{"id":1375,"date":"2023-05-05T10:00:45","date_gmt":"2023-05-05T04:30:45","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=1375"},"modified":"2023-05-06T09:19:22","modified_gmt":"2023-05-06T03:49:22","slug":"construct-tangent-to-circle-from-external-point","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/construct-tangent-to-circle-from-external-point\/","title":{"rendered":"How To Construct A Tangent To A Circle From An External Point"},"content":{"rendered":"<h2><strong>How To Construct A Tangent To A Circle From An External Point<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12489\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point.jpg\" alt=\"\" width=\"1280\" height=\"720\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point.jpg 1280w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point-300x169.jpg 300w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point-768x432.jpg 768w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point-1024x576.jpg 1024w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/construct-tangent-to-circle-from-external-point-358x200.jpg 358w\" sizes=\"(max-width: 1280px) 100vw, 1280px\" \/><\/p>\n<p>Steps of construction<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-135248 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-1.png\" alt=\"How To Construct A Tangent To A Circle From An External Point 1\" width=\"275\" height=\"189\" \/>1. Take given circle and a point P outside the circle. O is centre of the circle.<br \/>\n2. Joint OP.<br \/>\n3. Bisect OP and get its mid-point M.<br \/>\n4. Draw circle with centre M and radius = PM = MO.<br \/>\n5. Circle drawn meets the given circle at Q above PO and at Q&#8217; below PO.<br \/>\n6. Join PQ and PQ&#8217;.<br \/>\n7. PQ and PQ&#8217; are the required tangents drawn to the circle from the point P.<br \/>\nWe observe that PQ = PQ&#8217;.<\/p>\n<p><strong>Read More:<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/www.aplustopper.com\/number-tangents-to-a-circle\/\">Number Of Tangents From A Point On A Circle<\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/lesson\/rs-aggarwal-class-10-solutions-circles\/\">RS Aggarwal Class 10 Solutions Circles<\/a><\/li>\n<\/ul>\n<h2><strong>Construction Of Tangent To A Circle\u00a0<\/strong><strong>From An External Point<\/strong><\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/HEdUc-q9_es?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Example 1:<\/strong> \u00a0 \u00a0Draw a circle of radius 3 cm. Take a point at a distance of 5.5 cm from the centre of the circle. From point P, draw two tangents to the circle.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 \u00a0Steps of Construction<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-135249 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-2.png\" alt=\"How To Construct A Tangent To A Circle From An External Point 2\" width=\"196\" height=\"196\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-2.png 196w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-2-150x150.png 150w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-2-100x100.png 100w\" sizes=\"(max-width: 196px) 100vw, 196px\" \/>Step I: Take a point O in the plane of the paper and draw a circle of radius 3 cm.<br \/>\nStep II: Mark a point P at a distance of 5.5 cm from the centre O and join OP.<br \/>\nStep III: Draw the right bisector of OP, intersecting OP at Q.<br \/>\nStep IV: Taking Q as centre and OQ = PQ as radius, draw a circle to intersect the given circle at T and T\u00b4.<br \/>\nStep V : Join PT and PT\u00b4 to get the required tangents.<br \/>\n<strong>Example 2:<\/strong> \u00a0 \u00a0Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius\u00a06 cm and measure its length. Also verify the measurement by actual calculation.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 In order to do the desired construction,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-135250 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-3.png\" alt=\"How To Construct A Tangent To A Circle From An External Point 3\" width=\"288\" height=\"248\" \/>we follow the following steps:<br \/>\nStep I: Take a point O on the plane of the paper and draw a circle of radius OA = 4 cm. Also, draw a concentric circle of radius OB = 6 cm.<br \/>\nStep II: Find the mid-point C of OB and draw a circle of radius OC = BC. Suppose this circle intersects the circle of radius 4 cm at P and Q.<br \/>\nStep III: Join BP and BQ to get the desired tangents from a point B on the circle of radius 6 cm.<br \/>\nBy actual measurement, we find the<br \/>\nBP = BQ = 4.5 cm<br \/>\nJustification: In \u2206BPO, we have<br \/>\nOB = 6 cm and OP = 4 cm<br \/>\nOB<sup>2<\/sup>= BP<sup>2<\/sup> + OP<sup>2<\/sup> \u00a0 \u00a0 \u00a0 \u00a0[Using Pythagoras theorem]<br \/>\n\\( \\Rightarrow BP=\\sqrt{O{{B}^{2}}-O{{P}^{2}}} \\)<br \/>\n\\( \\Rightarrow \\sqrt{36-16}=\\sqrt{20}=4.47\\simeq 4.5\\text{ }cm~ \\)<br \/>\nSimilarly, BQ = 4.47cm \u2248 4.5 cm<\/p>\n<p><strong>Example 3:<\/strong> \u00a0 \u00a0Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other an angle of 60\u00ba.<br \/>\n<strong>Sol.\u00a0<\/strong>\u00a0 \u00a0In order to draw the pair of tangents, we follow the following steps.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-135251 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-4.png\" alt=\"How To Construct A Tangent To A Circle From An External Point 4\" width=\"352\" height=\"293\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-4.png 352w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-4-300x250.png 300w\" sizes=\"(max-width: 352px) 100vw, 352px\" \/>Step I: Take a point O on the plane of the paper and draw a circle of radius OA = 5 cm.<br \/>\nStep II: Produce OA to B such that OA = AB = 5 cm.<br \/>\nStep III: Taking A as the centre draw a circle of radius AO = AB = 5 cm. Suppose it cuts the circle drawn in step I at P and Q.<br \/>\nStep IV: Join BP and BQ to get the desired tangents.<br \/>\n<strong>Justification:<\/strong> In OAP, we have<br \/>\nOA = OP = 5 cm (= Radius) Also,<br \/>\nAP = 5 cm (= Radius of circle with centre A)<br \/>\n\u2234 \u2206OAP is equilateral.<br \/>\n\u21d2 \u2220PAO = 60\u00ba \u21d2 \u2220BAP = 120\u00ba<br \/>\nIn \u2206BAP, we have<br \/>\nBA = AP and \u00a0\u2220BAP = 120\u00ba<br \/>\n\u2234 \u2220ABP = \u2220APB = 30\u00ba \u21d2 \u2220PBQ = 60\u00ba<\/p>\n<p><strong>Example 4: \u00a0 \u00a0<\/strong>Draw a circle of radius 3 cm. Draw a pair of tangents to this circle, which are inclined to each other at an angle of 60\u00ba.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Steps of construction<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-135252 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-5.png\" alt=\"How To Construct A Tangent To A Circle From An External Point 5\" width=\"326\" height=\"263\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-5.png 326w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/How-To-Construct-A-Tangent-To-A-Circle-From-An-External-Point-5-300x242.png 300w\" sizes=\"(max-width: 326px) 100vw, 326px\" \/>Step I: Draw a circle with O as centre and radius = 3 cm.<br \/>\nStep II: Draw any diameter AOB of this circle.<br \/>\nStep III: Construct \u2220BOC = 60\u00ba such that radius OC meets the circle at C.<br \/>\nStep IV: Draw AM \u22a5 AB and CN \u22a5OC.<br \/>\nLet AM and CN intersect each other at P.<br \/>\nThen, PA and PC are the desired tangents to the given circle, inclined at an angle of 60\u00ba<br \/>\n<strong>Proof:<\/strong> \u2220AOC = (180\u00ba \u2013 60\u00ba) = 120\u00ba<br \/>\nIn quad. OAPC, we have<br \/>\n\u2220OAP = 90\u00ba, \u2220AOC = 120\u00ba, \u2220OCP = 90\u00ba.<br \/>\n\u2234\u2220APC = [360\u00ba \u2013 (90\u00ba + 120\u00ba + 90\u00ba)] = 60\u00ba.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How To Construct A Tangent To A Circle From An External Point Steps of construction 1. Take given circle and a point P outside the circle. O is centre of the circle. 2. Joint OP. 3. Bisect OP and get its mid-point M. 4. Draw circle with centre M and radius = PM = MO. 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Take given circle and a point P outside the circle. O is centre of the circle. 2. Joint OP. 3. Bisect OP and get its mid-point M. 4. Draw circle with centre M and radius = PM = MO. 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