026(i)<\/p>\n\n\n\n<p>We have to prove that CD passes through the center O. Let\u2019s assume that the line CD does not pass through the center O.&nbsp;<\/p>\n\n\n\n<p>(Let\u2019s imagine it as a different line as shown in the diagram below.)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"533\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem.png\" alt=\"Perpendicular bisector of a chord theorem - proof\" class=\"wp-image-32841\" title=\"Perpendicular bisector of a chord theorem - proof\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-theorem-300x258.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>We know that a line that joins the center of a circle to the midpoint of a chord is always perpendicular to the chord. OC joins the center of a circle O to the midpoint C of a chord AB.<\/p>\n\n\n\n<p>Thus, \u2220OCA = \u2220OCB $= 90^{\\circ}$ &nbsp; \u2026(ii)<\/p>\n\n\n\n<p>This is a contradiction to (i).<\/p>\n\n\n\n<p>Equations (i) and (ii) are possible only if C, O, and D lie on the same line.&nbsp;<\/p>\n\n\n\n<p>Thus, the perpendicular bisector CD passes through the center O.<\/p>\n\n\n\n<p>It proves that the perpendicular bisector of the chord will pass through the center of the circle.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-facts-about-perpendicular-bisector-of-a-chord\">Facts about Perpendicular Bisector of a Chord<\/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>The diameter is the longest chord of a circle.<\/li>\n\n\n\n<li>The radius of a circle that is perpendicular to the chord bisects it.&nbsp;<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about the perpendicular bisector of a chord\u2014a line that cuts a chord into two equal segments at right angles. Understanding this geometric concept and related theorems is vital in various applications, such as circle geometry and construction. Let&#8217;s now reinforce our knowledge with examples and MCQs for better comprehension.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-solved-examples-on-perpendicular-bisector-of-a-chord\">Solved Examples on Perpendicular Bisector of a Chord<\/h2>\n\n\n\n<p><strong>1. Find CM if the radius is 20 units and the length of the chord AB is 20 units.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"313\" height=\"312\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example.png\" alt=\"Perpendicular bisector theorem example\" class=\"wp-image-32842\" title=\"Perpendicular bisector theorem example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example.png 313w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example-300x300.png 300w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example-150x150.png 150w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example-250x250.png 250w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example-120x120.png 120w\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" \/><\/figure>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>According to the perpendicular bisector theorem, CM is the perpendicular bisector of AB.<\/p>\n\n\n\n<p>$AC = BC = 10$ units<\/p>\n\n\n\n<p>In the right triangle BCM, we have<\/p>\n\n\n\n<p>BC<sup>2<\/sup> + CM<sup>2<\/sup> = BM<sup>2<\/sup><\/p>\n\n\n\n<p>20<sup>2<\/sup> = 10<sup>2<\/sup> + CM<sup>2<\/sup><\/p>\n\n\n\n<p>CM<sup>2<\/sup> = = 300<\/p>\n\n\n\n<p>CM = 17.3 units&nbsp;<\/p>\n\n\n\n<p><strong>2. O is the center of the circle. Find the missing length and the length of the chord AB.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"313\" height=\"312\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem.png\" alt=\"Perpendicular bisector of a chord example\" class=\"wp-image-32843\" title=\"Perpendicular bisector of a chord example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem.png 313w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem-300x300.png 300w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem-150x150.png 150w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem-250x250.png 250w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-problem-120x120.png 120w\" sizes=\"auto, (max-width: 313px) 100vw, 313px\" \/><\/figure>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>OM passes through the center of the circle and it is perpendicular to the chord AB.<\/p>\n\n\n\n<p>Thus, it is the perpendicular bisector of AB.<\/p>\n\n\n\n<p>In right triangle OMA, we have<\/p>\n\n\n\n<p>OA = 5 in, OM = 4 in<\/p>\n\n\n\n<p>Thus, OM<sup>2<\/sup> + AM<sup>2<\/sup> = OA<sup>2<\/sup><\/p>\n\n\n\n<p>4<sup>2<\/sup> +&nbsp; AM<sup>2 <\/sup>= 5<sup>2<\/sup><\/p>\n\n\n\n<p>AM<sup>2 <\/sup>= 25 &#8211; 16<\/p>\n\n\n\n<p>AM<sup>2 <\/sup>= 9<\/p>\n\n\n\n<p>AM = 3 in<\/p>\n\n\n\n<p>Here, since OM is the perpendicular bisector, we have<\/p>\n\n\n\n<p>AB = AM + BM = 3 + 3 = 6 in<\/p>\n\n\n\n<p><strong>3. What will be the length of the chord of the circle if radius is 15 units and the distance between the chord and center is 12 units?<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>Use the Pythagoras theorem to find the length of the chord.&nbsp;<\/p>\n\n\n\n<p>Hypotenuse = Radius of the circle<\/p>\n\n\n\n<p>15<sup>2<\/sup> = 12<sup>2<\/sup> + Chord length<sup>2<\/sup><\/p>\n\n\n\n<p>225 = 144 + Chord length<sup>2<\/sup><\/p>\n\n\n\n<p>Chord length<sup>2<\/sup> = 225 &#8211; 144<\/p>\n\n\n\n<p>Chord length = $\\sqrt{81}$<\/p>\n\n\n\n<p>Chord length = 9 cm<\/p>\n\n\n\n<p><strong>4. Define the following terms: perpendicular bisector, chord and diameter.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p><strong>Perpendicular bisector: <\/strong>Perpendicular bisector of a segment is a line that meets the given segment at a right angle and divides the given segment into two equal parts.<\/p>\n\n\n\n<p><strong>Chord:<\/strong> Chord of a circle is a line segment that has its endpoints on the circumference of the circle.<\/p>\n\n\n\n<p><strong>Diameter: <\/strong>Diameter of a circle is a line segment that passes through the center of the circle such that its endpoints are on the circumference of the circle. It is the longest chord of the circle.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-practice-problems-on-perpendicular-bisector-of-a-chord\">Practice Problems on Perpendicular Bisector of a Chord<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Perpendicular Bisector of a Chord: Definition, Properties, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">The perpendicular bisector of the chord meets the chord at ________.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">a right angle<\/div><div class=\"spq_answer_block\" data-value=\"1\">a straight angle<\/div><div class=\"spq_answer_block\" data-value=\"2\">a 60 degree angle<\/div><div class=\"spq_answer_block\" data-value=\"3\">45 degree angle<\/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: a right angle<br\/>The perpendicular bisector of the chord meets the chord at a right angle. <\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">If a line segment bisects the chord and passes through the center, then it is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">parallel to the chord<\/div><div class=\"spq_answer_block\" data-value=\"1\">perpendicular to the chord<\/div><div class=\"spq_answer_block\" data-value=\"2\">equal to the chord<\/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: perpendicular to the chord<br\/>A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">What is the missing length?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/08\/perpendicular-bisector-of-a-chord-example-i.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">1 unit<\/div><div class=\"spq_answer_block\" data-value=\"1\">1.5 units<\/div><div class=\"spq_answer_block\" data-value=\"2\">2 units<\/div><div class=\"spq_answer_block\" data-value=\"3\">3 units<\/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: 3 units<br\/>The line segment is the perpendicular bisector of the chord.<br>\r\nThus, missing length $= 3$ units<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The line segment that is perpendicular to the chord and passes through the center of the circle _______.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">doubles the chord<\/div><div class=\"spq_answer_block\" data-value=\"1\">bisects the chord<\/div><div class=\"spq_answer_block\" data-value=\"2\">bisects the radius<\/div><div class=\"spq_answer_block\" data-value=\"3\">is equal to the chord<\/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: bisects the chord<br\/>The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.<\/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\": \"Perpendicular Bisector of a Chord: Definition, Properties, Examples\",        \n        \"about\": {\n  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\"encodingFormat\": \"text\/html\",\n                                \"text\": \"a straight angle\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perpendicular bisector of the chord meets the chord at a right angle. \"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a 60 degree angle\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perpendicular bisector of the chord meets the chord at a right angle. \"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"45 degree angle\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The perpendicular bisector of the chord meets the chord at a right angle. \"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"a right angle\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The perpendicular bisector of the chord meets the chord at a right angle. \"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The perpendicular bisector of the chord meets the chord at a right angle. \"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If a line segment bisects the chord and passes through the center, then it is\",\n                    \"text\": \"If a line segment bisects the chord and passes through the center, then it is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"parallel to the chord\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"equal to the chord\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.\"\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\": \"A line that bisects a chord and passes through the center of a circle must be perpendicular to the chord.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                    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             \"text\": \"The line segment is the perpendicular bisector of the chord.<br>\r\nThus, missing length $$= 3$$ units\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"3 units\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The line segment is the perpendicular bisector of the chord.<br>\r\nThus, missing length $$= 3$$ units\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The line segment is the perpendicular bisector of the chord.<br>\r\nThus, missing length $$= 3$$ units\"\n                      }\n                    } \n\n         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\"text\/html\",\n                                \"text\": \"doubles the chord\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"bisects the radius\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"is equal to the chord\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"bisects the chord\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The line segment that is perpendicular to the chord and passes through the center of a circle bisects the chord.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-frequently-asked-questions-about-the-perpendicular-bisector-of-a-chord\">Frequently Asked Questions about the Perpendicular Bisector of a Chord<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-f14faca6-5e56-4860-b2ef-268fff553bf2\" 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-f14faca6-5e56-4860-b2ef-268fff553bf2\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f14faca6-5e56-4860-b2ef-268fff553bf2\"><strong>What is the intersecting chord 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-f14faca6-5e56-4860-b2ef-268fff553bf2\">\n\n<p>The interesting chord theorem represents the intersection property of chords. It says that the product of the length of segments of one chord is equal to the product of the length of the segment of another chord.<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f14faca6-5e56-4860-b2ef-268fff553bf2\"><strong>What will be the properties of two parallel chords?<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\">\n\n<p>The two parallel chords in the circle will be at an equal distance from each other and their length will be the same.<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f14faca6-5e56-4860-b2ef-268fff553bf2\"><strong>What is the secant line of a circle?<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\">\n\n<p>The secant line of a circle is the extended form of the chord. It extends the circle beyond the points touching the circumference of the circle.<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f14faca6-5e56-4860-b2ef-268fff553bf2\"><strong>How do you find the perpendicular bisector of a chord?<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\">\n\n<p>A line that is perpendicular to the chord and passes through the center bisects the chord. Thus, to find the perpendicular bisector of a chord, draw a perpendicular to the chord from the center of a circle.&nbsp;<\/p>\n\n\n\n<p>To find the length of the perpendicular bisector of a chord, we can use the Pythagoras theorem in a triangle, where the radius acts as the hypotenuse and the length of the perpendicular bisector acts as one of the legs.<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-f14faca6-5e56-4860-b2ef-268fff553bf2\"><strong>What is the midpoint of the chord?<\/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-f14faca6-5e56-4860-b2ef-268fff553bf2\">\n\n<p>The midpoint of the chord is the point of intersection of the perpendicular bisector and chord. The fact that resultant segments are equal in length proves it to be the midpoint.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Perpendicular Bisector of a Chord? The perpendicular bisector of a chord is a line passing through the center of the circle such that it divides the chord into two equal parts and meets the chord at a right angle. A circle is a 2D figure without corners or edges such that all &#8230; <a title=\"Perpendicular Bisector of a Chord: Definition, Properties, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/perpendicular-bisector-of-a-chord\" aria-label=\"More on Perpendicular Bisector of a Chord: Definition, Properties, 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-32831","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\/32831","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=32831"}],"version-history":[{"count":8,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32831\/revisions"}],"predecessor-version":[{"id":39966,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32831\/revisions\/39966"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=32831"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=32831"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=32831"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}