{"id":134815,"date":"2023-03-21T16:34:46","date_gmt":"2023-03-21T11:04:46","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=134815"},"modified":"2023-03-21T16:34:46","modified_gmt":"2023-03-21T11:04:46","slug":"prove-that-the-lengths-of-tangents-drawn-from-an-external-point-to-a-circle-are-equal","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/prove-that-the-lengths-of-tangents-drawn-from-an-external-point-to-a-circle-are-equal","title":{"rendered":"Prove that the lengths of tangents drawn from an external point to a circle are equal."},"content":{"rendered":"<h2>Question :Prove that the lengths of tangents drawn from an external point to a circle are equal.<\/h2>\n<h3>The correct answer :Let O be the center of a circle, and let A and B be two points on the circle. Let P be an external point to the circle, and let PA and PB be the tangents from P to the circle. We want to show that PA = PB.<\/h3>\n<h3>First, we draw a line segment OP from the external point P to the center O of the circle, and let the point of intersection of OP with the circle be C.<\/h3>\n<h3>Since OA and OB are radii of the circle, they are equal in length. Also, since OC is a radius of the circle, it is perpendicular to the tangent PA at point A, and to the tangent PB at point B.<\/h3>\n<h3>Therefore, we have two right triangles, PAC and PBC, where PC is the hypotenuse of both triangles. By the Pythagorean theorem, we have:<\/h3>\n<h3>PA\u00b2 = PC\u00b2 &#8211; AC\u00b2 PB\u00b2 = PC\u00b2 &#8211; BC\u00b2<\/h3>\n<h3>Subtracting these two equations, we get:<\/h3>\n<h3>PA\u00b2 &#8211; PB\u00b2 = (PC\u00b2 &#8211; AC\u00b2) &#8211; (PC\u00b2 &#8211; BC\u00b2)<\/h3>\n<h3>Simplifying, we get:<\/h3>\n<h3>PA\u00b2 &#8211; PB\u00b2 = BC\u00b2 &#8211; AC\u00b2<\/h3>\n<h3>But AC = BC, since they are both radii of the same circle, so AC\u00b2 = BC\u00b2. Therefore, we can simplify further:<\/h3>\n<h3>PA\u00b2 &#8211; PB\u00b2 = 0<\/h3>\n<h3>Which means that:<\/h3>\n<h3>PA\u00b2 = PB\u00b2<\/h3>\n<h3>Taking the square root of both sides, we get:<\/h3>\n<h3>PA = PB<\/h3>\n<h3>Therefore, we have shown that the lengths of the tangents drawn from an external point to a circle are equal.<\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Question :Prove that the lengths of tangents drawn from an external point to a circle are equal. The correct answer :Let O be the center of a circle, and let A and B be two points on the circle. Let P be an external point to the circle, and let PA and PB be the [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-134815","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134815","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=134815"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134815\/revisions"}],"predecessor-version":[{"id":134816,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134815\/revisions\/134816"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=134815"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=134815"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=134815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}