{"id":155894,"date":"2022-03-26T01:09:08","date_gmt":"2022-03-25T19:39:08","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/tangents-in-geometry-definition-derivation-applications-and-faqs\/"},"modified":"2024-09-23T11:32:05","modified_gmt":"2024-09-23T06:02:05","slug":"tangents-in-geometry-definition-derivation-applications-and-faqs","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/","title":{"rendered":"Tangents"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangents_meaning\" title=\"Tangents meaning\">Tangents meaning<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangent_definition\" title=\"Tangent definition\">Tangent definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangent_of_a_Circle\" title=\"Tangent of a Circle\">Tangent of a Circle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Point_of_Tangency\" title=\"Point of Tangency\">Point of Tangency<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangent_Properties\" title=\"Tangent Properties\">Tangent Properties<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangents_theorem\" title=\"Tangents theorem\">Tangents theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangent_Radius_Theorem\" title=\"Tangent Radius Theorem\">Tangent Radius Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Two_Tangents_Theorem\" title=\"Two Tangents Theorem\">Two Tangents Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#Tangents_of_Circles_Formula\" title=\"Tangents of Circles Formula\">Tangents of Circles Formula<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/tangents-in-geometry\/#FAQs_on_Tangents\" title=\"FAQs on Tangents\">FAQs on Tangents<\/a><\/li><\/ul><\/nav><\/div>\n<p>The meaning of the word \u201ctangent\u201d is \u201cto touch\u201d. It is similar to the word \u201ctangere\u201d in the Latin language. A tangent refers to the line that touches the circle at only one point on its circumference and does not enter the interior part of the circle. A circle can have an infinite number of tangents. They always make right angles to the radius.<\/p>\n<p>Read the following article to get more information on the tangent, its meaning and theorems.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Tangents_meaning\"><\/span>Tangents meaning<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In geometry, a tangent is a line that touches a curve at exactly one point without crossing it to enter the interior of the circle. A tangent can be drawn from an external point to a point on the curve, forming a line that just touches the curve at that single location. One practical example of a tangent is when you ride a bicycle. In this case, every point on the circumference of the wheel touches the ground at a single point, creating a tangent line with the road. This interaction between the wheel and the road is a perfect representation of how tangents work in the real world.<\/p>\n<p>Consider the following figure showing an arc S and a point P located outside the arc. A tangent line is drawn from point P to touch the arc S at a single point.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-734628\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/tangent.png\" alt=\"tangent\" width=\"219\" height=\"194\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/tangent.png 219w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/tangent-150x133.png 150w\" sizes=\"(max-width: 219px) 100vw, 219px\" \/><\/p>\n<p>This setup shows us the interaction of a tangent line with a curve.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangent_definition\"><\/span>Tangent definition<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>In geometry, point tangency is the condition where a line touches some curve or curved surface at one point only. In simpler words, a tangent is a line that touches a curve at exactly one point without crossing it to enter the circle&#8217;s interior.<\/p>\n<p style=\"text-align: center;\"><em><strong>Also Read: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/coordinate-geometry\/\">Coordinate Geometry<\/a><\/strong><\/em><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangent_of_a_Circle\"><\/span>Tangent of a Circle<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>We know that if a straight line meets or touches the circle at a single point, then it is called a tangent of the circle. A tangent is that line which actually is concurrent with the given circle but does not intersect with it. The diagram given below depicts a circle with a point P marked on it Any line drawn through P that is tangent to the circle is marked as S. Below is an example of a tangent to the circle.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-734629\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-of-a-Circle.png\" alt=\"Tangent of a Circle\" width=\"212\" height=\"205\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-of-a-Circle.png 212w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-of-a-Circle-150x145.png 150w\" sizes=\"(max-width: 212px) 100vw, 212px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Point_of_Tangency\"><\/span>Point of Tangency<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The point of tangency is the unique point where a straight line, such as a tangent, touches or intersects a curve without crossing it. It is the exact spot where the curve and the tangent line meet. In the figure given above, point P is highlighted as the point of tangency. This is the specific point where the tangent line just touches the circle, making P the precise point of contact between the line and the circle.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangent_Properties\"><\/span>Tangent Properties<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The tangent has two important properties<\/p>\n<ul>\n<li>A tangent line touches a curve, such as a <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/areas-related-to-circles\/circles-areas-related-to-circles\/\"><strong>circle<\/strong><\/a>, at exactly one point. This unique point is called the point of tangency.<\/li>\n<li>A tangent line does not cross into the interior of the circle. It only grazes the outer edge.<\/li>\n<li>At the point of tangency, the tangent line is always perpendicular to the radius of the circle, forming a <a href=\"https:\/\/infinitylearn.com\/surge\/a-pentagon-is-a-five-sided-polygon.-the-word-pentagon-comes-from-the-greek-word-pentagonon-meaning-five-cornered.-the-pentagon-is-the-simplest-polygon-that-can-be-constructed-by-connecting-five-points-with-straight-lines.-the-pentagon-is-also-the-only-regular-polygon-that-has-five-sides.\/right-angle\/\"><strong>right angle<\/strong><\/a> (90 degrees).<\/li>\n<\/ul>\n<p>Aside from the above-mentioned properties, a tangent to the circle has mathematical theorems related to it. These theorems are discussed below in detail.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangents_theorem\"><\/span>Tangents theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Tangents to circles are governed by specific theorems that are crucial in geometric calculations and proofs. These theorems help solve problems involving tangents, circles, and angles. The two most important theorems related to the tangents are discussed below.<\/p>\n<ul>\n<li><strong>Tangent-Radius Theorem:<\/strong> This theorem states that the tangent to a circle at any given point is perpendicular to the radius drawn to the point of contact.<\/li>\n<li><strong>Two Tangents Theorem:<\/strong> If two tangents are drawn to a circle from the same external point, they are equal in length. This theorem is often used in proofs and calculations involving circle geometry.<\/li>\n<\/ul>\n<p style=\"text-align: center;\"><em><strong>Also Read: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/euclidean-geometry\/\">Euclidean Geometry<\/a><\/strong><\/em><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangent_Radius_Theorem\"><\/span>Tangent Radius Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The tangent to a circle at any given point is perpendicular to the radius drawn to the point of contact.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-734630\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-Radius-Theorem.png\" alt=\"Tangent Radius Theorem\" width=\"215\" height=\"164\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-Radius-Theorem.png 215w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Tangent-Radius-Theorem-150x114.png 150w\" sizes=\"(max-width: 215px) 100vw, 215px\" \/><\/p>\n<p><strong>Given:<\/strong> A circle with center O and radius OA. A tangent line PL touches the circle at point A.<\/p>\n<p><strong>To Prove: <\/strong>The tangent PL is perpendicular to the radius OA at the point of contact A.<\/p>\n<p><strong>Proof: <\/strong><\/p>\n<p><strong>Assume:<\/strong> Point P lies on the line PL, but outside the circle.<\/p>\n<p><strong>Construction:<\/strong> Join P to the center O, forming the line segment PO.<\/p>\n<p><strong>Observation:<\/strong> Since P is outside the circle, the length of PO is greater than the radius OA. Mathematically, PO&gt;OA.<\/p>\n<ul>\n<li>This condition PO&gt;OA holds true for every point on the line PL except the point A. This is because A is the only point on PL that lies exactly on the circle, making OA the shortest possible distance from the center O to any point on the tangent line PL.<\/li>\n<li>Since OA is the shortest distance from the center O to the line PL, it implies that OA is the perpendicular distance from O to the tangent line PL.<\/li>\n<li>By the definition of perpendicularity, the shortest line segment from a point to a line is always perpendicular to that line.<\/li>\n<\/ul>\n<p><strong>Conclusion: <\/strong>Therefore, OA is perpendicular to the tangent PL at point A.<\/p>\n<p><strong>Hence proved:<\/strong> The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Two_Tangents_Theorem\"><\/span>Two Tangents Theorem<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If two tangents are drawn to a circle from the same external point, they are equal in length.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-734631\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Two-Tangents-Theorem.png\" alt=\"Two Tangents Theorem\" width=\"236\" height=\"187\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Two-Tangents-Theorem.png 236w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/Two-Tangents-Theorem-150x119.png 150w\" sizes=\"(max-width: 236px) 100vw, 236px\" \/><\/p>\n<p><strong>Given: <\/strong>Two tangents CA and CB are drawn to a circle from an exterior point C, touching the circle at points A and B, respectively.<\/p>\n<p><strong>To Prove: <\/strong>The lengths of the two tangents are equal, i.e., CA=CB.<\/p>\n<ul>\n<li>The two tangents subtend equal angles at the centre of the circle, i.e., \u2220COA=\u2220COB.<\/li>\n<li>The angle between the tangents is bisected by the line joining the exterior point C to the centre O, i.e., \u2220ACO=\u2220BCO.<\/li>\n<\/ul>\n<p><strong>Proof:<\/strong><\/p>\n<p><strong>Construct Triangles:<\/strong> Draw radii OA and OB to the points of tangency A and B. Join OC.<\/p>\n<ul>\n<li><strong><strong>Identify Key Properties: <\/strong><\/strong><\/li>\n<li>OA=OB (radii of the same circle).<\/li>\n<li>OC=OC (common side).<\/li>\n<\/ul>\n<p><strong>Tangent-Radius Perpendicularity:<\/strong><\/p>\n<p>\u2220OAC=\u2220OBC=90\u00b0 (the tangent to a circle is perpendicular to the radius at the point of tangency).<\/p>\n<p><strong><strong>Triangle Congruence:<\/strong><\/strong><\/p>\n<p>In \u0394CAO and \u0394CBO:<\/p>\n<ul>\n<li>OA=OB (radii of the circle).<\/li>\n<li>OC=OC (common side).<\/li>\n<li>\u2220OAC=\u2220OBC=90\u00b0 (perpendicularity of tangent and radius).<\/li>\n<\/ul>\n<p>By the RHS (Right angle-Hypotenuse-Side) criterion, \u0394CAO\u2245\u0394CBO.<\/p>\n<p><strong><strong>Conclusions from Congruence:<\/strong><\/strong><\/p>\n<p>Since \u0394CAO\u2245\u0394CBO, it follows that:<\/p>\n<ul>\n<li>CA=CB (corresponding sides are equal).<\/li>\n<li>\u2220COA=\u2220COB (corresponding angles are equal).<\/li>\n<li>\u2220ACO=\u2220BCO (corresponding angles are equal).<\/li>\n<\/ul>\n<p><strong>Hence proved:<\/strong> The lengths of the two tangents drawn from an exterior point to a circle are equal, the tangents subtend equal angles at the center, and the angle between the tangents is bisected by the line joining the exterior point to the center.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Tangents_of_Circles_Formula\"><\/span>Tangents of Circles Formula<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Now, let&#8217;s look into the equation of tangents. Tangent is a line and to write the equation of tangent we need two things, slope (m) and one point on the line.<\/p>\n<p>The general equation of the tangent to a circle is:<\/p>\n<ol>\n<li>The tangent to a circle equation x\u00b2 + y\u00b2 = a\u00b2 for a line y= mx+c is given by the equation y = mx \u221a a [ 1 + m2 ]<\/li>\n<li>The tangent to a circle equation x\u00b2 + y\u00b2 =a\u00b2 at (a1, b1) is xa1 + yb1 = a2<\/li>\n<\/ol>\n<p>Therefore, the equation of tangent is xa1 + yb1 = a2, where (a1, b1) are coordinates from which it has been drawn.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"FAQs_on_Tangents\"><\/span>FAQs on Tangents<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"&#8221;What\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"&#8221;A\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"&#8221;What\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"&#8221;The\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"&#8221;How\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"&#8221;The\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n for a circle centred at the origin with radius a.&#8221; image-2=&#8221;&#8221; count=&#8221;3&#8243; html=&#8221;true&#8221; css_class=&#8221;&#8221;]\n","protected":false},"excerpt":{"rendered":"<p>The meaning of the word \u201ctangent\u201d is \u201cto touch\u201d. It is similar to the word \u201ctangere\u201d in the Latin language. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Tangents","_yoast_wpseo_title":"Tangents - Definition, Derivation, Theorems and FAQs","_yoast_wpseo_metadesc":"Learn about tangents in geometry topic of maths in details explained by subject experts on infinitylearn.com. 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