{"id":26602,"date":"2023-03-21T19:54:10","date_gmt":"2023-03-21T19:54:10","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=26602"},"modified":"2024-02-05T15:28:38","modified_gmt":"2024-02-05T15:28:38","slug":"tangent-of-a-circle-definition-properties-formula-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle","title":{"rendered":"Tangent of a Circle: Definition, Properties, Formula, 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-fb672d2a-3bb7-4bfc-baad-349212a57e67\" 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\/tangent-to-a-circle#0-what-is-the-tangent-of-a-circle>What Is The Tangent of a Circle?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle#6-tangent-of-a-circle-formula>Tangent of a Circle: Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle#8-condition-of-tangency>Condition of Tangency<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle#11-solved-examples-on-tangent-of-a-circle>&nbsp;Solved Examples on Tangent of a Circle<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle#12-practice-problems-on-tangent-of-a-circle>Practice Problems on Tangent of a Circle<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle#13-frequently-asked-questions-on-tangent-of-a-circle>Frequently Asked Questions on Tangent of a Circle<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-the-tangent-of-a-circle\">What Is The Tangent of a Circle?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The word \u201ctangent\u201d originated from the Latin word \u201c<em>tangere<\/em>,\u201d which means \u201cto touch.\u201d The tangent of a circle refers to a line that touches a circle at a single point.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">In the image shown below, the line l is a tangent to the circle with the center C.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"302\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-1.png\" alt=\"A tangent to a circle with center C at point T\" class=\"wp-image-26610\" title=\"A tangent to a circle with center C at point T\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-1.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-1-300x146.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">A straight line that touches or intersects the circle at only one point is called a tangent of the circle. A tangent touches the circle externally.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"350\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-3.png\" alt=\"Tangent of a circle\" class=\"wp-image-26611\" title=\"Tangent of a circle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-3.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-3-300x169.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"1-tangent-of-a-circle-definition\">Tangent of a Circle: Definition<\/h3>\n\n\n\n<p class=\"eplus-wrapper\">Tangent in geometry is defined as a line that touches the circle at only one point. The point of contact of the tangent with the circle is known as the point of tangency.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"271\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-2.png\" alt=\"Tangent of a circle and the point of tangency\" class=\"wp-image-26612\" title=\"Tangent of a circle and the point of tangency\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-2.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-2-300x131.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">Here, the line PQ is the tangent to the circle with center O. The line PQ touches the circle at only one point, A.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The point A is the point of tangency.<\/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\/count-upto-10-objects-in-circle\" 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\/count_comp_hide_and_seek_10_gm.png\" alt=\"Count upto 10 Objects in Circle 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\">Count upto 10 Objects in Circle 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-circles-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_circles_rectangles_pt.png\" alt=\"Identify Circles 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 Circles 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\/match-rectangles-and-circles\" 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_3_6_gm.png\" alt=\"Match Rectangles and Circles 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 Rectangles and Circles 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=\"2-tangent-properties\">Tangent Properties<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">A tangent touches a circle at only one point.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">A tangent is a straight line that never enters the interior of the circle.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The tangent makes a right angle at the point of tangency with the radius of a circle.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Tangents drawn from an external point to a circle have the same length.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">A circle can have infinitely many tangents.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">We can draw exactly <strong>one<\/strong> tangent to a circle passing through a point that lies on a circle.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">We can draw exactly <strong>two<\/strong> tangents to a circle passing through a point that lies outside the circle.<\/li>\n<\/ul>\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\/circle-equivalent-fractions\" 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\/circle-equivalent-fractions.jpeg\" alt=\"Circle Equivalent Fractions 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\/squares-and-circles\" 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\/squares-and-circles.jpeg\" alt=\"Squares and Circles 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=\"3-tangent-theorems\">Tangent Theorems<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">There are some theorems based on the tangent of a circle, let\u2019s discuss their statements and proof in detail.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"4-theorem-1-tangent-radius-theorem\">Theorem 1: Tangent Radius Theorem<\/h3>\n\n\n\n<p class=\"eplus-wrapper\">The tangent makes a right angle with the radius of a circle at the point of tangency.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">(At the point of tangency, the tangent is perpendicular to the radius.)<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"366\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-4.png\" alt=\"Tangent AB perpendicular to the radius OP\" class=\"wp-image-26613\" title=\"Tangent AB perpendicular to the radius OP\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-4.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-4-300x177.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>To prove:<\/strong> The radius OP is perpendicular to tangent AB.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Proof:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The points A and B lie outside the circle.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, we can write&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$OA \\gt OP$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$OB \\gt OP$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, for any such point on the line AB, we will get the same relation.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">It means that OP is the shortest distance between line AB and the point O.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $OP \\bot AB$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">&nbsp;$\\angle OPA = \\angle OPB = 90^{\\circ}$.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"5-theorem-2-two-tangents-theorem\">Theorem 2: Two Tangents Theorem<\/h3>\n\n\n\n<p class=\"eplus-wrapper\">The lengths of two tangents from a common external point to a circle are equal. The two tangents will subtend equal angles at the center. The line that connects the exterior point to the center will divide the angle between the tangents into two equal angles.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s consider two tangent lines PA and PB are drawn from an external point \u201cP\u201d as shown in the figure, then<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">$PA = PB$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The two tangents subtend equal angles at the center of the circle. $\\left[\\angle POA = \\angle POB\\right]$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The line that connects the exterior point to the center will divide the angle between the tangents into two equal angles. $\\left[\\angle OPA = \\angle OPB\\right]$<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"450\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-5.png\" alt=\"Two tangents drawn from a common external \" class=\"wp-image-26614\" title=\"Two tangents drawn from a common external \" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-5.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-5-300x218.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-tangent-of-a-circle-formula\">Tangent of a Circle: Formula<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"344\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-6.png\" alt=\"Tangent segment and external secant segment\" class=\"wp-image-26615\" title=\"Tangent segment and external secant segment\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-6.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-6-300x166.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">How can we find the tangent of a circle? The \u201c<strong>tangent-secant theorem<\/strong>\u201d explains the relationship between a tangent and a secant of the same circle.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s consider a circle with P as any exterior point, PQ is a tangent, and AB is a secant drawn from external point P.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, PA is the external secant segment.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The formula for tangent-secant relationship is given by<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PQ^2 = PA \\times PB$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-general-equation\">General Equation<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">If the equation of the circle is $x^2 + y^2 = r^2$, and a tangent is drawn from any external point (a, b) as shown in the figure below, then the general equation of the tangent to a circle is $xa + yb = r^2$.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"388\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-7.png\" alt=\"General equation of the tangent to a circle\" class=\"wp-image-26617\" title=\"General equation of the tangent to a circle\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-7.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-7-300x188.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">If the general equation of a line is y=mx+c and that of a circle is $x^2 + y^2 = r^2$, then the general equation of the tangent to the circle is given by $y = mx \\pm  r\\sqrt{1+m2}$.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The tangent to a circle equation $x^2 + y^2 = a^2$ at $(a\\; cos\\; \\theta,\\; a\\; sin\\; \\theta)\\; is\\; x.\\;cos\\; \\theta + y.\\;sin\\; theta = a$<\/li>\n\n\n\n<li class=\"eplus-wrapper\">The tangent to a circle equation $x^2 + y^2 + 2gx + 2fy + c = 0$ at $(x_1,\\; y_1)$ is&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"eplus-wrapper\">$xx_1 + yy_1+ g(x + x_1) + f(y + y,) + c = 0$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-condition-of-tangency\">Condition of Tangency<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>When the distance between the center of the circle and the straight line is the same as the radius of the circle, then the given line will touch the circle<\/strong>.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">There are exactly two tangents to circle from a point outside the circle. We can draw only one tangent to a circle through a point that lies on the circle. A tangent cannot be drawn from a point lying inside the circle.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-facts-about-tangent-of-a-circle\">Facts about Tangent of a Circle<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">If a particle or object moves in a circular path, and we abruptly stop its motion, then it will move in a direction that is tangent to the circular path. That means the direction of velocity will be tangential to the circular path.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Circle and tangent of a circle can be applied in our daily life, such as in construction, architecture, and engineering.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Real-life example: When the wheels of a bicycle roll along a road, the roadline can be compared to a tangent to the wheel at each point.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"317\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-8.png\" alt=\"Road as a tangent on a bicycle wheel\" class=\"wp-image-26618\" title=\"Road as a tangent on a bicycle wheel\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-8.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-8-300x153.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learned about the tangent of a circle, its properties, theorems, formula, general equations of tangents, condition of tangency, etc. Let\u2019s solve a few examples and practice problems.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"11-solved-examples-on-tangent-of-a-circle\">&nbsp;Solved Examples on Tangent of a Circle<\/h2>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>In the given figure, if <\/strong>$AC = 15$<strong> inches and <\/strong>$BC = 20$<strong> inches, find the length of the tangent.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"458\" height=\"224\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se1.png\" alt=\"Tangent to a circle theorem example\" class=\"wp-image-26619\" title=\"Tangent to a circle theorem example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se1.png 458w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se1-300x147.png 300w\" sizes=\"auto, (max-width: 458px) 100vw, 458px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AC = 15$ inches and $BC = 25$ inches.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">As we know, the radius and tangent of a circle are perpendicular to each other.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">In ABC, applying Pythagoras&#8217; theorem<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AC^2 + AB^2 = BC^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$15^2 + AB^2 = 25^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AB^2 = 25^2 \\;\u2212\\; 15^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AB^2 = 25^2 \\;\u2212\\; 15^2$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$AB^2 = 400$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\therefore AB = 20$ inches.<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\"><strong>&nbsp;Find the value of \u201cx\u201d in the figure given below.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"467\" height=\"248\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se2.png\" alt=\"Tangent to a circle theorem example\" class=\"wp-image-26620\" title=\"Tangent to a circle theorem example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se2.png 467w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se2-300x159.png 300w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Two tangent lines PA and PB are drawn from an external point \u201cP.\u201d Thus, the two tangents have the same length.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PA = PB$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Therefore,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x^2 + 10 = 46$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x^2 = 46 \\;\u2212\\; 10$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x^2 = 36$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$x = 6$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the value of \u201cx\u201d is 6.<\/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, if <\/strong>$PA = 9$<strong> feet, <\/strong>$AB = 7$<strong> feet, find the length of the tangent PQ.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"397\" height=\"209\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se3.png\" alt=\"Tangent-secant formula example\" class=\"wp-image-26621\" title=\"Tangent-secant formula example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se3.png 397w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se3-300x158.png 300w\" sizes=\"auto, (max-width: 397px) 100vw, 397px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Given, $PA = 9\\; feet,\\; AB = 7\\; feet$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, $PB = PA + AB = 9\\; feet + 7\\; feet = 16\\; feet$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the tangent-secant formula,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PQ^2 = PA \\times PB$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PQ^2 = 9 \\times 16$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PQ^2 = 144$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$PQ = 12\\; feet$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the length of tangent PQ is 12 feet.<\/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, if <\/strong>$DC = 14$<strong> inches and <\/strong>$BC = 11$<strong> inches, find the radius of the circle.<\/strong><\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"484\" height=\"220\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se4.png\" alt=\"Tangent-secant formula example\" class=\"wp-image-26622\" title=\"Tangent-secant formula example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se4.png 484w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-se4-300x136.png 300w\" sizes=\"auto, (max-width: 484px) 100vw, 484px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Given, $DC = 14$ inches and $BC = 11$ inches<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, $AC = AB + BC = r + 11$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">According to the tangent-secant formula,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$DC^2 = BC \\times AC$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$14^2 = 11 (r + 11)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$196 = 11r + 121$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$11r = 196 \\;\u2212\\; 121 = 75$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$r = \\frac{75}{11} = 6.81$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Hence, the radius of the circle is 6.81 inches.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"12-practice-problems-on-tangent-of-a-circle\">Practice Problems on Tangent of a Circle<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Tangent of a Circle: Definition, Properties, Formula, Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">A line that touches the circle at _____ is called a tangent to the circle.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">two points<\/div><div class=\"spq_answer_block\" data-value=\"1\">one point<\/div><div class=\"spq_answer_block\" data-value=\"2\">at least one point<\/div><div class=\"spq_answer_block\" data-value=\"3\">at most two points<\/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: one point<br\/>A tangent to the circle touches the circle at only one point.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">The number of tangents that pass through a point on a circle is _____.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">one<\/div><div class=\"spq_answer_block\" data-value=\"1\">two<\/div><div class=\"spq_answer_block\" data-value=\"2\">infinitely many<\/div><div class=\"spq_answer_block\" data-value=\"3\">0<\/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: one<br\/>Only one tangent can be drawn through a point on a circle.<\/span><\/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\">How many tangents can be drawn from an external point to a circle?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">1<\/div><div class=\"spq_answer_block\" data-value=\"1\">2<\/div><div class=\"spq_answer_block\" data-value=\"2\">3<\/div><div class=\"spq_answer_block\" data-value=\"3\">Infinite<\/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: 2<br\/>From an external point, two tangent lines can be drawn to a circle.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Which of the following results is related to the figure given below?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-pp1.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">$PA = PB$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\\angle POA = \\angle POB$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\\angle OPA = \\angle OPB$<\/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\/>All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">What is the length of tangent AP in the given figure?<\/h3><\/span><div class=\"spq_question_image\"><img decoding=\"async\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-pp2.png\"><\/div><div class=\"spq_answer_block\" data-value=\"0\">6 inches<\/div><div class=\"spq_answer_block\" data-value=\"1\">10 inches<\/div><div class=\"spq_answer_block\" data-value=\"2\">12 inches<\/div><div class=\"spq_answer_block\" data-value=\"3\">14 inches<\/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: 12 inches<br\/>The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $AP = PB = 12\\; inches$.<\/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\": \"Tangent of a Circle: Definition, Properties, Formula, Examples\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Tangent of a Circle\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"A line that touches the circle at _____ is called a tangent to the circle.\",\n                    \"text\": \"A line that touches the circle at _____ is called a tangent to the circle.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"two points\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"at least one point\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"at most two points\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"one point\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"A tangent to the circle touches the circle at only one point.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The number of tangents that pass through a point on a circle is _____.\",\n                    \"text\": \"The number of tangents that pass through a point on a circle is _____.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"two\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"infinitely many\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"one\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Only one tangent can be drawn through a point on a circle.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"How many tangents can be drawn from an external point to a circle?\",\n                    \"text\": \"How many tangents can be drawn from an external point to a circle?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"3\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Infinite\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"2\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"From an external point, two tangent lines can be drawn to a circle.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following results is related to the figure given below?\",\n                    \"text\": \"Which of the following results is related to the figure given below? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-pp1.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$PA = PB$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\angle POA = \\\\angle POB$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\\\\angle OPA = \\\\angle OPB$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\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\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"All results are given in options a, b, and c are about the \u201ctwo tangent theorem,\u201d which is shown in the given figure.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the length of tangent AP in the given figure?\",\n                    \"text\": \"What is the length of tangent AP in the given figure? <img src=\\\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/03\/tangent-of-circle-pp2.png\\\"\/>\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"6 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"10 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"14 inches\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"12 inches\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The lengths of two tangents from a common external point to a circle are equal.<br>\r\nTherefore, $$AP = PB = 12\\\\; inches$$.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"13-frequently-asked-questions-on-tangent-of-a-circle\">Frequently Asked Questions on Tangent of a Circle<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-faf16a56-6cda-4acc-a772-54e521dfa147\" 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-faf16a56-6cda-4acc-a772-54e521dfa147\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-faf16a56-6cda-4acc-a772-54e521dfa147\"><strong>What is the point of tangency 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-0-faf16a56-6cda-4acc-a772-54e521dfa147\">\n\n<p class=\"eplus-wrapper\">The point of contact of the tangent with the circle is known as the circle&#8217;s point of tangency.<\/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-faf16a56-6cda-4acc-a772-54e521dfa147\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-faf16a56-6cda-4acc-a772-54e521dfa147\"><strong>What are the two important theorems related to tangents to 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-1-faf16a56-6cda-4acc-a772-54e521dfa147\">\n\n<p class=\"eplus-wrapper\">The two important theorems related to tangents to a circle are as follows:<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Theorem 1: The tangent makes a right angle at the point of tangency with the radius of a circle.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Theorem 2: The lengths of the two tangents drawn from an external point to a circle are equal.<\/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-faf16a56-6cda-4acc-a772-54e521dfa147\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-faf16a56-6cda-4acc-a772-54e521dfa147\"><strong>What is the tangent-secant formula?<\/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-faf16a56-6cda-4acc-a772-54e521dfa147\">\n\n<p class=\"eplus-wrapper\">If P is the exterior point of a circle, PQ is a tangent and AB is a secant drawn from external point P, then the formula for tangent-secant is given by, $PQ^2 = PA \\times PB$.<\/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-faf16a56-6cda-4acc-a772-54e521dfa147\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-faf16a56-6cda-4acc-a772-54e521dfa147\"><strong>How many tangents are drawn from a point within the 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-3-faf16a56-6cda-4acc-a772-54e521dfa147\">\n\n<p class=\"eplus-wrapper\">There is no way to make a tangent when a point is inside the circle. That means zero tangents are drawn from a point within a 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-4-faf16a56-6cda-4acc-a772-54e521dfa147\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-faf16a56-6cda-4acc-a772-54e521dfa147\"><strong>How many tangent lines are possible for a circle inscribed in a triangle?<\/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-faf16a56-6cda-4acc-a772-54e521dfa147\">\n\n<p class=\"eplus-wrapper\">When a circle is inscribed in a triangle then all three sides of the triangle become the circle\u2019s tangent. Hence, three tangent lines are possible, i.e., an incircle of a triangle has three tangents.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is The Tangent of a Circle? The word \u201ctangent\u201d originated from the Latin word \u201ctangere,\u201d which means \u201cto touch.\u201d The tangent of a circle refers to a line that touches a circle at a single point. In the image shown below, the line l is a tangent to the circle with the center C. &#8230; <a title=\"Tangent of a Circle: Definition, Properties, Formula, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/tangent-to-a-circle\" aria-label=\"More on Tangent of a Circle: Definition, Properties, Formula, 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-26602","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\/26602","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=26602"}],"version-history":[{"count":11,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26602\/revisions"}],"predecessor-version":[{"id":39922,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26602\/revisions\/39922"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=26602"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=26602"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=26602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}