Class 10th Mathematics Gujarat Board Solution
Exercise 11.1- A and B are the points on ⨀(O, r). bar ab is not a diameter of the circle.…
- A, B are the points on ⨀ (O, r) such that tangents at A and B intersect in P.…
- A, B are the points on ⨀(O, r) such that tangents at A and B to the circle…
- ⨀(O, r1) and ⨀(O, r2) are such that r1 r2. Chord AB of ⨀(O, r1) touches ⨀(O,…
- In example 4, if r1 = 41 and r2 = 9, find AB.
Exercise 11.2- P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle…
- Two concentric circles having radii 73 and 55 are given. The chord of the…
- bar ab is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which…
- P is in the exterior of a circle at distance 34 from the centre 0. A line…
- In figure 11.24, two tangents are drawn to a circle from a point A which is in…
- Prove that the perpendicular drawn to a tangent to the circle at the point of…
- Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and…
- bar pt and bar pr are the tangents drawn to ⨀ (O, r) from point P lying in the…
- vector ab is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the…
- P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to…
Exercise 11- A circle touches the sides bar bc , bar ca , vector ab of ΔABC at points D, E,…
- ΔABC is an isosceles triangle in which vector ab congruent bar ac . A circle…
- ∠B is a right angle in ΔABC. If AB = 24, BC = 7, then find the radius of the…
- A circle touches all the three sides of a right angled ΔABC in which LB is…
- In â ABCD, m∠D = 90. A circle with centre 0 and radius r touches its sides bar…
- Two concentric circles are given. Prove that all chords of the circle with…
- A circle touches all the sides of â ABCD. If AB = 5, BC = 8, CD = 6. Find AD.…
- A circle touches all the sides of â ABCD. If vector ab is the largest side then…
- P is a point in the exterior of a circle having centre 0 and radius 24. OP =…
- P is in exterior of ⨀(O, 15). A tangent from P touches the circle at T. If PT…
- vector p_si , pb touch ⨀(O, 15) at A and B. If m∠AOB = 80, then m∠OPB =A. 80…
- A tangent from P, a point in the exterior of a circle, touches the circle at…
- In ΔABC, AB = 3, BC = 4, AC = 5, then the radius of the circle touching all…
- vector pq and vector pr touch the circle with centre 0 at A and B…
- The points of contact of the tangents from an exterior point P to the circle…
- A chord of ⨀(O, 5) touches ⨀(O, 3). Therefore the length of the chord =A. 8…
- A and B are the points on ⨀(O, r). bar ab is not a diameter of the circle.…
- A, B are the points on ⨀ (O, r) such that tangents at A and B intersect in P.…
- A, B are the points on ⨀(O, r) such that tangents at A and B to the circle…
- ⨀(O, r1) and ⨀(O, r2) are such that r1 r2. Chord AB of ⨀(O, r1) touches ⨀(O,…
- In example 4, if r1 = 41 and r2 = 9, find AB.
- P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle…
- Two concentric circles having radii 73 and 55 are given. The chord of the…
- bar ab is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which…
- P is in the exterior of a circle at distance 34 from the centre 0. A line…
- In figure 11.24, two tangents are drawn to a circle from a point A which is in…
- Prove that the perpendicular drawn to a tangent to the circle at the point of…
- Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and…
- bar pt and bar pr are the tangents drawn to ⨀ (O, r) from point P lying in the…
- vector ab is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the…
- P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to…
- A circle touches the sides bar bc , bar ca , vector ab of ΔABC at points D, E,…
- ΔABC is an isosceles triangle in which vector ab congruent bar ac . A circle…
- ∠B is a right angle in ΔABC. If AB = 24, BC = 7, then find the radius of the…
- A circle touches all the three sides of a right angled ΔABC in which LB is…
- In â ABCD, m∠D = 90. A circle with centre 0 and radius r touches its sides bar…
- Two concentric circles are given. Prove that all chords of the circle with…
- A circle touches all the sides of â ABCD. If AB = 5, BC = 8, CD = 6. Find AD.…
- A circle touches all the sides of â ABCD. If vector ab is the largest side then…
- P is a point in the exterior of a circle having centre 0 and radius 24. OP =…
- P is in exterior of ⨀(O, 15). A tangent from P touches the circle at T. If PT…
- vector p_si , pb touch ⨀(O, 15) at A and B. If m∠AOB = 80, then m∠OPB =A. 80…
- A tangent from P, a point in the exterior of a circle, touches the circle at…
- In ΔABC, AB = 3, BC = 4, AC = 5, then the radius of the circle touching all…
- vector pq and vector pr touch the circle with centre 0 at A and B…
- The points of contact of the tangents from an exterior point P to the circle…
- A chord of ⨀(O, 5) touches ⨀(O, 3). Therefore the length of the chord =A. 8…
Exercise 11.1
Question 1.A and B are the points on ⨀(O, r).
is not a diameter of the circle. Prove that the tangents to the circle at A and B are not parallel.
Answer:Given that A and B are points on (O, r) and AB is a not a diameter of the circle.
We have to prove that the tangents to the circle at A and B are not parallel.
Proof:
Using the method of contradiction,
Let l and m be two parallel tangents to the circle have centre O drawn at the points A and B.
∴ OA ⊥ l and OB ⊥ m
Consider OA and OB perpendicular to l and m respectively and O is a common point.
∴ since l and m are two parallel lines, A – O – B
Hence, AB is a diameter, which contradicts with our assumption.
∴ Our assumption is wrong i.e. l and m are intersecting lines.
∴ Tangents to the circle at A and B are not parallel.
Hence proved.
Question 2.A, B are the points on ⨀ (O, r) such that tangents at A and B intersect in P. Prove that OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Answer:Given that A and B are the points on (O, r) such that tangents at A and B intersect in P.
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SONhYjfColJFE51FUFEG2KgF0EmEcS37pjxw5T4sOJhhWXEiJWhA9+cer8xJrBp0JC5ufnm0AA1A5dL9CJIRU/58CxTxQO9genCGmhxxzV2+CDSktKV6wYfCokJOrqyZMnBSetNkXAXwSQkqRbUebFaUOMd4TPwoULjQqLe8Tp6/j7zE7sXyEhkY6oqxr06wTM8XcOarcSDkf+YyiIwlyVTBMMPrNmzZIXXnjBBK5Eu8GnXEJiXeWtQ/6aqqvxRyYnnph41szMTLOshNNqq/f9MafE4EMFi1gw+JRLyOzsbLOKLiFQ2hSBQBHAGEiQAKuhhbLuEgYfUgKRllaEDwYf/o62Vi4hceoSZaHqarR1p7vuF0JS7uXQoUPGlx3q5m3weeWVV4wPM9oMPuUSEpMyE2cnfEih7gQ9v3sRIHeW+FRiUsNBSJBg3FoRPhh8mMNi8ImWwJYyhCRMjjca7o54TxZ171CPjjvDcU/UzoEDB8J6w7hALIMPET4UbcZbQMCC21sZQuLqOHPmjDRu3Njt9673FwUIMI6YR54/fz7skTUYfEjpwuDDGpcYKqlo4OYInzKEzMrKMpKRFXS1KQLBIoATn0oCGAoj8ZK3DD5WhA91oVBh3WrwKUPIo0ePGotYPJdRCHYQ6vE/IsDcDdcZ4yoShLTuhOTp0ildbjT4lCEk7g5ADDSXTQejIuCNQHp6uqkeACEj3VgCgRo+BL671eBTgpCFhYVy/PjxsFnEIt1BTl//880nZcnWk1JUVCS929SWYZ3rOn2JqDsfL3YidRhXbmiWwQcVlggftxl8ShAyNzdXWLcD07E2/xB4a9lhmb8hW8YPaCBJnjn49GVZsu94njxxkxrHGE+EYmLBd4vmxRwSgw/hfRh8cI8MGzYs4gafEoSkXAdSkhhWbfYRyMkrkFmrjslf7mslLa5IMwd2bFJDfvLyt3Jnjyukfu0U+yeLwT2RkIwt1oNxCyGBGVV61KhRJqVr7ty5Yhl8rIVoI9EVZQhJHKsS0r+uOHzqoqRUSZSmdasVH1g3PUXq1UqR3UfOxT0hmUciHSFlRet/+Ie4s3vjn2zUqJHMmTNHXn31VePDvP766yOS0lWCkPgfiaIHQG32EUivliz5BZckL/+SVE9N+ueBRZJzvkAyavgsW2T/QlG6J9E6jCvGF9E7bmzeKV0kV+OzJHvEqenbxAUHZMfhs1ItJVE8Mk9u61pX+l91WRkoSoyW77//3rg8NMPDvyGDStoqM02en79ffjG0iSQnJciUzw9KrbRkadOgun8ni8G9LUJio3Bzw/9OuRqqEBDh8+KLLzoS4eOx8cnirSfk9q71pFOTmrI/O0/+MOc7ubJO1RJaFdiUICRR+UQxaDFk/4fNb+5oJs/O2yc/f2OHR9URSUtJkj+MbilJiQn+nyzGjmBhJsYU0TrR0DD4eNfwCdbg4+GjoEUNujpDGmdUlU5Na8oMjxHw4IkLvglJ7dVYKYcQzs4H8N/d3UIOnswzKglvP20/IFClShWzRQshuWcEEwYfMlaYW1o1fAIx+HgEr2dKUyR//Xi/0aR2ZJ2VmlWTpVOzmpWrrJaE1IEUOAKNPG9AbSURQDoiJRlf0daoQIDBh4VmSenC4DNw4ED/hZaHlExhsDFc3z5DbvBIy7Rie8OPqJRQWSlsxZIB2nwgUFQoX3y+UP6x75B0aNdauvceoJBVggAaFxKS8RWNjQifn/70p8JyjJbBh6LNdg0+zCEvXSqSMX3rS+v6P7jFKmolCInLQ+ePlQ+ZvLPfy2O/eEbeXJsvRTUaSZWcN+UXQ/4uv3vmNyKJloU1GoddaO8ZUuLjjtbG/VsGH5Y4oGgzKV3EyNppDS5LlYJCz1zGR1NC+kKo1P+nvjFDpq1Jloxe4ySpqEAuFt4gv5/zkvTsMk+GDL/Dz7PFz+4EBOCLjPbmbfCZOXNmcUpXZSVKsOv9/u6WUtXjq/bVigmJdFQJ6QsukS/X75HkZn0kqcAzH7qUL6nJVSWlaX95fvI0yTp8RC56/JEMPjQNPu1s3vuW/m797f3p/Z03N+Z6Nu/vvp8kvHtwz4yvWGgYPu+8887iCB87Bp8f/dOVI1BMSKtTCYzWVjECl6dXlYJDpyWhfhMpuuiZEyWlysXcbGnVprEnx66FnMnJNZKADRWNeRPWRetv65P/WRuYe29c3btaQ+k+8SYfJIT01qf3y6A0mSt7UVT2AqnoJVH6PqwXgvVZGkXmkbHUMPiQUmYZfFBpqePDi3rtquWm77t06yEpafYDbUoQMtr1/HB09oOjh8jMh5+XYynVJS0jU87t8yxGdGGlPPPvz0pm46aV3oJFOiSF93dvcnqT2fq99G/W36U/2Z/fGAje//P+2zon98B3616876n0C8L623pRWERkvEBWa7MIX/o3ft+1a5cJOMGFYB3H75DUm/D8bWkVFWkH/G6dg8+KtnCMB1IVSemimBdGn53fbpKFa3bLF9+let6qSdKh1psy5fdPyDWdu9u6nRJzSCWkb8y6dO8t8yaclz+/9J4cP1VFmnkKKzzzr7/xScbSg9n3lUKzhzfZLFJ6vxB8ffeW9JUR3/of57t48aLZuDaLN/E/62VgTZW8P9mv9AvCGz/rhWARszQ5vQnr/bLwnj54E996MZSeXpSeJlh/e1+P3zgXJVMz610uI8c/JVtqjpTLO/cRzwPL6n3fysO/nSSLZ06Q6ukZPju1BCE5eTRbwnw+rUM79B0wSPoOGCj5585IlbTaDp01PKexpiZczRpM4bjy5MmTTRwr5TNKk9CbnNZ378/SmoIl/Uu/EMrTDrxfIHl5ecUvAu8Xj7e24H1v4GJNF6wpROm5uiWdkf5nTx+T48nNJKN1T0m8eMbAWqdpO9m4drXs2LZVOvfwvap0CUIyWeWmtdlBICHqyGjnqUKxD4Oc1CuruJQ1iENxLTvntCRwedK5ohdCRdME63ek6+H9uyTBE5hujKOE5+B/9GxJRfm248NLEBLTLYujcMNaAtJO1+o+dhBAkvGi54XvhsbYtlRZp+6HtTCzDu6XRgn7Ze2m+VKj3U2GQzmbFssdnhrRbdpdbetSZQhJTVZ1f9jCTneyiQDSEcnj5vKLNh+lzG7Mjb/88ktTWe+yyzJkyv/9Wd6Z/YnMXTVVPHJSbmuTIs/97j+kSmrlETrWiUsQkjQZ4g0RwxqxE2gX6XGlEcDtAyFDub5HJFCnLAklQCgM169fP5PUzDN27tZTfn1kvxR6eFSvkX/ri5QgJJUCUC2oq0MwsDZFwAkEyIOMpdIwp06dkkWLFsm6detMcMBjjz1mSkx6t8vrl/zbLo4lCEmlAPReEpW1LqtdCHU/XwhQuoMW7ZUosK2sXLnSBJjDEyzGLLXnpL2lBCHJ9MACxgRVmyLgFAKMJ+aP0ayyEh7HitAsHNStWzcZPHiwZ85YtgRHsJiVIST+FNb30KYIOIUAcyxWUovG0jBM34jAWbZsmfGjPvjgg8Lq0KFqZfyQsB4AtSkCTiFAkWRyB51U7Zy6t8rOs3HjRlmwYIEwByZGFcNNqONxy5REi8TyYeEAV68RGQSQMNgkOnToEJkbCOCqLHsAEbds2WLu++abbzbL6oWjlSEk2dHcCG8F3CDaFIFgEGD+iNsjXAM6mHslgAGfIioq9WPvvfdeueaaa4I5pd/HliEky9Dh7OQtoYT0G089oBQCBJNjKLRb7iJSAO7YscP4FBn3ffv2NT7FSBR1LkNIJq5MvonYCaTCVqQA1eu6EwGsk9glMOq4sSHB8SmuXbvW1GN99NFHpUmTJhG71TKExDSNesE6B9oUgWAQIAQTCYnW5bbSotzb6tWrjU8R/yJFq/ApRvo+y61zT9TBpk2bjOoajabqYAaRHuscAkgfkhV69erl3EkdONP+/fuNT5ECyF27djUW1FD4FAO51XIJiar61VdfmbdbJMV3IA+kx7gHAdRVpI9bxhDGJRZrxafInJbSjqH0KQbSE+USksKwqK68QdwCZiAPp8dEFgHKdlDigi3S7ZtvvjFSkTA+ih1juHFjvHa5hCTMCbUVQCnco00R8BcBpjtIyKuvvjqiAQEEJUDEzZs3S7t27WT8+PGuXYELjCtcK61169Yyf/58MwdgwU1tioA/CDBPY/k5xlEkGj5Fpl1Lliwx2t7YsWOlU6dOkbgVv65ZKSHxy5Dz1aePp2CPNkXADwS2bt1qDCWRmPKg2TF2s7KyzNhlLY5o8alXSEjAbNq0qYnaUUL6MRJ1V2Od37Ztm7Rv3z7ksZ/ecCORcWOsWbPGjF18inxGU6uQkDwE+j91NAk2d3ukRTSBHuv3SooS8asQMhwNSy4kXLhwoSk/w8rH+BSjsepFpYTEJIwfEgsVliltioAdBNavX29e4KWz6O0c6+8+Bw4cKPYpdu7c2fgUozm5vlJCkrDMW440lOuuuy6s6oe/HaP7uwMBggGwOzBvozRiqBo+RYLAMdzgVvnJT34ibdu2DdXlwnZen4jx1iHOj4kyZmNtikBlCOBeQIUMZZYE18Bog1oM8aka7kafYiAjxSchsZI1bNjQxP0pIQOBOH6OoVohczkkVSiCyfEpkqfIFIqxOG7cOFf7FAPpeZ+EJNiWWMRZs2bJwYMHzfLO2hSB8hDAsgppRo4c6ShA+BSXL19uwt4IWrnnnnsEzS0Wm09C8tBkTX/++eem4taoUaNiEQd9piARQE2FNKQwOelqIHwT9ZR0wN69exsVFdtGrDZbhEQ/79mzp/HxkLjphtjEWO2QaH0uiIO747777nPkEfApIhGZKmGtfeSRRwzZY73ZIiQgkKaCRYsSB9Sj1KYIWAggHbF4Mp256qqrggKGc2FEZK7InHTo0KFGMkajTzEQIGwTknIGVN3C+UqkvAYKBAJ3bB6zc+dOkxmEkSUY4mCjQD3Fos8c8aabbopqn2IgvW2bkJy8e/fuZp7A2/Duu+8O5Hp6TIwhQGQMqiXzxkCt8PgUWYGYjenQAw88EPC5oh1evwiJhYtJNWuqE5rk5OQ92oGM1/vfsGGDSbNiWe9A6q4ShE5WEQEF2CcGDBgQMz7FQMaEX4S05pJMtNHxH3744YA6IZAb1WPchwCSDUNfx44dpVUrzyKIfjTcIxSXIgoMvyUlF6m9E+/Nb0ISDnXLLbfI3/72N+HtGKv+oHgfGHaeHwMfGfjEj9pt+BRxn+FGw3rP1AeDobYfEPCbkBxE0inJnkzAeTPGsl9IB0r5CFBviUVKUTPtGvj27NljxgzGG9xoJCxES55iuMZBQITk5m699VaZMGGCUV01WCBc3eWO62DIIS2PKvfM+Xw1Yk4x/Kxatcq4RpjqNG/u30Kmvq4RK/8PmJDEKkLK999/30TyBOt/ihVA4+E5UDn37t1rnPW+grotnyJJy0OGDDHJ7sG4RmId34AJCTCsk4eVbPbs2fLUU0+p6hrro8XzfKiqWEXJsKissj2hbhSXokQ/0xsWrNHaTL4HSFCExMxNdvbEiRNl7ty5xlKmLXYRwCBDkgGqKk778lpeXl6xT5FEYYIFomnlq0j3XlCE5OapvUMo3euvv27mBYQ5aYtNBLAXIPmeeOKJcivaoy1htDl16pSRoBh8fKm0sYlU4E8VNCG5NG9AJvdIyczMzLgIAg4c8ug8EhcXkTSsgVE6BS87O7vYp4gFfsyYMZqmF2A3O0JIro1vkrfn22+/LU8++aSkp6cHeEt6mNsQoF9RVQmd9K5AWFhYKCtWrDA+RVYWxtqOXUFb4Ag4RkgCBnDyTpo0SWbOnGniEUO9/HPgj61H2kUAx/9bb71lVkS77bbbig/Dyop6SkFky6eo/mi7qFa8n2OEtOaTGHamTJliLK8agB58B0XyDBhx0Hgw1Dz44INmPghB8Sni+kB1feihh6Rly5aRvM2YurajhAQZkkhHjx4tM2bMMAYff8KqYgrZKH8Y8hJRUwkcZwqCy2LdunUmEOTChQvGB00anvoUne1oxwnJ7eF3IuN73rx5ZllorXzubKeF42wfffSRKSbFkm1Vq1aV1157zVQjp5ocLg9cH9qcRyAkhOQ2sbqeO3fOqK7ML0nX0hYdCJCETvU4svUJBMClhZHu/vvvN9XstYUOgZARklvG8mo5kyFlly5dQvckemZHECD5nKBxonBIjYKQqKYUyiYfVltoEQgpIbn14cOHC+ZxLK98YjrX5k4EWLqNcDfmhcwdSUAnXrVx48buvOEYvKuQExLMcCbTye+++66RmDqndN9Igogff/yxuTGIiERUn2L4+ykshLQkJQv3MKck03zw4MHhf1q9YhkESKUiYwcyYkmlRAvz/1q1ailaEUAgbITk2Yj4x2JH5xPvSAysms0j0Ov/vCR1bKZNm2aibXDuo8lonmLk+oMrh5WQXBBViLfve++9ZwobEfeoWeP+D4JN+3Plq22nPDWNRFKrJMnQznXkiloptk+EFZUInLNnz5qEYfollKtV2b6xON8x7IQE72uvvdaQkgFBqB2kjMTS19Hc97NWH5EdWefllk515MCJPPnljF0y8YE2Uiut8i49evSoSQLAgEMg+NNPP63Yu2ggRISQPD+q0eOPP24k5csvv2ziJFGbtNlDAMk4pk99ub3rDw76e17YLIdPX6yQkBjTqDzPdAFXBtOHu+66S9LS0uxdUPcKCwIRIyRPRwIrkSBkoH/wwQemLMSwYcO08oCNrq/mUVNf/uyAfL3njOw9el4ya6dK2wblk4vK4lhRCX1DM8GVYacWjo3b0F0cRiCihDSTWE/AANIRlRVV6sUXXzRVCGJhNVyH+6rE6QovFUm7RjVkUPsM6d6iUGavOSordpyW3m1qF+9H+CKpUUhGjGjUT2WpuHhYtCaU2Ify3BEnpPVwxL+yyhHVzKZOnWrWpKRMoKb0lN/9FwouSe/WtWVghwyzw5ItJ2Xt3hxDSFwZX3/9tSlijNZBPDEvOSKnsHJrcy8CriGkpcKyVjxmeOIpCWamUhlk1VYSgbSUJHl9aZbsOnxWsnPy5eS5S/LvfRqamqdkZJDhT6U31tuAiO3bt1cIowABVxHSwou6PG3atDEDi3w85j5ISySoth8QuK9fA+nUNF0KL4mk16gqV9UT2bTmM1n42RKzijGL1pAihTsDCaktOhBwJSGBjqiRsWPHGhcJ0vKll14yoVwUT9IFY0UaZqSajbZn51aZ/up82bx1m7GaEgxOJJS+wKKDhN536VpCWjeJysVyBSzwQxYCqhgB6kjReKrzuXn9atm6fZc0qF9X+g+8wQNPohw7dsyU0Vi2bJkJ3Cc1atCgQSZnUVt0IuB6QgIrtXl46yMtKR3BRkVsFvphoZaGDRtGJ/q27rpI/jphojwzY73kpLWQ1POr5P5ei2XEzf1kwaLFhpRU/YOI4KF1jGyB6tqdooKQFnrMhVDFkJBYEQn/YmO+iTqLJI218K/lX34hv56+UVK6PCr1UpOkMKGKvLp+tqxb8T9y18jbTd0i8kwJ3NcW/QhEFSEtuMleRyKgtlJmAnV2+vTpZm6JFMXfFivzzFXrtsrF2tdI7ZQEKbp4VhITkyS1cXfJKDgiv/zlLyUxWYkY/TT88QmikpDW7ZPBTrgd0hF/G9ZY1iwk6x3nN6Z+pCYRQdHWUEX37/tONm1cKwlF7USSqojk50lCUopcyDkljZvWVjJGW6fauN+oJqT1fKRwQTw2MkgoaW8tlc36I2S8E0gNSams7ka1lkpuWVlZwhqKvFxwXeTm5njScQok4+giOV6zsaTVaSTnTx2RtIOfyvhfPGCje3WXaEMgJgjpDTrL5FGRgI1BzepLbFhoWUKbWE4IiksAYxCLjYa7VgwlFikARuYFjvx9+/aZqu+kQlE6k3Q0om3y8wukz4BBMnpMPXnnk5Wybu8KaVgrSZ7+33ul/3Wa4B1tZLNzvzFHSO+HplQhGxZaFg2lyjYSCAJA0oKCAkNG5pts1v6QFgMSYWbBGEuQelRHgGhIbl4QqKInTpwwGwWIOT/XxVpMPC+/L1++3JPnmGCqvhFCSIHiG24eKue/z5Zq6bU9jxjT3WZn3MbsPnHTsxiCcA+wIaEgyJEjR0wqEhtk/fbbb00RYBpEgZRsEAJyQl7cCqi8qMmJiYlGkkFs/ICEqkFAiMYGEZGE/M6+nIf7QCozv0V9btCggbkGgeBWyBuL35IeRfl+71YtvU7MDkR9sB8QiBtCenc40gfVkM175WdIgSSlXD6E5e/c3FxDKr4j3SCg9wbRrA2SQjqIC+lQPVGhkbgEyUPG0sHynAt1mvL8RNngxsBSrC0+EYhLQlbU1RCnsuJOkAdJiITlO5+QG0LyaUlNu0Np165dJhcUCc2cF1eOxp3aRS8291NC+tGvliT045Byd0XakhpFUAMlFx977DEtoxEsqDFyvBIyjB2JRCXCiLki30nMxmgD0bUpAnE7h4xE12M0oozG7t27TagbC9Ywh9WmCHgjoBIyxOMBgxBGG8po4N4gAdvbkBTiy+vpowwBJWQIO2zz5s0mPQrLrVURPBi/ZghvVU/tEgSUkCHoCAIAmCdu2rTJSMNx48ZJ/fr1Q3AlPWWsIaCEdLBHqX1KlA1FiPFHUgCaHEVtioBdBJSQdpHysR8+RdRTAsRJC8OnqEskOARuHJ1GCRlkZ1u1T8nJtNZT1LqnQYIax4crIYPofBz7FOAilpWFaZGM6lMMAlA9ND5jWYPtd1Km8CmiphJ3euONN8ZVwa1g8dPjK0ZAJaQfo4PsDfyJVCWgCgHWUy1A7AeAuqtPBJSQPiH6cQfWycCKSqEtFqvBkqpNEXASASWkH2j26NHD1O8hh1GbIhAKBJSQfqCqzn0/wNJdA0JACRkQbHqQIhAaBJSQocFVz6oIBISAEjIg2PQgRSA0CPw/mGkqYZS3/R8AAAAASUVORK5CYII=)
We have to prove that OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Proof:
In circle (O, r), AP is a tangent at A and BP is the tangent at B.
⇒ ∠OAP = ∠OBP = 90°
Considering ΔOAP and ΔOBP,
⇒ OA = OB (radius)
⇒ OP = OP (common segment)
∴ By RHS theorem, ΔOAP = ΔOBP i.e. OAP and OBP is a congruence.
∴ ∠APO = ∠BOP and ∠AOP = ∠BOP
Here, O is in the interior part of ∠APB and P is in the interior part of ∠AOB.
∴ OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Hence proved.
Question 3.A, B are the points on ⨀(O, r) such that tangents at A and B to the circle intersect in P. Show that the circle with
as a diameter passes through A and B.
Answer:Given that A and B are points on circle (O, r) such that tangents at A and B to the circle intersect in P.
![](data:image/png;base64,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)
We have to prove that the circle with OP as a diameter passes through A and B.
Proof:
We know that a tangent drawn to a circle is perpendicular to the radius drawn from the point of contact.
⇒ OA ⊥ AP and OB ⊥ PB
⇒ ∠OAP = 90° and ∠OBP = 90°
∴ ∠OAP + ∠OBP = 180° … (1)
For OAPB,
∠OAP + ∠APB + ∠AOB + ∠OBP = 360°
⇒ (∠APB + ∠AOB) + (∠OAP + ∠OBP) = 360°
From (1),
⇒ ∠APB + ∠AOB + 180° = 360°
∴ ∠APB + ∠AOB = 180° … (2)
From (1) and (2),
OAPB is cyclic.
Here, OP makes a right angle at A.
Then OP is a diameter.
∴ The circle with OP as a diameter passes through A and B.
Hence proved.
Question 4.⨀(O, r1) and ⨀(O, r2) are such that r1> r2. Chord AB of ⨀(O, r1) touches ⨀(O, r2). Find AB in terms of r1 and r2.
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP0AAAD5CAYAAADspDPqAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAADvUSURBVHhe7Z0HmFVVtucXOQdBgkoQBSSriGQVUYKIWUQRUUBFFH1jv55+X09Pv+k3Pf1NO/2+fiq2gihKEkXFgAFRJCdBJCMZFZRQ5FBQQDH3t+hTFlU333PvPefsvb/vWCV17zn7rL3/e+W1Sp8NDbHDUsBSwBgKlDbmTQP6oqdOnRLnOnnypJw4cUJyc3P14v/PnDkjp0+f1p/O7/n5+VKqVCm9SpcuXfA7/1++fHmpUKGCXvxetmxZvcqUKaOftcP/FLCr6IM1zMvLk6NHj8q+ffvkwIEDcvDgQdm/f7/+zr8DdD4DqB3BrUSJEuJcJUuWPO93/h/gc/F55ye/O//vfBegA3oOgCpVqsgFF1wgNWrU0J/O75UrV7YHgg/2kTNFC3oPLRaAO3z4sAJ67969smvXLv3JBbjh2IAREFaqVEmqVq0q9erVk2rVqknFihULLgDqcGm4NyB3Luf/OSAc4PPTkQQ4PDhEuI4fP67XsWPH5NChQ3pt3bpV1qxZo9IF80UCYB4XXnih1K5dW+rUqSO1atXSg4FDwg7vUcCCPotrAojh3j///LP8+OOPeu3Zs0cBBUgBNmBq1qyZgqlmzZoKJP4dkGdD3GZuzkHAAZWTkyO7d+/WeS9fvlz/xmFQrlw5nXODBg2kYcOGctFFF6lkwKFjR3YpYEGfYfojkv/www+yadMm2bZtm4rocFpE5Isvvliuv/56/QnYq1evruDx0oCzI1lwMc/CA+mA9+Eg2LFjhx5iy5Ytk7lz5+oBxTtddtll0qRJE6lfv75KCHZkngIW9GmmOdwcAAByLrg63BKuBwfs0qWLAgCRGOOZnwcqBRydq3Xr1voqqAZIAY4ks3btWlm4cKG+K6oJB8Dll18ul1xyiUo3dqSfAhb0aaAx+jGbnA2+fv161cnRw9nk3bt3l8aNGysw4JpBH6gijRo10ouBR2Hnzp2yefNm2bhxo8yYMUPtCXXr1pWWLVtK8+bN7QGQ5k1hQe8SgdFjt2/fLuvWrdMLoKN3w8l69OihXB3ubvpAXUHE5+rZs6eqAqg5HJDz5s2Tr776Sg/EVq1a6QHAQWmHuxSwoE+RnrjPsGZ/++23KsYjtgL0m2++WTm6tWBHJzB6Pte1116rrki4P/ScPXu2fPnll3LppZfKNddco1IAdg87UqeABX0SNMTwtmXLFjVSwdURWeFc9913n1xxxRXWQJUETfkKhsv27dvrhdsS2n733Xfy3nvvyfTp09VO0LZtWz0I7EieAhb0CdAOd9SqVatk0aJFqpfii+7YsaNcddVVqofa4R4FoG3Xrl3V0Im3A/CvXr1aaQ/oO3furNwfW4kdiVHAgj4OeuGGWrp0qXzzzTcqgjZt2lQeeugh9Z97zaUWx+v46iMEIwFyLmwAcP/FixfLpEmT1OPRqVMnufrqq60alcCqWtBHIRZBJ/Pnz1cug4W5TZs2ymEwytmReQrgCUD3b9eunbo/FyxYIJ988ol8/fXX+u8cAEgIdkSngAV9GPrgV8aSDHeHkyPCo2cSZmpH9ikA90fa4sJ4umTJEj0AEP0BPioB9gE7wlPAgr4QXXAfOWAnXLRbt266iYg+s8ObFMClx0UkI8BHMuMQcMBvo/6Kr5sFfYgmR44ckTlz5iingIvAKSy38CbII80K/f7OO+9U9YuDm/VE97/uuuv0sraXXylnNOjR0+EKBIQQLsqGYYNYvdBfgC88W1Swe+65Rw9tgI+rjxgKjIAY/OwQMRb0GzZs0A1BFN2VV16pm6JoAondIP6lAGG9/fv3V1sM6zx+/Hj1vvTu3dt4Q6xxoCfo4/PPP9fTn0SXxx9/XMM97QgmBYj5f+KJJ9QDQ5z/yJEjVQogNNrUCD+jQI/ODuCJqLvrrrvU2JONnPRgwsu7b4Wdhki+Fi1aaJrvzJkz1d/ft29flfJMG0aA/pdffpFp06bpQqPX3XrrrRrvbYdZFCD1FzWOSD72w5tvvqlx/X369DHKjhNo0JP5RuLGF198odVmHn74YWvMMQvnYd+WkOlhw4aptwbJD/sOXL9Dhw5GUCewoMfnPnXqVOXuWOR79eplrA5nxE5O8CUR+fHWkCD16aefyuTJkzXH//bbbw98SG8gQU+ttg8++EDrsQ0ZMkTDZ+2wFAhHAeoODho0SKP7PvroI83tx+UXZONuoEBPjTZ0NaKyMNBgrLOFKyzY46EAodbkVLz//vsyZswYrYeAdBjEQp6BAT21595++23BaMdJTVimHZYCiVCAij3o+gRr4d6j5Bm+/qAxjkCAfuXKlTJlyhQ11j355JMF9dgSWXD7WUsBKEDdwltuuUW5Pno+fv0HHnhAqyEFZfga9PjbscxzUVWlX79+tmpNUHZmlt8Dn/4zzzyjwB89erTccccdahAOwvAt6ImVf+edd7SSDboXly2hHIQt6Z13IIkHcf/jjz/Wkl2ojnfffbfvA7p8CXoqzU6YMEErzj7yyCNarsoOS4F0UIDsPCRI8jJwAVNFacCAAb526/kO9CTIjBs3TlNgOYVtkcR0bHV7z6IUIF4fg97EiRNl1KhR6uajbZcfh69AjyhPbTQyqCA6PlY7LAUyRQH0/OHDhyvTefnll7VOImXO/TZ8A3paIb377rvaBOH+++9XS70dlgKZpgCZmU899ZSm6r766qsycOBA3wV/+QL0s2bNkg8//FBLImNICWLARKY3r31e8hRAzCcl+6233tKkHXR8inX6ZXge9LjjiI2mBxzlkOywFPACBehkhIpJQBh6fl5ensby+2F4GvSE1NLaiHRIsqDssBTwEgUI5IHLU5MB9zHAp5iq14dnQY97BLGe3HdKHNlhKeBFCqBqYmPiJ3uWNuRU5fHy8CToyXYC8KQ5ep2AXl5cO7fMUAD3MX0M4fhIp/y88cYbM/PwJJ7iOdBT1ICEBzi8BXwSK2q/kjUKYGQ+ffq0pnUj+mN49uLwFOgBO6BHh7civRe3i51TLArce++9KuITtktzTarxem14BvTUKCfG+YYbbpDbbrvNa3Sy87EUiIsC5H+QjgvwSdaB43ut3r4nQE85aowguDwQkeywFPAzBdDpsepjzceXTyAZlXm8MrIOeuqS4et0UmMxithhKeB3CiDaE61HWi7JYdR5oEiHF0ZWQb9r1y4NZ4QYFCqwkXZe2BJ2Dm5RoGLFihqfT5w+8frE7XuhGWrWQH/48GENYUQUIrLJxtK7tdXsfbxEAforkP79yiuvaOTeo48+mvVmmlkBPW4NdJ2DBw/q6WcbT3hpm9q5uE2BBg0ayIMPPihjx47Vwpvo+9kcWQE9sfTff/+9DB482PhmgtlcfPvszFGA7FCqM5MpijqbzeCdjIOezqH0EiP4xsQ+YpnbZvZJXqMANfao2kzUHpV4aLSRjZFR0FNSmKAFylvZaLtsLLd9ZrYpQKYotfZQb59++umsqLYZA/3Ro0e16g25yNQcs0Uss7397POzQQFq7uGporQ2rmry8nHvZXJkDPTEI1NUcMSIEb4uKpjJxbHPCiYFqK1HZt5rr70m06dP18SyTI6MgJ7uoEuXLtVMJCyZdlgKmE4BDHu0ziLf5PLLL9f22ZkaaQc9ATikyhJ/7NWso0wR2z7HUqAwBUgs27Jli7rx6tWrl7HAnbSCHn88Lory5curu8IOSwFLgV8pgC5PVh76PYyRILVMjLSCnlJXnGREIVWvXj0T72OfYSngKwpccsklqtNj1KNfXqdOndI+/7SB/ocfflB9hVRZ9Bc7LAUsBcJTgOzSjRs3yieffKLZeOnu55AW0CPWI64weVsMw251S4HYFKCGxN///ncN3CFWP50jLaCfN2+ebN26VR577DGhVLAdlgKWAtEpAIMkSpWqum3atJG2bdumjWSug3737t3aOrpDhw4ZdUOkjUL2xpYCGaIA+vzKlSuV29Muq2rVqml5squgP3v2rJa8IuqoT58+aZmwvamlQFApQAGZO+64Q55//nkN2iGuJR3DVdBzSq1Zs0bTCL1QLCAdBLP3tBRIJwVIxCFohwKxtMq67LLLXH+ca6A/ceKEnk5YH/3U18t1imb5hkhbJ0+elNzcXDl27FjBT9YnPz+/4CL3gYtqRdhdqPLCT4qZ8BNpzY7sUOD6668X6kYCfMpsuV1CzjXQz58/X/bs2WOTaTK4T86cOSP79u1TuhP5iD0lJydHSG4C9HhRGA7A+Z0NxMXhwMXgPs7vVG8F9FWqVNEMMNqCc9WuXVtq1Kjh+gbMILl88yiC2fB6UVlq+fLlcs0117g6d1dAv3//fu1IA4cnjtiO9FEAUOMZ4dqxY4dAe0ALZyYACqA2atRIf8cQBAfn4u8O+B3QO5wfKeD48eN6HTp0SCsakRzF/detW6dVXSlrxr1p1cwaX3rppZoxaUd6KED6ebNmzVR6btGihateMFdATxAOG4hYYjvcp8DOnTu10hAAJBcbkAM4AIiXhKguXD6VK1d2tbgokgJSw969e4U5UA9hw4YNQiEUJAKez4ZkcyIJ2OEeBTiYb7nlFg3RnTt3rvTq1cu1m6cMejYDm4CiGLbWnWvrohwco+iKFSsUcOjecFcOVjg5JZfSnYcNd0di4CJElIFUQPUXJA0Oos8++0wv5gZ34hBIl6vJPer6407Q9NprrxXiXjjc3QplTxn0X3/9tXKYLl26+IOSHp/lpk2bZMmSJbJ+/XrVyQE4bhwMpF44VNE3sShzYWXGlgD4V69erVWRMAQSdk07p4YNG3qc2t6fXvfu3eW7775T4LvV+Skl0CPuwYnwyQN8O5KjAOI6XH3BggWaoIToTqAGnJOUSy8Px9BHjsX27dt1g+K6RfpD7CeuvHnz5l5+BU/PDbWJA5SaFDBWjKmpjpRAj/EOf3zHjh1TnYex31+1apXMnj1btm3bpgC/5557FOwY3/w00EGRSrhuuukmtTovXrxYXn31VQU/1V+91NrJT7TFhUcRGrg9Ul+qI2nQc6qzYRE5bKOKxJcBkM+YMUNFY8BOCySqA6NH+33ACAA50sqyZct0s44aNUoPMw4EDI92xE+BWrVqqU7PIQq3T1XNS3qH0WUWwwKTsSN+CuASo84A4hqiGrXSSK7AGh60gf5PtSRcudgpsP+sXbtW0605FGwyVvwrDrdHZUIFTJXbJwV63EbooAQQ2IWLf+Hgeli6iZTDCMZCmiAlAX6ATvYYKiF9D9g/ffv2VWu/HbEpgEuWIB32EAdmKh6SpEAPlyLYw4bbxl4sPgF3J3MKvQyjFpvd6wa6+N4ssU9hoKQVOeCHHmPGjNGDD380B4Md0SmAURSJiRDdVDrkJAx6orV4KIC3STWxtykBNVOnTtUgF+qhIe66HUsdexbe+gRpo5RCR9zHroHHAtrgl7YjMgVIxkEygulyACSbH5Ew6OFWp06dykgtL79vACIVEefxV1MNxUTuHmkNsWEQZYa/n2qwtHOmeGomasT5eV/BNOiAixGdwJ1kRkKgJxoLY0Lr1q01CcOO8BQg2YVAFQ5IdFk8HOmOnvPrWhDpB9cH+JMnT9YwYwpFBsGLkY41QUrCLQq3R8dPplNUQqDHvURWV7qS+9NBpEzfE/rQh/ynn35SyzximB3RKUBgF+WfEV/paEwIMq2fTDByJro3UA2Ji6F6LsVnOQASHQmBHs6FjzWZByU6MT9+nhj5N954Q/PZ6VFmg1HiX0U2s5O/AccfPXq0qkRuRKDFPwt/fJJuOFjvseQng8W4QU+MNXHhhNxa0av45oCzY43GCj18+HDlWnYkTgE6IZHLP27cOI3mGzp0qBCcYsevFCBak0Auoh7xfCQaAh836ImpBuy2p3zx7Ud0Is0I8WYMGTIk7XXLgw4A9FYkpbFjxyrwqapsU3fPX3X0eQJ1SHRK1PgZF+gRVwE97gJbOOF84sPhATx0gSu5lf4YdGDHej9y9QE+tEWCeuKJJ+xhWohoeIJwcSLio+Mn4gaOC/SI9RiobD+687cqlmaHw9O6y8YtxIJyYn+nZgB0Rb9//fXXZdiwYZbG/yQhIIfb4/WgwhGHZLwjLtAjQhDkb0th/UpWouww2uGKg8NbwMe75RL7nAN8fNPo+XB/G713joZkLxKgQ0izq6AnTpw+W4ROWl/zOWITrzB+/HitKUe1UmthTgzIiX4aURZLPhyfDjAPPfRQUv7pRJ/r9c+jSsKISWKiohLVleIZMTk9IZJHjhyxTSgLURORCh8pXMda6ePZZql/hiAefPcTJkxQ3Z78BTtEA+U4CBHx461UFBP0jmgf7w2DvhBkiZH0MGDAAOuHz/Bio8NSpJN68By26ez3luFXS/pxxIKg7sDt48VoVNCTJIIRD+Ja0V5UzSE7rFu3brZaUNLbNLUvEq+PAXXKlClaHJTL5EGQDi5OR8SPJ4YmKuip7gLwiQAyfUAHNhqnqRUts7cbsFqTkffCCy/oemBTCWIBkkQojIhPFCNVihs0aBDzq1FBD5fH/2yzw0Q++ugjPQCx1FupJ+a+SusHiNgj/4MSXKTm0uLZ5EEoLnty8+bNqYGe8ssY8QgASDZvNygLQWAS2YX9+/c3Xpz0ypqiy1J9iPRlCpOko9GjV9411jxgzNg4YNKUzI41InJ6py8a+qvJA88F7bdRcRINdzSZbpl4d0BPkZIPP/xQnn76aaPFfLwbFCAlfiRWzEhE0MPlydVNJosnEwueqWcgPuKPJ8c7kVDHTM3P5Ocg0lIkksAdWj9RadfUgTEPqQdXMjE10UZE0GOpRmTAJ2rqoJnHwoULtcKLLRrizV0AhyP2nNJblNg2db+S8g6HdwLpEgY9nI3ccOrgmcrdaN2MP5hNRPFGO7xLAcR8ykfB6bC7mDiwu+FZgtPTMSladF5YTo8+D/DjMf8HlcDoilQKogmFjfX29ioTBo3tibbOVCpKJA7d22+W2OwAPfuWNuPRGmKEBT0hfZwUpgY+0HYbcZFDD5HRDu9TALCjihExSektEweudbg8/vqEQY+IwJdMzZ0nuonApIcffjjuJAYTN5mX3plqMhQhJWISW4yJUmqdOnW0ByLvH82YV4zTU94aTo/VPt6sHS8tfqpzQZfH9YGISMtlO/xDAbq70m5t/vz5mhth2gDwSOcUdknIkEexjMOHD8cdvB80wm7dulWDkjAImXjo+Xk9abFGb0VUM4x7JpbYQsKhPDbRo5Fq5xXj9Bjx0AtMTRl1GkvG8nX6GRxBnjvcHp89EZQm5kig19N3IScnJzHQY602UZ+n3jrWT1x01mLvz6OBfUsCCqHTBOuY1mAVFzOZdnv27InYJiwsp4dwJjYaWLlypUo5lGG2w78UIO+eHg24XE1bS7CLbo/EHmmcB3pcVRQpwAqYTLsc/24TEd59xYoVQoQX72+HfylA8g3qKXXhTQM9EipxC3GDnuQSutKaWNsejwX+zX79+vl3t9uZKwUwwCLiY8knUMU0VRWmRcYdnrhwtQbO4/TotHl5eUZyOnzzhDJeccUVFjoBoAA9GgjLRcQ3LTuSPBGkHLxw4XIRzgM9pyLDtOquiPZsDsTCWGmJAcCDEa+AeI/Pev369caBnsA66mEgtccEPR8iXTHR3lh+30VYOtGBcPfYEQwKYJOi0Aauu2g+62C87flvQWUhVBxy68ONYpweq71pbg5CbonEM7n6ShA3Pznms2fP1rBUxH1TBkwbXd6R3Iu+93mgh9NTXdO0QoNE4dEZ1cQIriADgUAVAMD6mgR6XHZcMUEPp0McMC2zDt0Hyz21AG3YbbCOAKRWjFqxYtGD9daiwTkwb5h4VPGe/Hl0H9O6rhKuCHHibRQQtA3C+5zJPyuT5v8iq348KhdUKiP3daojTepWDMSrsq6LFy82Tq8Hx/QHCFdQo0C8J14Xd51p1msMeFjvKTdk4jhxKl+efH29VCxbSm5uU1N27jshj726Tv5yf2Pp0rS670mCiA9DI+jMJAM1OEatAdNFbXQFoKcHPZvfNCMeoOedTXNTOmievGCXnDp9Vl56olkBwBtcWF6e+2i7TP3XK6V0yRK+Bj7uKycW3aQir+xpAB8V9HRiRa83DfS7du1SwGP4MHEs2XxIOXzhcXPrGvLSjJ/kp5wT0qh2BV+TBTE3Viy6r18wwuR5Z+xVgD6i9d4BvUnZZRxy1A/Aam9yAdBw3LyElJCzAUCDkzGKeG/SgHmjzyPBRwQ9Oj0b3yTQc9CRb2By6G37xtXky1X75MGudQv2xpz1B1Wsr1+zfCBwArcH9KivpiSSOZIruI4Ieowd6D4mtbA6duyYnoSmeSwKb4IBXS6SuesPyLPjNkgPDHkHTsqkebvkT/0ukzKl/K3PO+9Jwg1GLQ55U9Q4cAwTB9dRxXtAb1JgDi5KTn98mqaOCmVLymvDWsiEub/IjBDHr1G5jLw8tJm0qFcpMCThUAfwcD1TQE84PVJNVPEe+Z8PmSL+sKNNNV4WRXOZUiVlyI3BdVk6+i3rbcpwsAyuI3J6E0Hv2DFM8ljknzouB/flSI06F4mUKGMEBlhfjLYmgZ7oUsT7qKDHvG8ap0f04Z1NMV6+/dZ4ef7N6bL7ZHlpWOWk/OGp+6XHLbcFHviOfhtO1A3qy7OvY4LeCdczSbznoIMwJrzzlBDgH/jjVCnb4QkpX6ma/HRgt9z6zMvy1eiycn33XkHd+/pe0bheUF/cAT17PKp47xAnqIQo+l6mSDf5p07ICxO/kNLthqqn4uzpk1Khbj3ZfeJBeXHseyHQ9wyRJhiW+nB711TQA/yooMeKDdczKUjFlHc+eyZPDp2uEOLwIcCfOReskX8qV8pVqyW7DsMJMPZE7Fruex7g7GnW25ThcPpw71yw0pyGfACDhynDeeegb4afd+dIiUNbJbfSz1KpfhPJP3lcSpavIic3fictmuOuDC7g9YD7J9hNUOMc7DpYDpcufh7o0etNAj1xCUE+6EgZpsUTRRJvbn+F7J47Xvbk3iFV6jSSwxuXyqWHZ8jvnvlb4M94Z41Zb1MG78wV7p2Lcfqgc73Ciw5BOOTC6T1+3hy8DznkX375pbqp6NjTq1cvGbJ6ufz5uZGyfPVk6de2qfzh3/4ujZo09/OrxjV36ME6m1QkJS5O73A9k0CPq473DZL/lqq+06dPF9qN07+gZ8+eBX0JW1/dXv72f/8kr45+Rf77v/1eql9wfnZdXAjy4YeIxwD0prhmWSJHao8p3jsigQ/XNakpE7SBkScIoKei74wZM1SUpyDI0KFDw7baPno8lEJdsqwcPnLMGNCzvibFYzh2DA46K94XORqck5/EG78OuNi8efNk1qxZusC33367dO7cWUuZhxtsBJPsNtCA9YU2JnH6uMR7uB5tcMIl3fsVELHmTfkkNgOdQPw46Mz6xRdfaNoofdnpyW5qBaBo64dBk/1tSrINtCD6EBE/3EFXYMiDIJGS7v0IiHjmDOghSqRSwfHcIxufQV8H7LTVpqED/fcuv/zybEzFF88E9DSAMClt3EkmC3fQFYDeSToJgn4b705EBGYz+AX0lCifOXOmLFq0SAuYDhgwQK699lqjAqriXVvnc6gyJjaxdPLoo3J654/hKm0kSmg/fZ4mF3SrDVcq2Cvv4bjgaMjI+nTr1k0v6rrbEZ0CbH44vWnVkaIVxSng9IAeS7ZpoKcZAmIyBTW8WP67qAsOf7tpDUlSOdjg8kivtG82aYDjSJWwCkCPvoNPL1x5nSATi82A0YOmF14CPaW5Ca7BBUcH1kguuCCvjRvvhisTSzYSnUkDHIPpcF6c8wx5fChSK5ygEgxOD2FobeUFYxhcac6cOXpxCMdywQV1Xdx6L1paYbeh/r1JAxyj/kUFPeJ9tP5XQSUYRRPp4U1n02wP64JzfwVYV9Qhk3z0UNFpRhsua/a8DATEWyzEWDxNSbHlPevXry+bNm3SGIVIQS3ub8df78jGJHQW20KTJk3kvvvus22zXSA4Gx/x/oYbbnDhbv65BfuY0u7s63DjPNBTYGHnzp26+U3yaSLWL1myRK34dK/N1OCAJZJuwYIFak944IEH1AVnUgpoOmnNYYq95rLLLkvnYzx3byIQURORYmOCng9h9eMyCfQAnffdsmVLRkCPexBfe2EXHNzIpAaLmUDKxo0b9TA1rTkpniiiayP1cyjG6fkwooFJDSA47NgYuMe6d++eVtVmw4YNKspv375d2rRpoymvWOftcJcCSKubN29WLm8SA4OKhJXjsYgL9MRto+PivoqkD7i7NN65W8uWLeXTTz9VHTAdPl3uiwvu22+/tS64DCw7ocrs4z59+mTgad56BLkY2KYiuaDP4/RwPKyc+IhNG82bN5fPP/9c1q9f7yroi2bB3XbbbdKlS5esGAxNWtO1a9eqq65x48Ymvba+K52Y4fK8f0ydHr8ewOdLpg2CNxo2bCgrV67USjNuGNOsCy47uwgj1po1a6RZs2ZGZdZBbTxvMG3iEiJVCipWNAzRFqunl2PR07WVrrnmGnnnnXe06kyjRo2Sfgz0IwsOboMLzmbBJU3KpL6IbYbw26uvvjqp7/v5S1jucVVy4EUaYUG/evVq44x5EAi9HpFo2bJlSYEeFxyFKBcuXKiBTmTBtWvXzhWpIdmNiOQC1yNABSOlGxJMsnPJ1PeWLl2qdhPTXHXQl8MOlTKaXaoY6GvXrq3m/n379hllwYdg5B5fddVVwqbp0aNH3O+PVOQUooTguN/Igsu2Cw7/P6m4FNdYtWqVBl7hMQhy4BVxJrjq+vbtG7ZUVKYOnmw9ByMeIn60XIOwnJ7sHK/EomeaeFSgASwO8GM933HBbdu2raAQZTb9wkgav/zyix5gGGVRLTBSoma8/vrrOkfakQe1ZBZSFrUh2rZtG2vpAvl3J9cgWgWlYqB3YtHRa00cJOC0bt1aOXfXrl11A4Ub4VxwfC/bg8hCRLvrrrtOGjRoUDAd3IVYsuHyTg+/bM/V7ecjnZKVSI3ASJZrt5/ptfs5uQbROjEXAz06Hz56otMQ8+EKXh6IcxSG7N+/v2tiK9b7FStWaGguYnrhgfg+f/58DZ+FVrjg2GReCQBhsxcF/NSpUzXE+JlnntF1DWqZc/YBg/UwcRCUg3jP+kcbYVt+wCEIIiG4wUsFG5yinYQZ4ockQWX8+PF6qrupp+K6Q/edO3eutG/fvsDtw0FANB2E5d/RlcnQ89KADg6oEeGnTJmiuvxvf/tbnWbQGns4tGevIp116tTJc2uSqf2Bqx13ZWEJL9yzw4Le0Unhol4CPUYpJ6INDstLEsaKi8btgaX7pZde0o1EqaXPPvuswAV37733ejbow6mCCj0+/vhjlYIef/xxteCj5xOpFUQLPrYM/NKxuJzb+8RL90OfR+JERU2Y02P5g3ui1+Ny8srAHYErZvDgwTolOC1GK1yMbg9UHPy877//vkoVAIYsODi8m1KF2/PG++AYcTDIYrjjoOQw4J0w6gGOIBny2KeoYjAAk0uAk8+B9w13ccKg57Qg8wyLNOKgVxr/sVkRvQuDjs3s9sAFxybC9YNxiIPm2Wef9UV0FxKKM2699dZipIHjY6vx8sGVyHpyeCGFoWaZzOVReeH08VRHjtjGk3rq+HYJ6cumC6rwBmCBwxmh3ORahbPg4Oro9t988426MKGJHd6iAOoX6t0jjzwS0dPirRmnZzZY7YnGwzUba0QEPZwezoq11yughwNzFT0I3DBO4YIjv92J5hoyZIi67hiITYj5cHvTyi7F2kDZ/DvqHpmRqGEmhtwWpj2Vn1DJ48mOjQh6AvYxCHAzXFheGDfeeGMxFyJzvOeee5KeHtZOpxec44LDP1+4bNadd94pI0eO1Cy8u+66K+ln2S+6S4GPPvpIbRNE35k8kH5hzqi+0fzzDo0igh6dj2AOdFv0hWyHlDLhcMUmeMlkY6xxwZEYA5dHlL/pppvCVk1F6sFIhO6I+NSqVSuT95gn3p2oSQJxHnzwQeMq3RZdAPYvV8eOHeNam4ig59tscIJQsI6SjBKUUTgLjoNt+PDhMV1wHAhUYnn33Xf18DHZSpztfYB9BS7PQU3YtOkDLo/EE28J96igx8lPWC4usSCAniCV2bNna0Qdbo37779fN048fmvsG/jnX3zxRXn77bflscce83y0YhDBgLFq0qRJui9Ru+wQxSe6PO66eEZU0OO6w6+LFZ+OGX5t9YvO42TBsWmwURBem2h8NkTloHjttddk2rRpcvfdd8dDY/sZlyjAOiJp4UYdMWKEJ1ROl14t6dsQiYgkjvoZrxs2KuiZCRZs9CdECC8klCRKHXzthM4Sc8D8e/bsKfXq1Uv0NgWfR5/HcES0G8ZOrxg5k34hH30RmwrViAYOHBgz1NRHr5XSVCnvxmEIc453xAQ9Ij6bGxHCT6AnPp7MMgpiYOEnig+fuxuDSMD9+/fLBx98oMUHiXqzI70UwMMyY8YMIeCIABQ7zlEAKRyrfSLFXGOCHtcVpwgnLKKx19sj44JDZycWG12dTYILzu0sOFx32AgmTpyovnvTWiFnEnQc3MRJUFAUMdaOcxTAYk8UXqIVf2OCnpvDITlpiXyijpxXR1EXXDqz4Eg5phzW2LFj5c0335RHH300buupV+nnxXlR7uutt97SikapxGN48d1SnRPSN3p8ixYtErpVXKBHfEAP5sT1Iug57fC3E1cerwsuISpF+DBSD+GfGPaoSkM76XjdJm48P+j3APATJkzQTY0/3is5IF6gO1GoJFIRGp5oR964QI+YDNixWFOMwSsdWSga4PSCwxKfjV5wPBewv/HGG/Lqq6+q7SBaJVIvbBg/zAEGA4fHVfzQQw9Z92iRRSNmhLyYREV7bhMX6B0RH25KFFS2QU/8PUkwGHawM5BdlYwLzq3Nj88f8X7cuHHK9Tl8vCgRufW+6b7PnDlz1EhKnTto6fXqTemmR7j7cyjC4ZNJAosb9FipOXXRm9GVs5V44rjgtm7dWtALzgsJQYj6JOkQuINIipGvcJprNjaG356J6+mTTz7RCr64Qgm+idSwwW/v5uZ88RxRNQoaJdNaPW7QM2kKanDC4BvMdFYTLjgnC45qPgDMLRecWwvCQTho0CB143344YdqXcXK77bnwK35euk+qGoE3mCcoioSjMWO8BSA8aLTJ4vBhEBPYgvhfpQZxpoabwRQKovnuODQ3RnpcsGlMsfC38X+AdARvYgPp6QXUXyxShi59Xw/3ofAqcmTJ2uDFQ5NU8tXx7N21IlEtcVulIhvvvC9EwI9ohaVRhFhidBLd3NArLfYEQCOU4gyUUtlPIRMx2ewMwB0aEW8/h133GGTQ4oQmiQRciGItGMDP/XUUylFS6ZjHb12TyQhR4JMdm4JgZ6HIFIT6UZobrpA77jg6AWHdEEWXDwVQZIlQrq+x5yffvppFfXhZNQmQFIhWcT0wUFOKDO6KUE30MWvuR2ZWktsHuCOVO9U8JAw6NFbydt1OLCbYit6nZMFR/7+fffdp9wxniy4TBE+0edQVBNfPotFlReAj5vF1JRQdFEnpBZbx8MPP5y0bproWvj987jpqOJEzEJhTJzlxUL/CcXpxDUSBj13JfYZcKLbu5Fp5rjgkCAo2JFtF1xclEvwQ3AzJCOAj8hPWDMhpal0x01wCln/OAZgmAVZYahrvXv3tlJPAqtCeDmZnk4OzJ5DefL0mxukZOgeJUL/qVC2lPy2b0NpfkmlqHdNCvT4pbHkU1WHZo2pNHyA81GGCmMOGWwAIZUsuARomPGPorfidSDWgc3/j3/8Qzc/rhc3JaaMv1iMB1K0BEMsNhoMwdThTyQrzGvvk435cFCi7mIbctx0+4+dEoD/yqPNpFyZkjJj1X75w9ubZcp/ayOlS0Vm+0mBnpcmiQXQ0wUmmbpxhV1wbmfBZWNREnkm1mlCSxFzoR+HAOBHGkjWIpvI8zP1WWwzvCNSDZGLSIWohjacNvEVIIEMW1DRDMMaVcpIs4vPcfZOTarK3HX7kfTd5/TcEe7OAi5atEgPgGitcQvPgDr1ThYc/+51F1ziyxPfN7CN0A4biQl9nyIfVOLFUEprJow1fhwYm9A9Uf0w0hG0RA0D3skLdRb9SFOkYKz2BCsVLnxZIcTdt+w6Lg+OXKOcfemWQ/I/7mokZaJwed4/KqefuWa//LTvhDxyw8VhaYXu7XD7eDKg/OyCS9dm4fSmKAdcHuDjgyWRAq8FobxIBIlW+EnXXKPdl3LUJDwxdzg8TOGWW25RzmTBntqKwOVxVRc1/uadPit1q5eVf+3bQMX7n3JOyD9m7JAuTatLo9rhuy1HBX3IhSpjvt4pa348Kr3a1JSLLihXbOZsWPz2iKiEnEZyRVHIkOo1hbPgUnE5pEZCb34b2gESDlIirgAPEWoAhkAM7B0cBKm6tapUqSzly5Vx5SDB2wJXZ12xzSDFIaEQjISxKVuh2t5c4eRmtTOEHSQmmGrRyM4z+WelaoXS0rbRuTZWLetVluc+/kG27c1NDvTfhESFGpXKSP/OdeTthbvl2Vt/7XVeePpdu14XWvTVMvGN0VLvknpSqXJF6da9h5StUEWKuuBoJ+13F1xySxf/twA56hKcn/wCDgDqGBD+jAEVaz8HJuDi9E9EP847fkTenzxRZs6aK/Uvqi1DHhsmUjL+VuREg2GLYV6O+4iEJ1Q7bBJUEIrVMTV+Spj7yY1rV8ikKR/Jkdw8OZbzk7S68iqlb9GBSL9u5zE13lUoGxL1d+fKJTXKSYfG1aISL6J4P+3bvdK95QVyY6sa8ujodTKiVz0pUxrnwPmjWrWqcvhgyGo46Rs5W7+LlDh1VDqMnCK/GdxXtv60S3L+2S+bRhVWzIt/IxPiTG4+F2BDr8PlRSQkJz9/R4Qm2YgLAyBluYn7D5eVlrN7p9wz+Dcyd299KV93gMwbtVI+mzNMJr72gpSvWKXYxHgmSUMUoSSQBmmNZqEke+AjxnXERsQKT70FmxgT/9pG++SMzz6W/r8bIwcv7CYlKlaTsz/uln4nV4WiFfOLfQ2A/6V/Y9l35JT+rXWDKtL7ypoq6kcbYUF/Ii9fZq09IBdVLydLNh2Sjb8cl/kbD8mNLYpHki1ZMFv+65OtUqnXX1X8OBtyGC7avlKe/P3/k+f+MEIGhXKhg2SRdmdpE7sLLhrKcXER3AIIceEQqIE7DCMPIa3QH2mACwMahyxX9aqV5T9fHCVzj7aTWp17S4nTuVKlUWd5f9Fb0vB//q9QvPtA2bf/oMZIcMG9kdKIhQf8HDAEGeFKRQKBm7OmzuHCs4m1SKanYKLfSfTzUDqZ7zgrlMx3k/kOB+mRA3vl2b+Ol6OthkudWnXl7Jk8yW/SSd6d+4rc9fZb8sDAR87bOPjl72hXK7HNFPp0WNDPXLtPzf67Qz5AriZ1K8i7i3aHBf206bMkt/Z1UrVcacnPOywlQt+seemVcnRXe9nx41YhN5qNFI4Q0YgTD+EifSbWd5N5bqx7Rttcsb4b798BHxdgY5MANMR7LkqUc1FYga60XPydz506flA+XXVUatz4mMiJA+fW4nSeVGvdU8ZP+50gBeSXLKubB47N/bk4bPiJHYHnwuUPHjyoLkYOHyz1ydCy8C5N5fux6BYv4FOZQzzPiGee0Hr7xjWy5Vg1uaDWxZJ/8mDo1iU08EYuaisLlq0JgT5hfIf9QjHQ54c2xNhZP8szvevL/Z3PNbffd/SU3PTnb+X7n48V+ASdu/H5c/F/v3oH9f9C/3syVKQSwMMtIo1wmXrRsvcK/y3S786zYmUBxvuconMv+r1Yz+H74b7DZkhmDoW/gzgPBz4vLDN0X+7t3D/v+CFZ9MM0yck7qbpfiIWEIrhKSt6J43Jp3VpyeyjgI3R86FyKXswdcDtXYVpEmnuiaxqJPtG2eDJ0C/ecaHsl0fdIdF8U/jwH98W1qsrk7+bJ6RC9SxNidzZfSpQKHcahQ7tm9cruID50l2Kgzw2J9p1DJv/eV9UseEjNymVkeI96knM4pDsU8d71ufk6+et7o+TYpe2lQvnQxEqWln0/rZeudXPl3//jBSlTPnpIoGtvYm8UlQI//rxX/vDWO1L6+qEhySB0IJ86I7nL3pLf/P4uuee+AZZ6nqBAD1m0YpO8vGyaVG/TN3SQl5B9v/woNfd/LQP6/cW1GRYDfaVypUJ+v4bFHjDs5vANIrp26yGv/Msm+f2Y5yWnYvOQvnhMqu+dLX8e838s4F1bptRv9Nvf/Ivs2vNHGT/nP+VIhYukyokf5X8/1FoGDR6S+s3tHVyjwLBH7pf5856SXcu3SF7JitK+2lF57oVn5IqW7vVWSDoMt/BbDhv+pPS86QZZvGRpyJhUVtZ/X042bQsFCYR0SmvVdW0/pHSjsiGJ68Xn/y4j1iyXHTt3qlegYePESienNAH75ZgUyM3NlS++mi1PDXtEeoXwlBeKe2jcNMRIy0QOtIl50zAfcAX03LdR05Z6MXApURmW8FLb9imZZUnfd5q2aitcdniPAhR6xRDbv/8zaY13cA30hUlI6Cjhl2TP4ceNNy7fe8tgZ2QpkBkKEPCEp4vagOkOcEoL6CETiTQbNmzQUsa0dY7Hwp0Z8pr7FLwwc9YdkKMnzvnUCa3u0bpm3MUXzKVcet+c8OWpU6dqevVNN92U3oeF7p420BPMQe7v+PHj9QSjLr0d2aXAks2H5I9TNku/jnVDHmCRactz5JPQ9V+DmkqpkKXYjuxQAImYJjJPPvlkRionpw30kI8sMWK0qRZDski6xZbsLJl/npp3Kl86Nqkuf7r3soJJd/73pRqz3fSiiv55kQDNlGQlqlCRbJWumpNFyZVW0PMwapijr5AxNmLEiIycZAHaE66+SslQ8M2JkH/+9JlzgVTLthJBKaFMrVKuPsfeLD4KEN343nvvaQJVJsR6Z1ZpBz0hnP369ZOXX35ZSx0nU2UnPhLaT8WiQMXypWTp5sPy8MtrJJSVKYePn9aYjLqhHAs7Mk8BqiSjz997770JZUumOtO0g54JIrYgvtCyiJTQZDtzpPqypn//xMkz0qZhFfnbwKYK+gtDpZbKx8jIMp1m6Xp/RHrKiFHZNtMJaRkBPYRDfCEj7J133tG0TC/0n0vXgnr1vlRaKRUK6b44TEEUr845iPOiPgI1/ymYEi5PPt3vnDHQkxBCHfuRI0fKxIkTVb8n/dOOzFHgwqq/FlHM3FPtkwpTICcnR1twUwzl9ttvzwpxMgZ63o5ab4gzlH7GgEGjAzsyR4Hrm18gXHZkhwKkO9PpiDFgwICkOs66MfOMgp4JU2UFwwXcnu6zVEq1w1LABAoQqEYFJOr+p9IrIlVaZRz0TBg9htJL+O/D1fJO9aXs9y0FvEaBmTNnaul3av9T6DSbIyug54Xx31M2GcMe5Z0oBWWHpUAQKUBRUwx31In0QmRq1kCPYY9SyWTjEapLm+KLLw5fXz+IG8G+kxkU2Lhxo/YuvOqqq7JmuCtK6ayBnolQF/2hUOFMAnfefPNNjT0mZt8OS4EgUAAVdsKECdq/DwbnldoSWQU9C4tOTytnB/hDhw51pRFDEDaNfQf/UgDX3NixYwsYW+F2VNl+q6yDHgIQqAPwX3vtNRk3bpwAfC8RKduLZJ/vLwpgq3r99de1IOywYcO0H4GXhidAD0FIOsBvz+mISMTvRdv4eIlwdi6WAuEoQL8A9jA/cc150U7lGdBDQCruoOPD7YlaGjhwYNhuLXa7WQp4kQKUe8c2ResvAE9MiheHp0APgeiH9sADD8ikSZO0lju/0wjADksBL1PAATwde1FPqR/h1eE50EMo6uvRnQUfPqGLcHzbAdWrW8jOi55/b7zxhla/wTaV7eCbWCviSdAz6Y4dO6poj5gPQQcNGmQTdGKtpv17xilAqy90eBp9DhkyxPOAh0CeBT2To9wWwMewhzWUU5ToPTssBbxAgT179ui+pOHno48+qr0E/DA8DXoI2KZNGxk8eLBG7Y0ZM0at+vRlt8NSIJsUoDYEBmcaeWK081P9R8+DnoXFqo9xBOATxIOFn3xkOywFskGBtWvXqqGZNuDsSy+65aLRxReg5wXw4w8fPlxF/dGjR2s+MlKAHZYCmaTAwoUL5f3339eybxiYiSj12/AN6CEsJ+oTTzyhpyzGPYps2rZZftty/p0vhV2nT5+utqb+/fv71qPkK9CzXejHjtGEktpU38Fq2rdvXxvE418seX7mx48f1w4033zzjdZ6JC2cGBK/Dt+BHkITrEPZLaqPcPKSzcTJm81qJH7dAHbe0SmwM9ThlxJX7DFKuVPM0u/Dl6B3iN67d2/t/wXXf+mllzR90Rbj8PuW9M78KVHN3iIwDLUSu1IQhq9BzwJQnICS2hQqoCAHopcXqpMEYXOY+g5EgyJB0jqarstIkX402EVaP9+D3jHwYdmnYwi61/bt27V5ZpAWylQAZvq9EeMpYEnFG/T3Pn36ZLT7TCbeNxCgh1Dk35OcQ2bTtGnT5MUXX9QihK1bt84EHe0zAkCBxYsXay07okCJ/kSKDOIIDOidxencubMCH45PiOQNN9ygLbVswk4Qt68770TuO4wC6zyxH7iCvVb4wp03PXeXwIGel6ISD4aXL7/8Ur766ittl003EWvkc3PrBONeK1asUMAfOXJEwW6CPSiQoGc7IqKhj5HXjMg2atQo6dq1qzbXoNOOHWZTgBp2n3/+uVCemkQZsji9WvTC7ZUKLOgdQpHbzGJ+/fXXMmvWLFm/fr3ceuuttnOu2zvJJ/c7e/asLFq0SK3z1Gq488471fdeunTgoVCwQka8KUY+gN6yZUsV5ShptGrVKuX6tNaywwwK4NUB7Bz86O7sCeI8TBtGgN5ZVJIkqK0/b948oc0Qi4/Ij7HPivzB3fqEaiPpLVmyRMO4iebMRotor1DYKNBDdBoOYKzhpGcjzJ49W4i8wifboUMHzzQk8MoG8fM8Tp48qf3jWGNEedpKccCTEmvyMA70zmLjkqF7brt27TTyasqUKeqycQ4EPydUmLyheXcAvnz5cgX77t271X5z8803W1XunxvDWNAXFvmpfILrBpEffR+LP+AnsKdEiRKmY8g3708VG8A+d+5cIVEGqzzNJqyr9vwlNB70DjmIvsLQB/jnzJmj+fqNGzdWcZB/t+D3Lvbh7CtXrlSwU4KaqkoUqWTdrMRWfN0s6AvRBN8+5bepvY+eD/iJ6mMTderUSf/d1uD3DvgpSMk64YKj/DSGWsJnW7VqZW0zUZbJgj4McQA2Rj24PxyETUVONYY/SnO3bdvWWvuziH3KThNUgzX+4MGDKpFZsMe/IBb0UWhFLz1cOxj7NmzYoOD/9NNP1UAE8DEQ1atXL35q20+mRIFt27bJt99+qyoY+js2Fw5hv5SeTunlXfyyBX0cxEQvJK+aC50RDgOnwd+P6E/NNCr2mu4KioOUCX8ETr5mzRo10EF7+h4ghXEY16lTJ+H72S8ENOEmnQtbv3594erRo4esW7dONyPVVQA8nAf/P2G/2AfsSI4CJ06cELg6HB0a0/IZbk5lJMKqK1WqlNyN7beUApbTJ7kRiOzCuMf1ww8/qEFp9erVqgLQjAOpAOsxTRDsARCbyACdMFm4OqoUHJ5YCuiLbcVvteVjv3H2PmFB7wLt4excxPJv3bpVwQ+XIhqsVq1a0rRpU73Q/224768EB9iI7ICcSjUHDhyQ6tWrKzfHAo81HruKHe5SwILeRXpWrFhRNysX7iQOADgXHgAOAFQAOD8FFhFX0UlN8iPjT9+1a5ds2bJFQb5jxw7Jzc1VoBNA4wDdFjxxcVOGuZUFfZroi96Jjs/FAcAGp5gHF00T8vPztaAn3J+DgJ+oBUHa8Lw3eetwc3q/QQOSX1B3yG4k2YnoR949SO+dpi3l2m0t6F0jZeQbcQDAybjI5967d6/aAeB4/EQSoAIroj+pnlT+QQrgUMB24HWvAO9E5RnEdTq5ws0JliHu/ejRoxrQhH4OwFGBOOSCXI4qA1sqpUdY0KdEvsS/TDgvYOYi+g+wE2xCFVa4IVwRdyBiL9IAHBCAYBvgQhTmIOCA4DCgVkAmVATmSacXuDc15Q4dOqQ6OAcYF4DHGEcWI4ccUgvvh6cDrk5lYhvKnPh+Scc3LOjTQdUE7glIHEA7DTkBPCBCNIZbOpzz+++/V/cVhwFA50AA+FzYEzgACv+Ew3L/ohffBcTchyAXfncu7g+4uZiH8xNODtcmXRXOzj0wsnEAIZUQosxBBtg5mKwBLoFNkOGPWtBnmODxPA7wcsEhnRLejggNp3VEabgth4PDfTkgACUGM0DMd4pePJ9/c7guP53LkRgoHYXeDXCdg8QBMyAvLGl4XfWIh96mfcaC3icrDjCJRuOKVOILMMOpuRzgO5y86E+4P+DmciQB53ckBOfyCXnsNBOggAV9AsTy+kc5GODOVrT2+kpld37/H2rTSz90mfqxAAAAAElFTkSuQmCC)
Given that circle (O, r1) and circle (O, r2) are such that r1 > r2.
∴ The circles are concentric.
Let chord AB of circle (O, r1) touches circle (O, r2) at P.
Thus, AB is tangent to circle (O, r2).
∴ OP ⊥ AB and P ∈ AB
Here, P is the foot of the perpendicular drawn from centre O on the chord AB of circle (O, r2).
∴ P is the midpoint of AB.
⇒ AB = 2AP … (1)
Consider right angle ΔOPA,
By Pythagoras Theorem,
⇒ OA2 = AP2 + OP2
⇒ r12 = AP2 + r22
⇒ AP2 = r12 – r22
∴ AP = ![](data:image/png;base64,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)
From (1),
∴ AB = 2![](data:image/png;base64,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)
Question 5.In example 4, if r1 = 41 and r2 = 9, find AB.
Answer:From example 4,
Given that r1 = 41 and r2 = 9
Length of chord AB = 2![](data:image/png;base64,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)
⇒ AB = 2![](data:image/png;base64,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)
= 2![](data:image/png;base64,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)
= 2![](data:image/png;base64,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)
= 2 (40)
= 80
∴ AB = 80
A and B are the points on ⨀(O, r). is not a diameter of the circle. Prove that the tangents to the circle at A and B are not parallel.
Answer:
Given that A and B are points on (O, r) and AB is a not a diameter of the circle.
We have to prove that the tangents to the circle at A and B are not parallel.
Proof:
Using the method of contradiction,
Let l and m be two parallel tangents to the circle have centre O drawn at the points A and B.
∴ OA ⊥ l and OB ⊥ m
Consider OA and OB perpendicular to l and m respectively and O is a common point.
∴ since l and m are two parallel lines, A – O – B
Hence, AB is a diameter, which contradicts with our assumption.
∴ Our assumption is wrong i.e. l and m are intersecting lines.
∴ Tangents to the circle at A and B are not parallel.
Hence proved.
Question 2.
A, B are the points on ⨀ (O, r) such that tangents at A and B intersect in P. Prove that OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Answer:
Given that A and B are the points on (O, r) such that tangents at A and B intersect in P.
We have to prove that OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Proof:
In circle (O, r), AP is a tangent at A and BP is the tangent at B.
⇒ ∠OAP = ∠OBP = 90°
Considering ΔOAP and ΔOBP,
⇒ OA = OB (radius)
⇒ OP = OP (common segment)
∴ By RHS theorem, ΔOAP = ΔOBP i.e. OAP and OBP is a congruence.
∴ ∠APO = ∠BOP and ∠AOP = ∠BOP
Here, O is in the interior part of ∠APB and P is in the interior part of ∠AOB.
∴ OP is the bisector of ∠AOB and PO is the bisector of ∠APB.
Hence proved.
Question 3.
A, B are the points on ⨀(O, r) such that tangents at A and B to the circle intersect in P. Show that the circle with as a diameter passes through A and B.
Answer:
Given that A and B are points on circle (O, r) such that tangents at A and B to the circle intersect in P.
We have to prove that the circle with OP as a diameter passes through A and B.
Proof:
We know that a tangent drawn to a circle is perpendicular to the radius drawn from the point of contact.
⇒ OA ⊥ AP and OB ⊥ PB
⇒ ∠OAP = 90° and ∠OBP = 90°
∴ ∠OAP + ∠OBP = 180° … (1)
For OAPB,
∠OAP + ∠APB + ∠AOB + ∠OBP = 360°
⇒ (∠APB + ∠AOB) + (∠OAP + ∠OBP) = 360°
From (1),
⇒ ∠APB + ∠AOB + 180° = 360°
∴ ∠APB + ∠AOB = 180° … (2)
From (1) and (2),
OAPB is cyclic.
Here, OP makes a right angle at A.
Then OP is a diameter.
∴ The circle with OP as a diameter passes through A and B.
Hence proved.
Question 4.
⨀(O, r1) and ⨀(O, r2) are such that r1> r2. Chord AB of ⨀(O, r1) touches ⨀(O, r2). Find AB in terms of r1 and r2.
Answer:
Given that circle (O, r1) and circle (O, r2) are such that r1 > r2.
∴ The circles are concentric.
Let chord AB of circle (O, r1) touches circle (O, r2) at P.
Thus, AB is tangent to circle (O, r2).
∴ OP ⊥ AB and P ∈ AB
Here, P is the foot of the perpendicular drawn from centre O on the chord AB of circle (O, r2).
∴ P is the midpoint of AB.
⇒ AB = 2AP … (1)
Consider right angle ΔOPA,
By Pythagoras Theorem,
⇒ OA2 = AP2 + OP2
⇒ r12 = AP2 + r22
⇒ AP2 = r12 – r22
∴ AP =
From (1),
∴ AB = 2
Question 5.
In example 4, if r1 = 41 and r2 = 9, find AB.
Answer:
From example 4,
Given that r1 = 41 and r2 = 9
Length of chord AB = 2
⇒ AB = 2
= 2
= 2
= 2 (40)
= 80
∴ AB = 80
Exercise 11.2
Question 1.P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle touch the circle at X and Y.
(1) Find OP, if r = 12, XP = 5
(2) Find m∠XPO, if m∠XOY = 110
(3) Find r, if OP = 25 and PY = 24
(4) Find m∠XOP, if m∠XPO = 80
Answer:Given that P is the point in exterior of circle (O, r) and the tangents from P to the circle touch the circle at X and Y.
(1) Here OX is the radius of the circle and PX is the tangent.
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)
∴ OX ⊥ PX
Given r = 12 = OX and XP = 5
For right angled ΔPXO,
⇒ OP2 = PX2 + OX2
= 52 + 122
= 25 + 144
= 169
⇒ OP2 = 169
∴ OP = 13
(2) Given ∠XOY = 110°
![](data:image/png;base64,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)
Here ∠XPY and ∠XOY are supplementary angles.
We know that two angles are supplementary when they add upto 180°.
⇒ ∠XPY + ∠XOY = 180°
⇒ ∠XPY + 110° = 180°
⇒ ∠XPY = 70°
Also, OP is a bisector pf ∠XPY
∴ ∠XPO = 1/2 ∠XPY = 1/2 (70°) = 35°
(3) OY is the radius of the circle and PY is a tangent.
∴ OY ⊥ PY
Now, OY = r,
Given that OP = 25 and PY = 24
![](data:image/png;base64,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)
In right angled ΔPYO,
By Pythagoras Theorem,
⇒ OP2 = OY2 + PY2
⇒ 252 = r2 + 242
⇒ r2 = 625 – 576
⇒ r2 = 49
∴ r = 7
(4) Given ∠XPO = 80°
Here, OX is the radius of the circle and PX is a tangent.
![](data:image/png;base64,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)
∴ OX ⊥ PX
Also, ∠XOP and ∠XPO are complementary angles.
We know that two angles are complementary when they add up to 90°.
⇒ ∠XOP + ∠XPO = 90°
⇒ ∠XOP + 80° = 90°
∴ ∠XOP = 10°
Question 2.Two concentric circles having radii 73 and 55 are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.
Answer:Given the radius of larger circle = 73 = OB and radius of smaller circle = 55 = OM
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)
Since AB is a tangent, OM ⊥ AB.
Consider ΔOMB,
Here, ∠OMB is a right angle.
By Pythagoras Theorem,
⇒ OB2 = OM2 + MB2
⇒ MB2 = OB2 – OM2
= 732 – 552
We know that a2 – b2 = (a + b) (a – b)
⇒ MB2 = (73 + 55) (73 – 55)
= (128) (18)
= 2304
∴ MB = 48
Now, length of chord = AB = 2MB = 2 (48) = 96
∴ The length of chord is 96.
Question 3.
is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which touches ⨀(O, 8) at D.
intersects 0(0, 10) in C. Find AC.
Answer:![](data:image/png;base64,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)
Given that AB is a diameter of circle (O, 10).
⇒ OA = OB = 10 = radius
⇒ AB = 20 = diameter
Also given a tangent is drawn from B to circle (O, 8) which touches the circle at D.
⇒ OD = 8 = radius
Since BD is a tangent, OD ⊥ BD.
And since the angle is inscribed in a semi circle, ∠ACB = 90°.
⇒ ∠ODB = ∠ACB = 90°
∴ ∠DBO ≅ ∠CBA
By AA theorem,
Thus, correspondence ODB ↔ ACB is a similarity.
Then
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒ AC =
= 16
∴ AC = 16
Question 4.P is in the exterior of a circle at distance 34 from the centre 0. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.
Answer:Given OP = 34, PQ = 16
OQ is the radius of the circle.
Since PQ is a tangent to the circle, OQ ⊥ PQ.
![](data:image/png;base64,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)
In right angled ΔOQP,
By Pythagoras Theorem,
⇒ OP2 = PQ2 + OQ2
⇒ 342 = 162 + OQ2
⇒ OQ2 = 1156 – 256
⇒ OQ2 = 900
⇒ OQ = 30
∴ Diameter of circle = 2r = 2 × OQ = 2 × 30 = 60
∴ Diameter = 60
Question 5.In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line 1 touches the circle at R and intersects
and
in B and C respectively. If AB = c, BC = a, CA = b, then prove that
(1) AP + AQ = a + b + c
(2) AB + BR = AC + CR = AP = AQ = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Answer:Given that AP and AQ are tangents to the circle.
A line l touches the circle at R and intersects AP and AQ in B and C respectively. And AB = c, BC = a, CA = b.
![](data:image/png;base64,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)
We have to prove that
(1) AP + AQ = a + b + c
(2) AB + BR = AC + CR = AP = AQ = ![](data:image/png;base64,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)
Proof:
By theorem,
AP = AQ, BP = BR and CQ = CR … (1)
(1) AP + AQ = (AB + BP) + (AC + CQ)
= (AB + BR) + (AC + CR) [From (1)]
= AB + AC + (BR + CR)
Since B – R – C,
⇒ AP + AQ = AB + AC + BC
= c + b + a
∴ AP + AQ = a + b + c … (2)
(2) AB + BR = AB + BP [From (1)]
= AP [∵ A – B – P]
= AQ [From (1)]
= AC + CQ [∵ A – C – Q]
= AC + CR [From (1)]
∴ AB + BR = AC + CR = AP = AQ … (3)
From (2),
⇒ AP + AQ = a + b + c
From (1),
⇒ AQ + AQ = a + b + c
⇒ 2AQ = a + b + c
⇒ AQ = ![](data:image/png;base64,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)
From (3),
∴ AB + BR = AC + CR = AP = AQ = ![](data:image/png;base64,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)
Hence proved.
Question 6.Prove that the perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.
Answer:Let l be a tangent to the circle having centre O. l touches the circle at P. Let m be the perpendicular line to l from P.
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)
We have to prove that m passes through O i.e. O ∈ m.
Proof:
If O m, then we can find such M m that O and M are in the same half plane of l.
T ∈ l is a distinct point from P.
∴ ∠MPT = 90° and ∠OPT = 90°
M and O are points of the same half plane so this is impossible.
Thus, our assumption is wrong.
∴ O ∈ m
∴The perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.
Hence proved.
Question 7.Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and B. Prove that
and
bisects
.
Answer:Let PA and PB be tangents to the circle drawn from point P which is in the exterior of circle (O, r). Given A and B are on the circle.
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)
We have to prove that OP ⊥ AB and OP bisects AB.
Proof:
Here PA and PB are tangents drawn to the circle from an exterior point P.
∴ OP intersects AB at C.
Also given that A and B are on the circle.
We know that the tangents drawn to a circle from a point in the exterior of the circle are congruent.
∴ PA = PB
⇒ OP = OP [common]
⇒ OA = OB [radii of circle]
∴ By SSS theorem,
ΔOAP ≅ ΔOBP
Then, ∠AOP = ∠BOP
⇒ ∠AOC = ∠BOC [C ∈ OP]
⇒ OA = OB [radii of circle]
⇒ OC = OC [common]
∴ By SAS theorem,
ΔAOC ≅ ΔBOC
∴ AC = BC and ∠ACO = ∠BCO = 90°
Now, as C ∈ OP,
⇒ OP bisects AB.
Also AC ⊥ OC and BC ⊥ OC
⇒ OP ⊥ AB [∵ A – C – B]
∴ OP ⊥ AB and OP bisects AB.
Hence proved.
Question 8.
and
are the tangents drawn to ⨀ (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively. Prove that m∠TPR = 2m∠OTR.
Answer:Given that PT and PR are tangents drawn to circle (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively.
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)
We have to prove that m∠TPR = 2m∠OTR
Proof:
By theorem,
PT ≅ PR
We know that angles opposite to congruent sides are equal.
∴ ∠PTR = ∠PRT
We know that sum of all angles in a triangle is 180°.
In ΔPTR,
⇒ ∠PTR + ∠PRT + ∠TPR = 180°
⇒ ∠PTR + ∠PTR + ∠TPR = 180°
⇒ 2∠PTR + ∠TPR = 180°
⇒ 2∠PTR = 180° – ∠TPR
⇒ ∠PTR = 90° – 1/2 ∠TPR
∴ 1/2 ∠TPR = 90° – ∠PTR … (1)
Then, OT ⊥ PT,
⇒ ∠OTP = 90°
⇒ ∠OTR + ∠PTR = 90°
⇒ ∠OTR = 90° – ∠PTR … (2)
From (1) and (2),
⇒ 1/2 ∠TPR = ∠OTR
⇒ ∠TPR = 2∠OTR
∴ m∠TPR = 2m∠OTR
Hence proved.
Question 9.
is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the circle intersect in P. Find PA.
Answer:Given AB is a chord of circle (O, 5) such that AB = 8.
Let PR = x.
Since OP is a perpendicular bisector of AB,
AR = BR = 4
![](data:image/png;base64,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)
Consider ΔORA where ∠R = 90°,
By Pythagoras Theorem,
⇒ OA2 = OR2 + AR2
⇒ 52 = OR2 + 42
⇒ OR2 = 25 – 16 = 9
∴ OR = 3
⇒ OP = PR + RO = x + 3
Consider ΔARP where ∠R = 90°,
⇒ PA2 = AR2 + PR2
⇒ PA2 = 42 + x2 = 16 + x2 … (1)
Consider ΔOAP where ∠A = 90°,
⇒ PA2 = OP2 – OA2
= (x + 3)2 – (5)2
= x2 + 9 + 6x – 25
= x2 + 6x – 16 … (2)
From (1) and (2),
⇒ 16 + x2 = x2 + 6x – 16
⇒ 6x = 32
⇒ x = ![](data:image/png;base64,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)
From (1),
⇒ PA2 = 16 + x2
= 16 + (
) 2
= 16 + ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ PA = ![](data:image/png;base64,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)
Question 10.P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to the circle which touch the circle in A and B. Find AB.
Answer:Given P lies in the exterior of circle (O, 5) such that OP = 13.
Let OR = x.
![](data:image/png;base64,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)
Consider ΔOAP where ∠A = 90°,
By Pythagoras Theorem,
⇒ OP2 = OA2 + AP2
⇒ 132 = 52 + AP2
⇒ AP2 = 169 – 25 = 144
∴ AP = 12
Consider ΔORA where ∠R = 90°,
⇒ AR2 = OA2 – OR2
⇒ AR2 = 52 – x2 = 25 – x2 … (1)
Consider ΔARP where ∠R = 90°,
⇒ AR2 = AP2 – PR2
= (12)2 – (13 – x)2
= 144 – (13 + x2 – 26x)
= – x2 + 26x + 131 … (2)
From (1) and (2),
⇒ 25 – x2 = – x2 + 26x + 131
⇒ 26x = 50
⇒ x = ![](data:image/png;base64,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)
From (1),
⇒ AR2 = 25 – x2
= 25 – (
) 2
= 25 – ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
∴ AR = ![](data:image/png;base64,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)
But AB = 2AR,
⇒ AB = 2 (
)
∴ AB =
= 9.23
P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle touch the circle at X and Y.
(1) Find OP, if r = 12, XP = 5
(2) Find m∠XPO, if m∠XOY = 110
(3) Find r, if OP = 25 and PY = 24
(4) Find m∠XOP, if m∠XPO = 80
Answer:
Given that P is the point in exterior of circle (O, r) and the tangents from P to the circle touch the circle at X and Y.
(1) Here OX is the radius of the circle and PX is the tangent.
∴ OX ⊥ PX
Given r = 12 = OX and XP = 5
For right angled ΔPXO,
⇒ OP2 = PX2 + OX2
= 52 + 122
= 25 + 144
= 169
⇒ OP2 = 169
∴ OP = 13
(2) Given ∠XOY = 110°
Here ∠XPY and ∠XOY are supplementary angles.
We know that two angles are supplementary when they add upto 180°.
⇒ ∠XPY + ∠XOY = 180°
⇒ ∠XPY + 110° = 180°
⇒ ∠XPY = 70°
Also, OP is a bisector pf ∠XPY
∴ ∠XPO = 1/2 ∠XPY = 1/2 (70°) = 35°
(3) OY is the radius of the circle and PY is a tangent.
∴ OY ⊥ PY
Now, OY = r,
Given that OP = 25 and PY = 24
In right angled ΔPYO,
By Pythagoras Theorem,
⇒ OP2 = OY2 + PY2
⇒ 252 = r2 + 242
⇒ r2 = 625 – 576
⇒ r2 = 49
∴ r = 7
(4) Given ∠XPO = 80°
Here, OX is the radius of the circle and PX is a tangent.
∴ OX ⊥ PX
Also, ∠XOP and ∠XPO are complementary angles.
We know that two angles are complementary when they add up to 90°.
⇒ ∠XOP + ∠XPO = 90°
⇒ ∠XOP + 80° = 90°
∴ ∠XOP = 10°
Question 2.
Two concentric circles having radii 73 and 55 are given. The chord of the circle with larger radius touches the circle with smaller radius. Find the length of the chord.
Answer:
Given the radius of larger circle = 73 = OB and radius of smaller circle = 55 = OM
Since AB is a tangent, OM ⊥ AB.
Consider ΔOMB,
Here, ∠OMB is a right angle.
By Pythagoras Theorem,
⇒ OB2 = OM2 + MB2
⇒ MB2 = OB2 – OM2
= 732 – 552
We know that a2 – b2 = (a + b) (a – b)
⇒ MB2 = (73 + 55) (73 – 55)
= (128) (18)
= 2304
∴ MB = 48
Now, length of chord = AB = 2MB = 2 (48) = 96
∴ The length of chord is 96.
Question 3.
is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which touches ⨀(O, 8) at D.
intersects 0(0, 10) in C. Find AC.
Answer:
Given that AB is a diameter of circle (O, 10).
⇒ OA = OB = 10 = radius
⇒ AB = 20 = diameter
Also given a tangent is drawn from B to circle (O, 8) which touches the circle at D.
⇒ OD = 8 = radius
Since BD is a tangent, OD ⊥ BD.
And since the angle is inscribed in a semi circle, ∠ACB = 90°.
⇒ ∠ODB = ∠ACB = 90°
∴ ∠DBO ≅ ∠CBA
By AA theorem,
Thus, correspondence ODB ↔ ACB is a similarity.
Then =
⇒ =
⇒ AC = = 16
∴ AC = 16
Question 4.
P is in the exterior of a circle at distance 34 from the centre 0. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.
Answer:
Given OP = 34, PQ = 16
OQ is the radius of the circle.
Since PQ is a tangent to the circle, OQ ⊥ PQ.
In right angled ΔOQP,
By Pythagoras Theorem,
⇒ OP2 = PQ2 + OQ2
⇒ 342 = 162 + OQ2
⇒ OQ2 = 1156 – 256
⇒ OQ2 = 900
⇒ OQ = 30
∴ Diameter of circle = 2r = 2 × OQ = 2 × 30 = 60
∴ Diameter = 60
Question 5.
In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line 1 touches the circle at R and intersects and
in B and C respectively. If AB = c, BC = a, CA = b, then prove that
(1) AP + AQ = a + b + c
(2) AB + BR = AC + CR = AP = AQ =
Answer:
Given that AP and AQ are tangents to the circle.
A line l touches the circle at R and intersects AP and AQ in B and C respectively. And AB = c, BC = a, CA = b.
We have to prove that
(1) AP + AQ = a + b + c
(2) AB + BR = AC + CR = AP = AQ =
Proof:
By theorem,
AP = AQ, BP = BR and CQ = CR … (1)
(1) AP + AQ = (AB + BP) + (AC + CQ)
= (AB + BR) + (AC + CR) [From (1)]
= AB + AC + (BR + CR)
Since B – R – C,
⇒ AP + AQ = AB + AC + BC
= c + b + a
∴ AP + AQ = a + b + c … (2)
(2) AB + BR = AB + BP [From (1)]
= AP [∵ A – B – P]
= AQ [From (1)]
= AC + CQ [∵ A – C – Q]
= AC + CR [From (1)]
∴ AB + BR = AC + CR = AP = AQ … (3)
From (2),
⇒ AP + AQ = a + b + c
From (1),
⇒ AQ + AQ = a + b + c
⇒ 2AQ = a + b + c
⇒ AQ =
From (3),
∴ AB + BR = AC + CR = AP = AQ =
Hence proved.
Question 6.
Prove that the perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.
Answer:
Let l be a tangent to the circle having centre O. l touches the circle at P. Let m be the perpendicular line to l from P.
We have to prove that m passes through O i.e. O ∈ m.
Proof:
If O m, then we can find such M m that O and M are in the same half plane of l.
T ∈ l is a distinct point from P.
∴ ∠MPT = 90° and ∠OPT = 90°
M and O are points of the same half plane so this is impossible.
Thus, our assumption is wrong.
∴ O ∈ m
∴The perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.
Hence proved.
Question 7.
Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and B. Prove that and
bisects
.
Answer:
Let PA and PB be tangents to the circle drawn from point P which is in the exterior of circle (O, r). Given A and B are on the circle.
We have to prove that OP ⊥ AB and OP bisects AB.
Proof:
Here PA and PB are tangents drawn to the circle from an exterior point P.
∴ OP intersects AB at C.
Also given that A and B are on the circle.
We know that the tangents drawn to a circle from a point in the exterior of the circle are congruent.
∴ PA = PB
⇒ OP = OP [common]
⇒ OA = OB [radii of circle]
∴ By SSS theorem,
ΔOAP ≅ ΔOBP
Then, ∠AOP = ∠BOP
⇒ ∠AOC = ∠BOC [C ∈ OP]
⇒ OA = OB [radii of circle]
⇒ OC = OC [common]
∴ By SAS theorem,
ΔAOC ≅ ΔBOC
∴ AC = BC and ∠ACO = ∠BCO = 90°
Now, as C ∈ OP,
⇒ OP bisects AB.
Also AC ⊥ OC and BC ⊥ OC
⇒ OP ⊥ AB [∵ A – C – B]
∴ OP ⊥ AB and OP bisects AB.
Hence proved.
Question 8.
and
are the tangents drawn to ⨀ (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively. Prove that m∠TPR = 2m∠OTR.
Answer:
Given that PT and PR are tangents drawn to circle (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively.
We have to prove that m∠TPR = 2m∠OTR
Proof:
By theorem,
PT ≅ PR
We know that angles opposite to congruent sides are equal.
∴ ∠PTR = ∠PRT
We know that sum of all angles in a triangle is 180°.
In ΔPTR,
⇒ ∠PTR + ∠PRT + ∠TPR = 180°
⇒ ∠PTR + ∠PTR + ∠TPR = 180°
⇒ 2∠PTR + ∠TPR = 180°
⇒ 2∠PTR = 180° – ∠TPR
⇒ ∠PTR = 90° – 1/2 ∠TPR
∴ 1/2 ∠TPR = 90° – ∠PTR … (1)
Then, OT ⊥ PT,
⇒ ∠OTP = 90°
⇒ ∠OTR + ∠PTR = 90°
⇒ ∠OTR = 90° – ∠PTR … (2)
From (1) and (2),
⇒ 1/2 ∠TPR = ∠OTR
⇒ ∠TPR = 2∠OTR
∴ m∠TPR = 2m∠OTR
Hence proved.
Question 9.
is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the circle intersect in P. Find PA.
Answer:
Given AB is a chord of circle (O, 5) such that AB = 8.
Let PR = x.
Since OP is a perpendicular bisector of AB,
AR = BR = 4
Consider ΔORA where ∠R = 90°,
By Pythagoras Theorem,
⇒ OA2 = OR2 + AR2
⇒ 52 = OR2 + 42
⇒ OR2 = 25 – 16 = 9
∴ OR = 3
⇒ OP = PR + RO = x + 3
Consider ΔARP where ∠R = 90°,
⇒ PA2 = AR2 + PR2
⇒ PA2 = 42 + x2 = 16 + x2 … (1)
Consider ΔOAP where ∠A = 90°,
⇒ PA2 = OP2 – OA2
= (x + 3)2 – (5)2
= x2 + 9 + 6x – 25
= x2 + 6x – 16 … (2)
From (1) and (2),
⇒ 16 + x2 = x2 + 6x – 16
⇒ 6x = 32
⇒ x =
From (1),
⇒ PA2 = 16 + x2
= 16 + () 2
= 16 +
=
=
∴ PA =
Question 10.
P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to the circle which touch the circle in A and B. Find AB.
Answer:
Given P lies in the exterior of circle (O, 5) such that OP = 13.
Let OR = x.
Consider ΔOAP where ∠A = 90°,
By Pythagoras Theorem,
⇒ OP2 = OA2 + AP2
⇒ 132 = 52 + AP2
⇒ AP2 = 169 – 25 = 144
∴ AP = 12
Consider ΔORA where ∠R = 90°,
⇒ AR2 = OA2 – OR2
⇒ AR2 = 52 – x2 = 25 – x2 … (1)
Consider ΔARP where ∠R = 90°,
⇒ AR2 = AP2 – PR2
= (12)2 – (13 – x)2
= 144 – (13 + x2 – 26x)
= – x2 + 26x + 131 … (2)
From (1) and (2),
⇒ 25 – x2 = – x2 + 26x + 131
⇒ 26x = 50
⇒ x =
From (1),
⇒ AR2 = 25 – x2
= 25 – () 2
= 25 –
=
=
∴ AR =
But AB = 2AR,
⇒ AB = 2 ()
∴ AB = = 9.23
Exercise 11
Question 1.A circle touches the sides
,
,
of ΔABC at points D, E, F respectively. BD = x, CE = y, AF = z. Prove that the area of ΔABC =
.
Answer:![](data:image/png;base64,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)
Given that a circle touches the sides BC, CA and AB of ΔABC at points D, E and F respectively.
Let BD = x, CE = y and AF = z.
We have to prove that area of ΔABC = ![](data:image/png;base64,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)
Proof:
BD and BF are tangents drawn from B. And D and F are points of contact.
∴ BD = BF = x
Similarly for tangents CE and CD drawn by C and tangents AE and AF drawn from A,
CE = CD = y and AF = AE = z
Sides of ΔABC,
⇒ AB = c = AF + BF = z + x … (1)
⇒ BC = a = BD + DC = x + y … (2)
⇒ CA = b = CE + AE = y + z … (3)
In ΔABC, 2s = AB + BC + AC
= (z + x) + (x + y) + (y + z)
= 2 (x + y + z)
∴ s = x + y + z … (4)
We know that area of ΔABC = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAAcCAMAAAD/XGWDAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///b8HgqLAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACj0lEQVRYR+1Xa1caMRBNVuzWB+oWWwW1gtpHWgu75P//t84rj9UYs3iO2h7yQUicm7lzZzIJSm3HVoGtAu9KAaP/kRGpZmfX70rDIjLt4TK2s3eHP4twyoy/e8PNUGV+UlarDz3O0+NeDLl92/1z+bd9DmX0jpMioFRpqLE87HHuPOPETk8GhN9+5MoqQBlPWjkUwEoFigIlh+tPcTWswt4l5M0upaUAFZFWgioI1XPwgfJKr6TttFcrz/LmvUpQMWlhUBBqICCByoLBeuhmWldHS9XWXCtunmRtf+9pfRTVhaCyILPz40pXXyKUhJpFqXYCxJBTX+oFaGWnu0t7CaleVVSkfp4iDc6Wds6Gao6ZYVQWpEwFvcZo1oRQHGoe1dZnqmvYDsWVQSW9bmAF60RI+3lSaVx0KZljUTMqDzJos264+opRZMiDApXBDW+hD77hp6s8N0djf12KuATskRZUFsQ2QiIi7V2nXJEOjnTgrwwnDMp0BLuK0n4egou+2fvTWrtEB6XzoMeknUDiOuXqCdLhDr+vQXNPWimapwbWdFA61DTaPgmSHDqlC1GPSHendCjpDqe/uKFk3c9TpCkwMeRdeZIFwUGkmqZ08wcLlEXFvRRr2ownlFmSkw7pBL6KlZ+nSLf1ieo+c2+M+3QWBI0Des4tHwtGSajOdTKrRp8pe+OPOfDFsuQ7vJtBkY5Rc274YZ7aagFd+k9DRR3fiHmQOb7SesR9klEsUB6lftV69DUEyh3y4sGLbsjVGjr+ZqiNbkQoswfPUoq/9BEDnMtfeVHGPGpIqP5ChFZPd3h/lD4X8Vi88D1dLlD0yjPV/tv+aCkVKJZn3Qx7iCYP+KsvmmEP0Vfn9z84/As5lVJNnXCA1wAAAABJRU5ErkJggg==)
Area = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Hence proved.
Question 2.ΔABC is an isosceles triangle in which
. A circle touching all the three sides of ΔABC touches
at D. Prove that D is the mid – point of
.
Answer:Given that ΔABC is an isosceles triangle in which AB ≅ AC and a circle touching all the three sides of ΔABC touches BC at D.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARkAAAD7CAYAAABe6+AqAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAADdpSURBVHhe7Z0JeFTlucdfAkkIAcIWFkFwAUUWFxAXFNwRFNkUBGURbLVWa21721pvbxe9tbW1tdqrVqksAuKuIAiKWhdEsQIqCCIoe9hkJ0ACSe78Pj0YQ2bmzMw5M2d53+cZg8lZvvP/vvmfd/9qVUREVBQBRUARcAmBWi5dVy+rCCgCioBBQElGF4IioAi4ioCSjKvw6sUVAUVASUbXgCKgCLiKgJKMq/AG8+Jl5RXy/opd0qFVXWmYr0somLPs3FPpCnEOy9Bcad7nu6TPHxfK7we3lZ/1bR2a59YHTQ4BJZnkcAv1WS98sEUGdCuUuZ/tkBsuain18mqGGg99+NgIKMnoCkkIgU07S2XBl7vlXzd2kF88sUJmffyVDDmrWULX0IPDhYCSTLjmO+WnfWnhVmldWFvatagjfU5tLM++v1lJJmVUg30BJZlgz6/jTzf9w61yfLM8+c8XuyUvp5b5uWLjPkM6KopAdQgoyei6sI3A0g3F8vnGYqmZVUP++OIqya5VQ2rUEHlm/ma5Y8Cxtq+jB4YLASWZcM13Sk87de4mObNtgTx+c6fD13nqvU3yp2mr5Ue9j5Z6tXU5pQRwQE/WVRHQiXX6sfYcKJPXP90u/z3wuxrLFV0K5c/T18jMhV/J0O7Nnb6tXi8ACCjJBGAS0/EIEQtJ7h91opzSpu53blcnt6ZMuqWTZNeMHKCiCFSDgJKMLgtbCORHyKTb8fWrPbZDy3xb19CDwomAkkw4512fWhFIGwJKMmmD2l83en32S/LW3HmSFQkfXXh+T+l5UW9/PYCO1jMIKMl4Ziq8MZCS/fvklltvlXFvFEl5izNEysvlD1Pukx/2nSn33nuvZOfkemOgOgrfIKAk45upSs9A/3LvPfKvN3ZJo153Si3jzK0hBw/1lQem/68ce8z9cttPf5GegehdAoOAkkxgpjL1B9m3e4dMmTlf8s78UaRl4kGpKC01F82ulSs53UbJ5Gnj5eabb5HsXM3uTR3t8FxBSSY8cx33SXfv2i57DkUIJa+uVJQfPHx8RdlByanTQHaV1pJ9e3dJgZJMXCz1gG8RUJLR1XAYgYZNmknT/HL5as92yW3SQioOlZi/1aiVIyW710vz+llSr6CRIqYIJISAkkxCcAX74NyIBnPD4IvkpgcmyaELfym1curhkpHSkgNycMkTcsMd/SQrYjqpKAKJIKAkkwhaITj2ezfcJC/NnCUvT/uxyNHdI9GlCqmx/h256pxWMnzUmBAgoI/oNAJKMk4j6vPrffTJYjnl1NOkR/dsadSkKYqMbNlcR/ZHXDSLl3wqJ598ss+fUIefbgSUZNKNuIfvt3v3bnnttdckt3ae/PDHP5H69b8uI9i2bZvcd999MmfOHGnXrp3k5eV5+Cl0aF5DQEnGazOSwfG88847UlRUJAMGDDhMMAyncePG0r17d5k9e7a8++67cvHFF2dwlHprvyGgJOO3GXNpvBs2bJB58+bJscceK2efffYRd+nZs6csXrxYICJMpqZNm7o0Er1s0BBQkgnajCbxPBUVFcZMOnDggFx00UXVmkN169aVCy64QKZOnSr//ve/5eqrr07iTnpKGBFQkgnjrFd55k8//VQ++eQT6dSpk3Tu3DkqIl26dJGFCxfKhx9+KPwb/4yKIhAPASWZeAgF/O9oL8bZm5trfC01aNobRWrVqmWO+fLLL805mFb8TkURiIWArpCQr4/3339fVq1aZcjj6KOPjotG27ZtpVu3bsYBjEZz1llnxT1HDwg3AkoyIZ5/QtM4cps1ayY4du3K+eefL0uXLpU333xTOnTo8J1IlN1r6HHhQUBJJjxzfcSTQhIQDU7cgoIC20gUFhZKjx49ZNq0aYakLr/8ctvn6oHhQ0BJJnxzbp4Yv8oHH3xgnLeYP4kKYe6PPvrIhL1PPfVUadmyZaKX0ONDgoCSTEgmuvJjlpWVGcdteaTrHb6YZJy3ZP0S7p44caK8/vrrMmLEiJhO4xDCrI/8DQJKMiFcCosWLZJly5bJGWecISeeeGLSCBDu5kP4mzA4IXAVRaAqAkoyIVsTe/fulTfeeEPq1atnNJFUJCsry1xjxYoVRpsh8lS7du1ULqnnBhABJZkATmqsR5o7d66sX79errjiCkdKA1q3bi1nnnmmIa758+fLeeedFzJE9XHjIaAkEw+hAP1906ZNJr8FYjjnnHMcezKIZcmSJSbShPnUqJF2z3MM3ABcSEkmAJNo9xEwaYqLi2XgwIFSp45zzcAbNGgg5M4888wzJndm0KBBdoekx4UAASWZEEwyj0jyHA7fjh07mpCz00IYnOsTFj/ttNNMyYGKIgACSjIhWAelka1N0GKs2iMctk5LTk6OcQI/9thj5l5jxowRN+7j9Lj1eu4joCTjPsYZvwPaxcqVK41J06ZNG9fGc9JJJxktZsGCBfLxxx+bf6soAkoyAV8DO3bsMH4SSgEgGbcFbWb58uUm2e+EE06Q/Px8t2+p1/c4AkoyHp+gVIcHwWzZskUGDx4sDRs2TPVycc9v3ry5iVzNnDnTRLJ69eoV9xw9INgIKMkEeH7XrFljHLEkyZHLki6BZDCXIBmczNqqM13Ie/M+SjLenJeUR0VdEibLwYMHTX0Sjtl0Ca06L7zwQpkyZYpxAg8bNixdt9b7eBABJRkPTooTQ0KTIEHu9NNPNz1f0i04fWnVyYdWnanUSKV77Ho/ZxFQknEWT09cbf/+/UaDwOmKRpEJqVmzptGgvvjiC6NRHXfccZKdnZ2Joeg9M4yAkkyGJ8CN2+MLWbt2rfTp00datGjhxi1sXRNiIUmPeiladVa31YqtC+lBvkZAScbX03fk4Ddv3mxqiFq1amW612Va2EalcqvORDrwZXrsen9nEFCScQZHz1yFauhdu3ZJ3759BQdspoXdJ+kf/MILL8jbb79tqr9VwoWAkkyA5vvzzz83jlYcvV7KtmVHA+qa3nvvPTMutCyV8CCgJBOQuSZUjYOVfZPIuk2mpaZbUNDIijFNmDDBOKRHjhyprTrdAtuD11WS8eCkJDMkHKtoMueee64cf/zxyVzC1XPoM8Me2rTqJLQea6dKVweiF087AkoyaYfc+Rvig2F/appF4Wj1oqBhEdKGCNFm2CVBW3V6caacH5OSjPOYpv2KOFTpekczKhytXhV8MfhnIBl2rkxHwaZXsQjTuJRkfD7b69atMw5VclL8kIcCsbCzAcSIyeRlUvT50vDM8JVkPDMViQ+koqLCNPAuKSkxjlU/mB/169c3zcaffvpp04LiyiuvTPzB9QxfIaAk46vp+u5gFy9ebKqdTznlFF/teWS16vzPf/5jQtpoYSrBRUBJxqdze+DAAePbsMLDOFb9ItQw4QQeO3asCbtff/31Qq2TSjARUJLx6bzih1m9erVccsklvkxuoyq7a9eupt8NiXpUi6sEEwElGR/O61dffWUcp3ShI2Xfr0KF+GeffWb8Su3bt/dEGYRfsfTyuJVkvDw7UcZGTgy9e4cOHSo4Uv0qzZo1+06rzksvvdSvj6LjjoGAkozPlge7DuAwxdwIgolBhjLOa9pB4MBGO1MJFgJKMj6az7KyMuPsJXTttfqkZGFkJ0vMpsmTJxuz6Zprrkn2UnqeRxFQkvHoxFQ3LPYzWrZsmUm6Y7uRoAjNxnH+8iGkzf5NKsFBQEnGJ3O5d+9eU59E0yev1iclCyXhazSzFStWGE2NAs90Nj5Pdtx6nj0ElGTs4ZTxo+h2t2HDBunfv38gtxhh72y2bXnrrbeMz4ltVVSCgYCSjA/msaioyOxhxBaz3bt398GIkxsi5QbUNUE0HTt2lAYNGiR3IT3LUwgoyXhqOqofDCbEvn37TJ1PXl6eD0ac3BCtVp3PP/+8yQPq169fchfSszyFgJKMp6bjyMHwZv/oo49MbRIO0qALrSB4XqtV59FHHx30Rw788ynJeHiKS0tLjSMUJyiOUT/VJyULa25urglp06qTuqZRo0ZJVlZWspfT8zyAgJKMByYh2hBo7MTmaHzp8MeERdDaSMxDo6FdZxg0uCDPrZKMR2d3+/btxgHKZvU4RMMkaGwQK606CduT3RxkX1TQ51ZJxqMzTEMnCiGHDBkSyigLrTpJOnz11VeNfyZT2+16dHn4alhKMh6cLlo40AKhbdu2ZpvXsAoV5phLRJown7RVpz9XgpKMx+aN+iQcnvyksVOYM1/r1atnspuffPJJU9c0ePBgj82WDscOAkoydlBK4zE4Owlbo8FoDY+YSnNqmqxWnWh3Kv5CQEnGQ/NFwh2OTvawVh/E1xPDTpiE72nVSTif8gNt1emhRWtjKEoyNkBK1yHUJ61du1Yuv/xy7atSCXQqztFocABTiX7GGWeka0r0Pg4goCTjAIhOXILN2ahPIsNViwOPRBTfzNKlS42m16FDB23V6cSiS9M1lGTSBHS82/Dl2bNnj6myxlxS+S4C5AvRRW/GjBmmi17v3r0VIp8goCTjgYmimfbChQuNo1ezW6NPCBoerTrR+Ahpt2jRwgOzp0OIh4CSTDyEXP77wYMHTcia+hy2N1GnZnTAadWJE/jxxx83TuDhw4e7PDt6eScQUJJxAsUUrkHSHR3hSDwjcqISGwFrt0xC/V26dDH+GRVvI6Akk8H52bVrl6lPatKkibARvUp8BND40GbYtYEEPfJmwpywGB+xzB+hJJPBOaA+afPmzaYZlabM25+IY445xoSxwW/+/PnSo0cP+yfrkWlHQEkm7ZB/fcN169YJrRzYbJ7etiqJIYDmt2TJEqMJdu7cOZRFpIkhlrmjlWQygH15eblx9tKUCtWfRk0qiSHQsGFD0wLjueeeMxrNgAEDEruAHp02BJRk0gb1tzdavHix8CFcTcPsdAubwx04cED2799vcnP4yf/zgfgOHTp0+MPYSO3nk52dbfwfkGLt2rWFaA9FjPR64Xfp7txnterEZGK/pjA19kr3mknlfkoyqaCXxLnUJ6HF8AWlytrtLyb327lzp2zbtk22bNlifrKPNk5n9nIqKSkRNCuIB2E81pj4af3eelSOtY4j3A7pQDTsyd2oUSPjWyJxjn+jbbjZbIp7g+G4ceMMpqNHj9ZWnUmsSbdPUZJxG+Eq16f+Zs2aNcLm8kcddZTjd0cr2bhxo7kH+zTxb7rsoaFAGmgcFilwf/6dn59/+MPfieBwLCQCyVgkxE+0HYgLgiouLpbdu3cbbYh7cE/uwzkQAFEzEuZatmwprVu3Nv9GA3JSCGET1iaZUVt1Oomsc9dSknEOy7hX2rp1q1AEyZfbyYgI16XR1apVq8xPOurRjwYtAs3Cyo4tLCw0X3zKFviyO9Wgm3tBPpAN92Y81GJBcLSt+PDDD42pxb2JDOHsxrTh/50QKtaXL19uQtoUU6IlqngHASWZNM4FXwJMl2HDhhnzIhXhOvTApWjwyy+/NBoFxIG2QI0PX+JmzZoZkuEL7qag8VjaUPPmzQ/fCq0G84wwPVoOJAjhzJs3zzw/29FSSkEP31TwQFNi07tXXnnFXBsTSsU7CCjJpGkuSB6jTQFfKNoWJCM4ZPmiUr+zbNkyY6KgraAdcF2+tBCL26Rid+yYTJAeH5zckA6EAxZoHtRs0ZAKjcYyeyDHZEor0Awxl9AU0dzQ2lS8gYCSTBrmAXLAMYnwlk30S4QZQjQKvwPmEEJLCDQWyMUN344bsEA6jJsPeS74jCAano3KarQQSiu6du1qNrNDO7IrVqvOqVOnGrPp6quvtnuqHucyAkoyLgPM5dFg+DKh0rdr1872HYkCYV5ALvg3MCnQgnhTo7X4OZ0exzI7EvBBC0G7oR4JnOjpiwlEbRKEU1BQYAszjkUzAm/OTQRrWzfQg5JCQEkmKdjsn4SvhDcr4Vy7LTUJL5P7QfEkjlRMoD59+phcEMLDQRMiWuQL8cFhDKlCFtOnTzdZ0WREU0aAthJLyOVBU3z00UeN5ohWxO9UMouAzoDL+OMjQAuhGVW8aArhZ8gFswHfBWbQoEGDDLnE+4K5/BhpuzyO48suu8x0B0QjgWQgGxqJ8zvIJlaGNAWTaDRWq04t2Ujb1EW9kZKMi3OAzwHCwDGLqRRLMBVIj8exi7ZCmjymUVjIpSo2mEhofhAGGh1kQwkBONGKE59NNOE8HONWq86wYuji0k7o0koyCcFl/2AS0misRP4I9UnRMl/RcubMmWO+PBxD4yre2JhXKmL8MWCCNodzGE1v/PjxhnwwjaozH4ks4edBA2JjOBqzq2QOASUZl7CnQphQMxXCJ5988hF3IeKEloO/Br8Nzlze0GTGqhyJAKYm2h04gRnm0xdffGEInBqmqomFbHELcWM2ET7HkaySGQSUZFzAnXogtBiiP3wJqtYn4W+hITZ5HTh1Sc4L83a0iUwBztwxY8YY8wnn7tNPP206C6KtVPZ5Wa06J06caOZixIgRrteJJfIcYTpWScaF2eYLgG8FgiEnpLLwBp41a5YpUuQN3KtXL21YleAcQNpoKoTxyfLFQbx+/Xrp27ev0XQsQYNEk4TMyYzORMV7go8WyMOVZByeVtLoceCiodDvxBK0m5dfftlkpOJnGDp0qDarShF7/DFoKGg3r776qmkwjsnJdimErjGh8NuQg4M2Ayk5XaCZ4iOE4nQlGYenGYIh3Z+MUyuJjILB559/3hQLkj7fr18/32TpOgyPK5cj8xlf1rRp0wzZ0NJi4MCBxnnO7wl7E2kiSkXDdpX0IqAk4yDeFCqykKkEtrZSpYjx2WefNZXJpNKTA6JvUwdB/+ZSkAm+mpkzZxqHOkmMgwcPNlpO5VadhL7pdaOSPgSUZBzCmnYHOCLpuULIFXUdX8ELL7xgusxdddVVuv2sQ1hHuwy1TkOGDDFhbUzTCRMmGNzxy5A789RTT5mewGg5doS+OfjWSJK0hPlFQ6VmTMUeAkoy9nCKexSp8DgXSbpDkyE/46WXXjK9W4geqdMxLoSOHYDmAhFgok6ePNnsBsG8ENK2WnWSIBlPaE/x97//3fS/obKd3CcqyfHtKMnEQ+/bvyvJ2Mcq6pF0icPmRw1HiyFp7MUXXxRS5HHwau6LAyAneAmS9ygopdiSMDdOYBzCjzzyiNE4Ma3iNe1id0/aVPzqV79K8O56eGUElGQcWA+QCpm7OHRJwsNEorr42muvNVEmlcwggMZx3XXXyZQpUwzZWKUapBhYO1DGGhmh8nhElJkn89ddlWRSnC/IhaxSHIxk7kI4NF5in2bdsC1FcB04nUzfUaNGyaRJk4xTmNwZok5onu3bt4/ZqhMTCYf9P/7xD+Njs8wlUhPQlFTsIaAkYw+nqEexWLHTcfxi76PBjBw5UmuPUsTVydPRJiEa8mhokIWDmDD3u+++a8zbaIKTl4pvfGxkb0MyOPE1OpXY7ISaZOi2T7gTDcQSFhFvO3rPxhMcvXxQq+mDgg8GDUaLG+Mhl/6/UzTJ3KDRQDAI6QbMdTSTlhcHfh12llBJHoFQkwyp/eSwWP1aLHUYh188QXsh8Q6iwm6n98s111yjJlI84DL4d8gER/wTTzwhRUVFhmwIaRP2rk5YDxANGo36ZpKfuFCTDLDhN8E5mGiCHDVIJNqh+UAwZPiqkzf5hZiuMzFnSdLDdIJoaG/KS6a6Vp2YSBDR3Xfffdgnw3w3aNBArr/+et16xeakhZ5keFvR7tJq02htahYLP7YjwRdD6JrEL7rXsXhV/IEATnoS8qjQhkQoRaiuVSe5ND/72c9MTyBLrI3r/NxfOd2zFGqSYdcAzJ2HH37YaDKYQORFoNnE2lYEgqGXCQ5EwtZEKVT8hQC+GOaO8DaJlHysUhDrSVgTOH1VUkMg1CSDrc1CohcJKrD1/7G2LFm7dq1pmoStToUv7RpU/IkAoWhSEGgXMXbsWJN+QB8a1kPVFh3+fEJvjDrUJIPqi5lEJIn0/3iCQ3j27NlGxSZNnWJHFX8jQG0T1dtkAeOfQZt98MEH5aGHHnJ0K2F/o5Ta6ENPMhAHNnc0ksHRd//99xsfDMcR7ia3gmxetctTW3xeOJvKeXJn0EpJPcAnx95PlBLwQrHz8vHCc3h5DKEmGbQYEquimUeYT7fddpupQyICQaNvFiCNqeJtb+LlSdexfYsACXmYvhCM5eClHIH0BNp6amZv6qsl1CRDZOiWW26Juh0qzkAqqWmRCSHxlmMxklvBXs5aiZv6Asz0FajWRpvFdGZ++aC9khejWowzsxNqkkGDibUnD6o0C43UctpnImgzLEQcwEoyzizCTF6FPjNkauOPsdpx8G+2pdFtbp2ZmVCTTDwICU3zVqNpEUQDuZBTgxmFSq3ifwSIKo4bN05+/vOfG98Mc02zqqrhbP8/aeaeQEkmBvbkUmAq0fyI3ryo1eTH/PSnPzWNjFSCgQC5MESY0FxJsJw6darZcoX0hMq7HwTjadP/FEoyMTDnjUbZAdmgOIjJocCHQ9hTJXgIWC8OGsGToMkOlLxcYiVmBg8F559ISSYGpmwfiwp94403mq03VMKBALsf0CNo0aJFZgsb/DYqySOgJBMFO5oVUdOCUxCTSSU8CBBJpOSAdAVyZfDPaKQp+flXkomCHbs8knI+evRora5Ofn359kwiTUSYKDmgjATSUUkOASWZanBbvXq1ScZq27atqsrJratAnMW2t2xrQ8kBJpR2xEtuWpVkqsENh9+ePXtMG00qrVXCiQB9gighIdqEf46eQSqJI6AkUwUzutjTkIocGTZ1Vwk3AtYWN9Su9ejRQ7cXTmI5KMlUAo2sXjrakw/DBmEk4KmEGwHKDi644ALTSY9dKb/3ve+FG5Aknl5JphJoFMstW7bMJNqdeeaZScCppwQRATQYnL+EtXEG22kyH0Qckn0mJZlvkCMBi4WEkOmpIctkl1TwzsPhi+nMpn1Em6hpstq1Bu9pnX8iJZlvMMXmJmRNM/CuXbs6j7Re0dcIoNnSOe/TTz81W6nQtEzFHgJKMhGcqFnB2UsBJKowUQUVRaAyAuxnTjCACm1eSJ06dTJ7MqnERyD0JEOVNWYShXGYSKeeeqohGxVFoDICtAWhgRWb+W3YsEHefvttIY9GJT4CoScZ6lPogEYrTbrdaQ+R+IsmrEfQP4hGZ3RGRPOlQlsbjsdfDaEmGRLu6HKH8Kai2pqQpYoiUB0CVkU+fZ7RfNGAKZzV3SVjr5dQkwwVthRCYibRrEi1GCWXWAhgRrNGSHPgpUQBJVX62nNGSaZaBIgk0ZiIKmurX4g2olKSiYcAa4QWrOzXRb8hnMCYUYlucxzvPkH6eyg1GZpGU/RGd3reQkQMIBuahKsoArEQKCwsNL471g6RSHwzJOmRFaxSPQKhJBkiBB9//LF06dLFOPIwmciN0aiSfk3iIWD57siZ6dmzp8mtItLUuXNn3SYnCnihIxl2CHz99deNiXTRRRcZmxphc3UVRcAOAgQIWEPUulHjNnHiRGM2DR482M7poTsmdCQzf/580wwcgiG7l43b6FiPRqOiCNhBANOarXRI4hw+fLjxyWByow2rX+9IBENFMtQnEbKGUOjbWlxcLJs3bzYZvrH2X7Kz8PSY8CDASwnfDEl57D556aWXyj//+U/j5xszZozWNVVZCqEiGbrd4X8ZMmSICVuThEfOg1Zch4cgnHhSfHck4eGXwSeD9oIWYzUf79atmxO3Ccw1QkMytNSksI2WmtYiQIvhTdSiRYvATKg+SHoQwGSi7xBriLomoksUT+KbIeqkVfzfzkMoSAYiQZWlTuniiy82JQTIli1bTH6D9m5NzxczSHch+5fgAWsIwQSn78yMGTPMNip9+vQJ0uOm9CyhIBnC1bxl0GAqNxxigfDG0VKClNZQKE8mp4rN/iySAQQaWrHW5s2bZwptVUP+emkEnmTIyiTciGO38iZdlBGwrzVOPFVtQ8kTKT00bR54Oe3cuVNIi0A7hnRYY5MmTTJrjsiTSghIBufcunXr5LLLLjNZvZYQWYKAWrZsqUl4+k1IGAE640EyRJh4YVkmOBrMwoULhYb0OIO1VWfASWbTpk0mAkAkAHu5skAyLA40GRVFIBkEIBkilKwly+SmIhu/H7lY+AGPP/74wwSUzD2CcE6gzSU8/bRzYPe/qvsnEbrGIaz+mCAs48w8AyYTESbWWGUhe5ytbUmZIKLJxnBhlsCSzPLly43airpKR7OqwtuHfAf1x4R5+af27FYCJ2upqlBusGTJEpP8SavOMGvMgSQZQtXs+IfqyuZcFLVVFUwlSEZL9FP7ooX5bGvtsJaqCtGn8847T5577jmj0QwYMCC0UAWSZKhPwlZmkilmq05w+kJCSjKhXfspPzh9ZXhRVUcyXJytdXAAsx6p+CdpL4wSOJIhLI2KSs8PVNZoQj8QIgRKMmFc9s48MyTDi4q1VJ0QcaIQd/z48cYJfN1114WyVWfgSAaCIao0aNCgmJm8mFQskOpMKWeWoF4l6AiwdlhDrKVo0rFjRzn55JONRvPJJ5+YJL2wSaBIZu3ataZIjbBhvM23iCyxQLQJdNiWvHPPa60f1lIsIaT9+eefmwQ92kKgAYVJAkMytNRkEsm+REW1kqOiTaaSTJiWuTvPCsngk4mlyXBnWonw0ps9e7Z5CVbOPHdnZN66amBIBlWULneoo6io8QRSQrTlZjyk9O/REGDtQDTWWoqFFMmgrE+rVSf9aMIigSAZvPtoMaihaDF2iAN7ury83HxUFIFkELDWjx2/Hjk1BCKmTp1q2kHQ0ygsEgiSoeqVfjG9e/c2tUh2hIXBG0hJxg5aekx1CCRCMpxPLRMJouxwQIJoWPb58j3J0OmO+iTs3qr1SbG+GqrJKHGkikCiJEPKBE7gsWPHGs2bHC5+F3Tx/ROieu7YsUOGDRuWUJ9eHMM4f6k9UVEEkkGAIANEEy/IUPnaaC9oNGwsiFZDjVPQxdckQ1Yvqmf79u3NxCUi+G8gmWjZmolcS48NJwKsHUzuREPSRJfY6pa9tKmtC3oTe9+SDGFDVE6cvDh7E1U7WRi8haJla4bza6NPnQgCrJ1kSIbIEpXZ06dPN6Z+0Ft1+pZk2OeGDc/JP0jGgQbJsECoYVJRBJJBwFo7iWoy3It1S9rFu+++a7ZKxqcYVPElydC/g8pWKl2T3YMYFRUtqGovkKBOtD6X8wjQkyjZdiFWq87HH3/8cKtOO6kXzj+F+1f0JcmQ0MR+N/379zebbCUj9JHBYUdBpYoikAwCrB20mGR7ElHTRK8Z6pqo0raTRJrMODN9ju9Ihp6q5MW0adNGzj777KTxo1MebxMaQasoAokigKkNybCGqnZdtHstsoXxJxLAwL/InmC5ubl2T/fNcb4iGas+Ca8+HvpkbGFrZnj7sDgqd5v3zazpQDOOgLXbBWZ3KuuQlyU7mJKKQavORHK9Mg6CzQH4imTYO4l9bVAzcZalIkSj2NSNTdOxrek/o6II2EWAlxP+PPr5plrJT7kBa5s2JZhMQdts0DckU1JSYlRKqxGQE06yZs2amaI1kvmUZOx+vfQ4EGDNEMJmDaUq9P/t2bOnPPvss6aAMmitOn1DMmRIonVgw7LFiRPC1qLk21CakEwY3Ikx6DX8iQA7R5JnlWzgoepTYzLhAGad00kADSko4guS2bZtm1ElIQUY3ylhgWBPr1+/3qlL6nVCggABCPx6TpGMpaGPGzfOaOyjR49O2QzzylT4gmQgmO3bt5vyeCe3lmCBYP8WFRUZjSbRrGGvTKKOI70IYCaxZjCxnfSfdOjQwWgxCxYsOBzWTu+TuXM3z5MMJhLd3jFnunXr5igKNBFnU3T2aEJbcsK+dnSAejFPIvDVV18Znwy5LdnZ2Y6OEXcA65G6JmryCJH7XTxNMhQw0uWdn5TIOz2hTB7+HapheTMpyfh9Oadn/JjXBCKc8g1WHjUvPUoOZs2aZfLBWPd+F0+TzKJFi0y1KuXwNGB2Q3CwQV7sXVzdTpNu3FOv6W8EWCv48shxcUOsVp3vvPOOSdfAF+ln8SzJsPUnCUokO7nZeLl58+ZmEumsx9spiBmXfl6gXhs765JdMVg3Tjl9qz4jDmWrVSdm09ChQ70GQ0Lj8SzJwOLr1q2Tvn37umrGQCpoM/h9MJmi7TiZEKp6cGARYE3ik+ncubOrgQL6I6HJ4wTG93PCCSf4FlNPkgybs1ECj817zjnnuA4uTmXux944SjKuw+3rG7BGELfzqmgPixMY04yQ9nHHHecqqbk5KZ4kGUDdu3evDBw4MOnis0RAYwIJR+L/SaYBViL30mP9iwChayI/mEpu+WMqo0PBJBFVXoD0T2JvbT+K50iGLzpqIjUc6drSExuYNxOtPFGHVZvx41J2f8xr1qyRzZs3m4TQdO2hTr+kpUuXGv8keTT169d3/0EdvoOnSIam3oSsra7uqRaeJYIVE0hK95IlS5RkEgEuRMdS54bQlzddgoYNqb344oumrgkfpd/EUySD83XlypXGs57u2g32zyZHgWpYolnJ9gjx2wLQ8dpDgN4xaNmtW7dO+9qkbxJ1TWxxS5qF3b3F7D2Z+0d5hmQonad8gLDgeeed5/6TV7kD6i85CSRBYXfj0VdRBCwEIBiywtEq3EgKjYU0a5MX38SJE40TeMSIEbZ2SfXK7HmGZOjZS2XrVVdd5Wg9SCJA06MGosMnxBvDiXYSidxfj/UmAlRbkxVeUFBgQteZEF6AfOinxNrM1DiSeXZPkAzJTZhKmCyUvGdKKCvA3qaLPMl56gDO1Ex46760xySUTHTHyYLIRJ7S2voHLRtthrwZvySOZpxkeEvg7MXpS/g4kd34Epkku8dafT1ohagkYxe1YB/HC5BgRKZ3eyRvDP8MJIN/Bt+lHyTjJIPWQETHK93a0aYIZxNJwDdEToRKeBFAyyaETEU0Tt9MC2uStWnVNWVKs0oEh4ySDJtjwcpEctBivCCEzVGL2TiOKthBgwZ5YVg6hgwhQCIcWjZrwgs+OvxCaDDPPPOMyZ258sorM4SM/dtmlGT4EpPg1Lt3bxM+9oqQCIg2Q90IZfeqzXhlZtI7DrQYQsf46dzqApDME2G2EZwgeZQap3SneyQ65oyRDJEkVD5i/l7bBgL7mzGNHz/ejHHw4MGJ4qrHBwABkt/oZUTYOp2JofGgI4ROn5nHHntM5syZI2PGjBFqnbwqGSMZStjZioQMRto5eE3IAMYOp2bk9NNPVyew1ybI5fFQCIkWQ6jYixXQrE38mAQoGCcajVclIyTDBJJ3YAHlRXB4M5AARQYyhBikxs5exNtLY6LfM75CNIZk91pPx/NYdU1Wq06vZqmnnWSs+iQmAZXPy2oeVbBoMfiOrCSodCwevUdmEcAXh+Ofl4wXIkrR0MBXeO6558rMmTNl7ty5cumll2YWuCh3TzvJMIFoMvSJIVzsdWGhEcLE9oV0vGjaeR1DP42P8hbmmm6JmShvSRQrSIY0EKJgdC3wYp/qtJIMRWaE3Yjte1kNrTzRVMGicbG7Hyp00Hb3S3RRB/34V155xdQoXXPNNY5uv+MWbphIvAgnT55s1ifj9pqklWSI1ND1ji+qn7aFJUcCbYa3hdfCmV5bUH4eD0luOFKpYfOyI7UqxtZeTYS1GbeXwu2MNW0kwzYSpEIT0yc12k9CSLtPnz4mp2f69Oly4403+rJ5kJ8wT/dY2UfppZdeMrtCMtdeClnHwwK/Jto29VWU6NDpMd2V4rHGmBaSqaioMA9P+0LASFdXsXiTk8jfqRvBsfb888/L7NmzzW6WKsFAgPWJ85TcLXYG8KJfIx7SEAtJeuT2kKRHEqlXJC0kQ20Sqiil6p06dfLKsyc8DpzVq1atMhoZUQe/9lxN+MEDfgJmPPlQfEkz2QUgVZgpN6DpGm1TyFqnBMEL4jrJoL3gkKIsnfokL9R/JAs8aukVV1whbLaOak0I0esp3ck+a1jOo43Dyy+/bDLPSQz18/ps3LixyU5+4YUXjEbDWvWCuE4yvPXZz7pXr17SqlUrLzxzSmMgMkZRGiUHzz33nHz/+99X/0xKiGbuZKJIzCEvD+bUj026q6KHdm216sQh7MZWuonOmKskwyZYMCrFjzBsUATv/eWXX24WKJ/hw4d7ytEWFJzdfA40bNIS2H1g2LBhxlkaBMHficUwYcIEY0GMGjUq49qZqyRDK0u89ldffXUg3hKVFyEFlFu3bjX2L7avtoTwz1eURml0/8d/gYad6WZUTiOHPwb/p9WrKdOtOl0jGWp+yDngrU9qfhAFm5ciT8iUTOBLLrkkiI8ZuGeiWTylIpgWhKuDJoTf0WbIrCeqS6Z6Xl5exh7TFZKhPB5VjTcGD+ulmL2TSPNcND6HaFi4OLeDZBY6iZVXrsWXjrIBquzZodTLtXOpYIb/k3y0V1991ewnlskMe1dIhgprtpDgTeHFMvlUJq/quSRvXXvttcYGJlGPxD0v5Sg4+ax+vxamLfkw+F/ww2Ty7Z4OLHnhkTqCXxSTKVNZ9o6TDHtYU5+En4KaijAIoUOcvxAN4UNEicZbM49JS9oB0RZeCl7JIXETJaJlFHk+/fTTxqTPVKtOx0mGxCbySMg5oJI1LEKWKEQzadIkkxWMyei1jn9hmYuqz4npjgZz1FFHGYLhpRAW6datmwlp4x8lpJ2JzgeOkszGjRtNESHZsJSgh01I6LruuutMRSwaTUlJiSmjUMkMApQLUAKCX4I1yc6LmTIZMoOAGH8oa3Ds2LHGCUzyaLr9UI6RDM2omNDiyA4EqGVBt3ejLRqygEeOHGmIZsaMGVJcXGy0unRPbKYWtVfuW1paanxkaNZEV9Bg/LB9iBv44Re1WnVSqZ3uaK9jJFOj7IDs3rVDOka89pTKh1kwE2nXybYVtEakjw55NDiJVdxHALxJksRMwEQgAhiEbN5UkMM/Src/q1VnomuxaEeJzFy4VTbuKpUTWuTLgNMLpXZ2lq0hJUUym3eWyu1PrpCdxYekZlYNkQqRc09qKCNGjpGK8kO+KpO3hVISBzVsGMEjop6T9EVpBUmJLHZMKhX3EGB7YQiG7UyIrvTr1y/ju5K697T2r4zPEBcGzm+0u0Tyg+Z8sl3+55mV0uWYetK0IFeemrdJpswtkn/d0FGaFeTEHURSJLOj+KB8sHK3/KJfG2keuemeA2XywKy1sr+0Qn7VP/O77MV96jQdgMlItjNqOvYwdnH//v3NhumZErKUqSXDMY3PgjR0EiYTfbNlavyx7otzExMVXxg4hyW6aXcu6CJAr2oSEVmDdvYT24RCMXWF3N7/GBl8VrPDtxpw78cy8a2iCAccE/f2SZFMZG1K0/o5cs05LSS7ZkST+UYeeW29kkwVyMm+pA8NbxK0milTpggNvEhnz8SG6fQa4U2PSQvJ4DN66qmnjHlH1z8/CjuR4g+kmTYaJL1+/NxSxK05qFOnjknKs1p14qeKJ7M/3iYN87O/QzCcM+GmjnLgYFm8083fkyKZmhFTbH9pmby2eLu0aJAtu/eXyYwFW2Vod903Ohrq+AaIbOCMRKtBnacsId3d8HGIQia33nrr4aFOmzZNxo0bJ7/97W99p9HQDQ4TgD4/EAtOdi/tRmrrW5jGg1iHOH/xV+EMjvdiWbN1v7RqnHvECBvkQx326MPeUVVuUSvCMrv3H5J7pq2SgvyciG+mVIq2l8jo849KI1z+uxWp3tdff70hGZKjHn30UaPSYyvn5MS3bZ14YvqloMFUFsgOvxF9jP1SLLh//36TyUoWL89DVTxYknGtEh0Bq1UntYWsQ/JmYq29unk1IxpLeUqQJjUjpYfKpXG9HBl3UwdpmFMm+fl15KVF2+Qnjy+XmbefFtFujmS+lEYZoJMxkfhCkK9AsyS0CLz+mE+EWjMhmHQkqNH2wA8CXuS+oMW0adPG4Om15tlexpG1x8uEFx3mM76aaHJ2uwJ5Yu5Go1TUz/uWLiZHfvfZhn3yv1fH39YoKZLhPVgrElVqXLeWZBVvk5zsetK3S6Hc8eRK2RgJdSnJxF9ilOMz2WSjksCIU5iJJw08EwljOIK93jwbEkRzoVUmSWa9e/cWWk6GNScr/iqLfkTVVp0NGjSo9uCzIiTTuXU9Gf3wpxFCaSvNG+TIvM93ye+e+VJ+PehYW0NIimSyIir35ki8/On3t0qLhnWkdNUWefb9zXJcYSRScVS+rRvrQRLRAPNNiBW7GLIhtEiPE8wnCMetiE/VFpP4aYg6eaGLWnXrgryX+fPnm6gI/6aCmizWY4+1t8h1rR2JABFPQvyUwKDREI2rTviuP3DdiXLXc6vkR+M/E4nEebIi/7lzyPGRwI89H2xSJNO0QZ5c0KmxzPlkm9SKRJcw8Y9rmid/6XWC5OfW1DlNEIF27dqZymDe0Ew4JhRqLKX6ZGcSFXBK8F+gtVhCdIkCOpLV2JvcS0KxLTjgL0KLgQRx7LK3kJ978XoFY6tVJ60gCGlHC0JgJv1leDtjMhVH0lWa1M+ORJXtJeLxrAmTDMWPS5d+KvcM6iZ1Cxp6BS/fjwOHHJ3yeUuT78HEE2rm3xS5sQicyFolLwafxt13320ww4HK7+hVnC7nc7zJYqtY2oVAMEVFRSafg94v4ODVTeXjPZMX/24196dfNU5g0hhikTdkU9kvY/eZEiYZ3rR8AXAUdunS1e599DibCNBhj0ZfvK0hGD4UW+K3gWjIb6GaOFnByYfmdOjQIROVwZ/BG8wLmgH5Q4RWCbHS5JvcIsxJTEfdgzzZGY99HmF/wtq06lweefm0dyFXyhbJbClaI+URFfur7TtNMyoa4HTufLI7T61XNQjgiLP6z/Klw5Rin2YSzih4g2yIRiXqt+HL6qUv7J49e4RtSchE5SeJdZhFOMBZ/E5ob7qkYiPQO9KC9IsVn8mvb/8vKa/dRMrKy6T7qe3kBz+8WQoaNkkZvpgks271SvndnX+UNz/ZKIdq1pacfeul20mt5OabfxjYlpopI+rwBSAbsjQxFXAK82WE6PnJmx6iwXFMKLeqKQEpkWZP1IgCTb64XhB8LdQY8Rzka+B0RnXnWSBPTEYn/VBeeGYvjyE34led9fo78sHO46XmcT0jWm2WTH9ynrz4yjB5dvKj0rJ1ag72qCRTtG61XDZ4jCyp0UPqdR4hNWrWkgPb1sjq9x6UIXPflAFXDfMyboEbGxoLPhs+mzZtMm0VSZ7Dd7FgwQIpLCw0Zg/JVRAOdvbf/vY38zt6LRNFuPPOO43dnW7BLGN7HLKcqZvi5/bt2w354W/hmdCOeQaV9CPwp3vukQ/2dpLCi2+SGgeLTcGzHNtV3n9novzm93+Qxx4bG/nFt+VDiY4wKsn848GHZEnpadL0/KFScWBn5MblkteqvezMuk3+596x0qPn+RHHb6PvZI9WzSStOphof6/6e/6/umMr/876d3W/q3zf6q5V3f04J9rvretVd8+q50Ubj53fV75WtLHgOyFHBO2FL+iaNWuMIxfi4ctLqBdNAS2GXAhMI87BBPnrX/9qTC3S7i2fjHXPeHhGmw/Ll2P9HULjQwSLcTAumpmhrfD/CFoLGhoRNUiQVpjW+Dm3Kt6x5ibWnDk9X3bWTax5joZhrLWeyJqPt8aq/h2SL923R56c9b7U7Xy7SOkeqSg7+DX8JYciv+stry36i+zYulEaFibvB6yeZCrK5P3FqySn7ZWRG0cWRoRgkPKSvVKv+bGyclkd+d1v/luOaddRyLGwA36sL1B1i6i6hZYsadk5L1mCtDPOWM9n9/yq1+DLTUSKNPrKX0zIAycephP+DDYxQ9AS0H6IKpEICAlwPguNa1X+xMLLIm3rJ9exPtyb9cCHJmb8v0WKRLD4EMHid/he2CMdf1PlkHrVe3vt/+04yDN5TNV7RxuLtX7279khuyNh6Vq5kS1TIr6Ybxm+XLJqZcvBGrlyIBKBTEWqJ5nIIsiJLN6K8u/WuKAykROTFcn2pdqVtxELqvKDRHvIWA9v53weMpnjkjmvuomxflfdc1Q3Ljs4VHfNeNeK9neLLPgSk0n8wAMPmC85RMI5zBNfbIiHVgjWvHEex1iEU5l0+HdlTdDSUvidRSwWwXEsGhZaCWuDZC80Lcw8iLAyoVnXtINR5fmLNv92fm9dJ9G5jbV+7F7LzbVv99pRnyOiQCxb/ROZuX6ZNDvxbCk/sNvwSVZ2vhRvXCqdG9eQpi1S64EUxVzKkgvP7CivTnldpPWPpUZJRI2KDKZGbl3ZsXKhnNH0oNx+x28kr279as2aWExuh+VTYU09V4yDl2gU9T2EwiEC/DYksv3yl780/hFyUYjsYMKQKwPx8EED4fjK5GJpORAJJMYHrYTwNySCSQZ5oS1BMvy/thv1z0q8/cffl9dG/lq25zWUei3aRl5KInu2rJWyhY/If91/i9TMrp3Sw0T1ydz4gx/IjDnXytw3HpbanfpEVKcc2bfiQ6mzcrL84bE7pU69AnNjJY2U8Hft5AcffFDuuusuk0HMHI0ZM0buuOMOQxA4hisLpAK5oN1YvhqLaKz5tTQctBWugXbi9Von18AN2IXPPf8SefK+bXLHHx6WtV80lYpIdOkoKZLf3jlSBlw5NOWnjUoyDRo1keefHC+///1d8s7isVJWI1uOqV8utz/+Jzn3fO3AnzLyLl8AbeLPf/6z8clYGki0W/J3nLGZaKLlMgx6eZsIQCYXXHCRLF38kdFkT+zQSZo0S81Msm4dM0+msNlR8n8PPRzxQO+OvOlKJb8g9cQcm8+shzmEAGaNiiJgB4GCRoVy9nnO7+duK+M3p059SU9LJTtQ6DGKgCLgJwRskYyfHkjHqggoAt5CQEnGW/Oho1EEAoeAkkzgplQfSBHwFgJKMt6aDx2NIhA4BJRkAjel+kCKgLcQUJLx1nzoaBSBwCGgJBO4KdUHUgS8hcD/A9T2c05yBupwAAAAAElFTkSuQmCC)
We have to prove that D is the midpoint of BC.
Proof:
We know that if a circle touches all sides of a triangle, the lengths of tangents drawn from each vertex are equal.
∴ AE = AF, BD = BF and CD = CE … (1)
Consider AB = AC,
Subtracting AF from both sides,
⇒ AB – AF = AC – AF
From (1),
⇒ AB – AF = AC – AE
Since A – F – B and A – E – C,
⇒ BF = CE
From (1),
⇒ BD = CD
Since B – D – C and BD = CD,
D is the midpoint of BC.
Question 3.∠B is a right angle in ΔABC. If AB = 24, BC = 7, then find the radius of the circle which touches all the three sides of ΔABC.
Answer:![](data:image/png;base64,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)
Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.
In the figure,
ID = IE = IF = r
Since ΔABC is a right angled triangle, ∠B = 90°.
Also ID ⊥ BC and AB ⊥ BC.
∴ ID || AB and ID || FB
Similarly, IF || BD
∴ IFBD is a parallelogram.
∴ FB = ID = r and BD = IF = r … (1)
∴ Parallelogram IFBD is a rhombus.
Since ∠B = 90°, parallelogram IFBD is a square.
By Pythagoras Theorem,
⇒ AC2 = AB2 + BC2
= 242 + 72
= 576 + 49
= 625
∴ AC = 25
⇒ AB + BC + AC = 24 + 7 + 25
⇒ AF + FB + BD + DC + AC = 56
⇒ AE + r + r + CE + AC = 56
⇒ 2r + (AE + CE) + AC = 56
⇒ 2r + 2AC = 56
⇒ 2r + 2(25) = 56
⇒ r + 25 = 28
⇒ r = 3
∴ The radius of circle is 3.
Question 4.A circle touches all the three sides of a right angled ΔABC in which LB is right angle. Prove that the radius of the circle is ![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Given that a circle touches all the three sides of a right angled ΔABC.
We have to prove that radius of a circle = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAAgCAMAAABto3VBAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDpmkLbbkNv/tmYAtmY6ttu2ttv/tv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///b2et8JgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABMklEQVRIS+1V21LCMBDdAEJFoFCrtkJFe8v//yF7ya2iM840U3xgH0rmnN2zJ0smAfi/0acrNNcmCuPh+N1ngHcHpWZZmCClAMzQolnPT/SD385wYb7D22Rb62I34KRUGFzol/f02WhBsawlV+eEcStKJ9xxTs2WOqbZQEFOqabdW4WhFuF9OrDEbaTUM+WbtG5oXjtjK/Rl8B+0TKljZLZoh+eS8yz7lDBGPW4rPGlLnVaFzjVJsLnql3khzklA6TZsqWHwn6Pm5Uw2anxdzZ7xSmWY7rV8qTCQK5JRavnJe326mpdshPEPXC7c+dKutBYmPCz39cQTkEMZJSZ2fm93wwl84cOwidO/37/CGS+TWNE+xtMKb9OR/s6RxkV3fDwpfAWhse/muB3KGxZH689OLuZdGdET3lTWAAAAAElFTkSuQmCC)
Proof:
Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.
In the figure,
ID = IE = IF = r
Since ΔABC is a right angled triangle, ∠B = 90°.
Also ID ⊥ BC and AB ⊥ BC.
∴ ID || AB and ID || FB
Similarly, IF || BD
∴ IFBD is a parallelogram.
∴ FB = ID = r and BD = IF = r … (1)
∴ Parallelogram IFBD is a rhombus.
Since ∠B = 90°, parallelogram IFBD is a square.
Now AE = AF
⇒ AE = AB – FB
= AB – r [From (1)] … (2)
And CE = CD
⇒ CE = BC – BD
= BC – r [From (1)] … (3)
Now, AC = AE + CE,
⇒ AC = AB – r + BC – r
⇒ AC = AB + BC – 2r
⇒ 2r = AB + BC – AC
⇒ r = ![](data:image/png;base64,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)
∴ The radius of a circle is
.
Question 5.In â ABCD, m∠D = 90. A circle with centre 0 and radius r touches its sides
and
in P, Q, R and S respectively. If BC = 40, CD = 30 and BP = 25, then find the radius of the circle.
Answer:Given that in â ABCD, ∠D = 90°. BC = 40, CD = 30 and BP = 25
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)
We know that tangents drawn to a circle are perpendicular to the radius of the circle.
⇒ ∠ORD = ∠OSD = 90°
Given ∠D = 90° and OR = OS = radius.
∴ ORDS is a square.
We know that the tangents drawn to a circle from a point in the exterior of the circle are congruent.
∴ BP = BQ, CQ = CR and DR = DS.
Consider BP = BQ,
⇒ BQ = 25 [BP = 25]
⇒ BC – CQ = 40
⇒ CQ = 40 – 25 = 15 [BC = 40]
Consider CQ = CR,
⇒ CR = 15
⇒ CD – DR = 15
⇒ DR = 30 – 15 = 15 [CD = 30]
But ORDS is a square.
∴ OR = DR = 15
∴ Radius of circle OR is 15.
Question 6.Two concentric circles are given. Prove that all chords of the circle with larger radius which touch the circle with smaller radius are congruent.
Answer:Given are two concentric circles.
Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius.
![](data:image/png;base64,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)
We have to prove that PQ ≅ RS.
Proof:
Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius at points M and N respectively.
PQ and RS are tangents to the circle with smaller radius,
∴ OM = ON = radius of smaller circle.
Thus, chords AB and CD are equidistant from the centre of the circle with larger radius.
∴ PQ = RS
∴ PQ ≅ RS
∴All chords of the circle with larger radius which touch the circle with smaller radius are congruent.
Hence proved.
Question 7.A circle touches all the sides of â ABCD. If AB = 5, BC = 8, CD = 6. Find AD.
Answer:![](data:image/png;base64,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)
We know that if a circle touches all the sides of a quadrilateral, then AB + CD = BC + DA
Given AB = 5, BC = 8, CD = 6
⇒ 5 + 6 = 8 + DA
⇒ 11 = 8 + DA
⇒ DA = 3
∴ AD = 3
Question 8.A circle touches all the sides of â ABCD. If
is the largest side then prove that
is the smallest side.
Answer:![](data:image/png;base64,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)
Given that a circle touches all the sides of ABCD and AB is the largest side.
We have to prove that CD is the smallest side.
Proof:
The circle touches all sides of ABCD.
∴ AB + CD = BC + DA … (1)
Given AB is the largest side.
⇒ AB > BC
∴ AB = BC + m
From (1),
⇒ BC + m + CD = BC + DA
⇒ CD + m = DA
∴ CD < DA
Hence CD is smaller than DA. … (2)
But AB is the largest side.
⇒ AB > DA
∴ AB = DA + n
From (1),
⇒ DA + n + CD = BC + DA
⇒ CD + n = BC
∴ CD < BC
Hence CD is smaller than BC. … (3)
AB is largest side, so CD is smaller than AB. … (4)
From (2), (3) and (4), CD is the smallest side of ABCD.
Question 9.P is a point in the exterior of a circle having centre 0 and radius 24. OP = 25. A tangent from P touches the circle at Q. Find PQ.
Answer:![](data:image/png;base64,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)
Given P lies in the exterior of a circle having centre O and PQ is a tangent.
∴ OQ ⊥ PQ
Also given OP = 25 and OQ = 24.
Consider ΔOQP,
∠OQP = 90°
By Pythagoras Theorem,
⇒ OP2 = OQ2 + PQ2
⇒ 252 = 242 + PQ2
⇒ PQ2 = 625 – 576
= 49
∴ PQ = 7
Question 10.P is in exterior of ⨀(O, 15). A tangent from P touches the circle at T. If PT = 8, then OP = ………..
A. 17
B. 13
C. 23
D. 7
Answer:Given P is in exterior of circle (O, 15) and a tangent from P touches the circle at T.
Thus, ∠OTP = 90°
![](data:image/png;base64,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)
By Pythagoras Theorem,
⇒ OP2 = OT2 + TP2
⇒ OP2 = 152 + 82
= 225 + 64
= 289
∴ OP = 17
Question 11.
,
touch ⨀(O, 15) at A and B. If m∠AOB = 80, then m∠OPB =
A. 80
B. 50
C. 10
D. 100
Answer:Given PA and PB touch circle (O, 15) at A and B and m ∠AOB = 80°.
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OxvZ4FAIDp5CbQcE9koA8bNkyefPJJeeedd0xWOmY7yaOADgW+movj9er//HqugM3ixYsFE4qEK0OIpCPjJBAONESd3KZ2pfQFPmOcxJ07dzbaDNclH4dE0uzs7BD51kUZtw5+umHHwYZFxSlM8lWfPn38dK86F48ksH79eqPR1A5x0mA6uQ009rbw3axYscKwPVI3RR4PNVREv9iTOTk5movj0R4o7TKOg83MmTNNa1zCjxUrVkzhremlUyGBVAEN90qCH9o09VMQsl1++eUmyQ8zCuf09OnTDeCQi8NndXgngYULFzrbXeH777+X+fPnS69evTQK4N06+uZK69atMz4SQtKEob3SaMIF0KpVK0PERsY6XNYNGzY0/0weDhoOPh0AB8ex8uJ4s3XoZAudjKOaDTzCVO/279/fm7vQq/hGAn4AGisMop/0HSMcTvjbDmhn0XCIWhGpwsSib7wO9yQA6L/99tvGj+YY2LC4EGKR1YmtriNzJBAONPho6qaYKxiNihfe1KlT5ayzzpJ27dodW4wuXboYXw6RKnJxlBfHvX1K7RpMnNSy0S3FEbApKCgwxXCNGzfWEKN7a+fLMwM0OIMBGD8AjRUSpjx+AvYluV7hBZrUUhEaf/XVVzUXx6VdhdwBe1IObLcUR8BmyZIlAn8sbwll3XNp9Xx4Wr8CDaKiGJPoFCYTGjcaTfggJQPAef311zUXx+G9RfoBzvjevXvL8OHDJSsry1whabAhcW/WrFmmVS52mY7MkMDatWvNQ4rJgjM41aZTaVLv1q2boTVhf8J1XTK5FEf26NGjTR4OB45jzXZPbv+SYoCvDKAfPHjwMaBxBGwWLVokP/zwg9x8882aKZzcOgXm2+FAg+nEQ+vHgZbNpkd7Wbp0qfEdlBzwHqORE6WCBsXy4minj/hWtLi42JQnAewQmZF2UHIkpdnk5+ebNwcJfFp/Et/iBPXTa9asMZXVaDR+BhorX8wnepOxT/m9tNIZ/nbVVVeZSNUHH3xgNBz8DMqLE9suLSoqksmTJx8jx6PwurSRFNisXLnSaDWEF5UhLbaFCfKnABpMJ6q2gwA0RnUPAQktnSlbQCOjSryswUMC4PDg2Fwc6rp0lC2BwsJCw1cFu8Nll10WsSQkYbBBbSKBD0cb9rCO9JaABRq6H+Cj8avpVNoqwBBJRjH7NRLY8F1ycQiNT5w40USqIFjXXJzS9zaWDSBO1A//TLT8uoTBhkI7UtOp6tYIVHoDzerVq43pRHcDfHNBAhpWBnOIECwREojWCWZEGjiWyTYmF8cScUX7TnrvgJ/fHYEhIn1Eook4EXmKNhIGGzhfUTFpnasjfSUA0GA6ATRHG8jVDuTN4hwmyWzBggVRwYYbpOzB0lQAONT6qQZ/dOkPHz4s48ePF9wo1Jn17Nkzpj2RENjs3bvXXOjMM8/UJnMxiTmYH1q1atUxjQYfTVCBBulTB0VqBpnu1EnF4otp1KjRz3hxqLvK5AGnONE9/F/wBoEBsY6EwIYFg2S6tFBirBfWz/lbAhZo8FcEWaMJlzJaOEWBtPAlwziWgckILw4+HMwGHMfw5mTiIC2ArGtMUXxZmJvxjLjBhtIEcmsgmIawSEf6SSAdgYZVwu9Cc0SI3VD9Y82lwWFMxJUolc3Fgcoi1u+nww45cOCAMaeJPpOXlEgCb9xgA6csFyQvQUf6SQDzmLdXgwYNjDM4yKZTydUBHNDGKQ6kvCYepy8hdPwThMZJx+ctzzOQCbw4uE0AGrqksCeiRfTKeiriBhvqTED6RC+Yfo9n+twRQEPUCQ4YTKd05Hth3xKVYh/HAzZ2lS+99FITqQKwMiEXh57qFNpCiIffLpkuKXGBTW5urhCdwFOvPaDSB2S4k0wAGu4THwwuAO6XrgyJsElCd8v+p57K8uIQrUu3sWPHDgM0OIUBGuSWzIgLbOj/Q4uWstKRk5mIfjd1EggHGjZVLJGa1M02+St36tTJJKNt3rzZtO5NZOAcRcMnMmN5cRLRlBK5thffwVUC0OCjxUHuxL3FBTaoniA6mo2O9JAABOH4aMgExx5Pd6Bh1TAFKMDk3hMFG3uecF4ccnHSwb2APwuggRqCqnh4qpwYMYMNoW74S1gcbFYdwZcALw+yZC3QpKOPprRV4j4pHqYEg5T7REwpe16bi4OGwwOK0xh2wKAOrBecwVQF4LdzshVTzGDz448/Cl5p5fsI6jY6ft6ZCjRWCmg3hPi3b9+edAoHXD5oAOThvPHGG8ZxHK1OyI+7iPIjgAZlAqBxuiYsZrChFgq0c8J286OgM2lOFmh4K2eK6VRyfdFsMBNI5XAiXwyif3Jx4MWB14XQOJwuQcnFsdQhACd7wsl+7Fb2MYENFd6YUIRE3ZhEJj3oqb5XgAYfDXY4b69MjSoSPeLNzb7u27evI8uCOUYuDpoBnUYAHHhx/J6LY3Or3M4WjwlsiLHjnSbFOyhI7cjuSbOTkKaPj4Y3+U033ZSxQMOywr+E/5FsYkDBSdAdOHCgOV94jyonz+/ktoTBkKJKuyfcDBDEBDbbtm0ztVBqQjm5zN6ei3o2nJhoNACNm5vK2ztL/GrsZxj82N9OgwG5OGg4lheHFH+4gPw0AFr8TJiUmIBuB35iAhtCYXCCkMKuI3gSsEDD2wt73OkHK3gSOTpj3AKUIWzdujWpzNiy7p/CTx5gQB6aCooXqc3yw4BqAyBs06aNmRc+J7dHTGBDW118NelUJ+O2YP1y/nA1WYHm+FXBGcq+5mXq1iDqdfvttx8j4vJDLg6V7yQ1whvOfLziWo4KNqQqo2YyMdv/xa2F0fM6KwGAhrcqb9NM99GUJln8NpiVRFopxXHroeMaAI7NxcGJHA8PjJO7Yu7cucaXBFXqtdde66nzOirYkFuDA82J8KCTQtNzRZaAAk1sOwQgWLx4sckhczKBreTV6UYRnovDM+V1Lg5tVgjLk3QI4JXWaSI2qSX2qahgQ1k5ESi/ObcSu93M+FY40GA6ue34C7JUCffSioTqZjfBBhlRS8V6vP3228dyceDFScZiWPBdtoz5cItUrVQudB+ha1QuL6PPbyQdmxyf5U/jOBrI0WmCLOdURJWjgg2Vn6iXfux4GORN7tbcIaAmlGlNJwWayJJmX5Osyj73YqBNXHPNNeYFAADYHlWJNg1YtuUn+X5Prvzv8KMV2Ys3ZsudL62Wt+/rLPVrVTJ/g1IDDh4iZEOHDk0J0DCPmMAGx7BuWi+2YnLXCAca1WhikyUpANRKeQU2dlb0WCIqCC8OJtX111+fYJQwSzo2rS7ntT/alZSf7y/eLet35hmwgVlw5syZhso0WS0qNomW/amIYEORmlUvU6F2JXtz6fb9nNxCqVg+SypXLPezW/v6669NszAS1Ygw6MshttVH08BFsGvXLmNOebnPyVy2uTgvvPCCCUHH665gP/ywN0+WbP7J3PDna/dLUXE56dC4ukyb+m4IaD41jeMAt1SPiGBz6NAho+bh3NKROgls2Z0rj07ZJDsOHBHJEukcssd/e3lzY6czaKvz2GOPGZ/A7373OwWaOJeK/Q23DS1KkKGXA5pSm4sDLw7JdU2bNo15ClVCL57lm3Pk8WmbpaCwWKpXrSRP3dpBVi2aKx98PEMuG3iZ6SbhhxERbAh7ExLU/JrULdXenHy5/YVVMqBbPfmfYafKoSNF8o/3NsmvXl0jY0e3la++/FIee/xfJguUhwX2uHvuuSd1Ew7gldnf7HNerF6DDeIisQ5eHChZbY+qdu3axSTJg3mF0q9jHXlhtG0rXCwfTZsiH8/6TAYPGuwrloaIYAOjOkWYCjYxrbsrH5r0+Q5pUKeS3HvJf992T9/aRi56eIksDEUipodUZaq3ARhM3r///e/mzTho0CBX5pOOJ8VngwmF78RpWoVY5UUIHsAhF+ell14yTuTu3btH/XpeQZEcCgEOo7AgX96cMNG0w8URfN5550X9vpcfiAg2FGBi02p6u5dLcvy1Vv6QI12b1Tjuj+VC1Ain1q8i3249JP3P6yP1G7cwEUNqfQhrkh3apUsXrWWLcdkAG8LPvFxTOchmvu2220y9EhFFwK9fv34Rp9SmYTWpGHpGaYfLdyhNYQ/Qs9xvIyLYZGdnGxazVKiWfhNUquZTtVJ5OXzk6Jvr6CgOHVnmbXZCyGfT95zzZOe+HGMG4NwkvEmSGhvvT3/6U6qmHajr4jMhm5j9nuoR3qPKRqrgxSkrF6d/hzrSryBPXnh5nCEDIyvYr107ozqIARu30rhTvbBBuP4lXevJXydvlHtCZhQJWwDNko0HZOPOXOndtrZkh/wMBQWFZjNi8vKTzfmvf/3LdH6E3FtHZAnAN0OeCwERPwzmAmhgUdhcnOHDh5s5lhz4maANoeSC8Hkspleq7jEi2OBw5AYTTThK1U2l03X7d6grX649INeP+VYu7lJPcAhOD+VR/GpgM2lQo1i27cg9DmjQcHAW03aDN6PWtEXfDbxQARy/gI2dMeFqtC5KDI4cyZfLB14s8z+bI8tWrZfmjRtIn9695aNP5xq2QcLm1Dv5eUQFG7QaL3MP/CysVM3twSEtZOHGwzJn5R6pUL6cPDuqnbSpX15y84sNTQLD5kSxVmTF0n6D7gHY8n5nikuVXO11LdjwcvXbwMlLgGbShNflsTFjZYV0k6Kap4ocXiuN/v6CDOvfRe6499fSNhTR8vuICjZe8Fz4XUipnt/nn8+XhV/Okysuvkhad+gkh0KOw9z8qqZh4JgxY0wkBcci/Ck8OCT3TZs2zWg4sCsCNtAKkLZO8V80p2Oq79fr6wPQtHbxI9ggCzSW50NJf8uz+spJva6XrIIQKJavLD+sPlVWbpkZCKDhPsoEm8LCQvO2zJT2Hl5v8FivN2/ePON/odapIKtyqBvAbhMhLJd/SO677z659957Deve73//e5k6daqhuSQ57KGHHpLJkyfLX/7yF3n00UcNJy5Vx4RVASVd1+NXALAhryzSwCfGcwG4lzzs30v+5HM0euPv4T/L+lvJv/P9vIMH5OOvN0uNTr8Ryd0X8s2FKi5D61/n1NNl6eJ5snn9SmnW0ubZxLqzvP9cmWBjBYuXXoc3Eli3apk89+J42bQjW9qderJ069Ba3n3/Q8MiRw4GkQroWVkT/AuADpoKwPHPf/7T/G3KlCmGZLtHjx7me1AJ8PcOHTqYpDHC40EpZSjtobZ/C3/owx9wfrdHtIfc/juf37Rp07GkSAsM/Hv4Z+x5mUNpoMMzY+dnf+cnR/jAiY82ZX/ye8n/Z435m3mxFOVJcVEoCED6uDmOjhDkhTTZ8ib0HYRR5iyt0BRsvFnGr+bPkStu/5vsPHGAVDzxLHln/laRZx6Wu6/sKb/+9a+NKQTQWP8LiXzY8/Sr5m+kvUORQI0P4W8GGg9mFflSABCk9XyvrBHp4S75gIW/qSM93CXf6KU9yPahjvZgl6VVlHywwx9uHmh7hD/Q4Q86DzQy4vqwUiIz+6Cz/5EvP/kcP+3v/L/9W/jPsn633+PcFkxK+2nnFg44u/beL498OFVO6n2zZBXlhsyoKnJg+UzpXvOAnNK8tTebNMmrKNgkKUAnvl4YypP44z+ek53NbpD6p3WS4iOHJOuUlrKnWj1Zu22aZO/fIz8dzD2mirP54ayhJgrTiM1NdS9aDb8TBv3kk08MOFkqA+p/eAgh+MZpXBIgLJhEe3uHv7lLvrVtLkj4A17WW7zkQxb+IBP9LOshjufvFhjsw13y/8P/DscMjJRkYhMUSYZjxok9EX4OOJKrnVBF2hfNkFVz9ktxrRCdxMHt0rJwsbTt0cbQR/DS8dOcS5NBRDOKjaWajdNb5+fnO7B7u6zcUSS1uraR4tyjiWXFedlSq1EbWfjZBHn8n/+QarXrmzcvgwcOSgSiTuRZsMnogwTdI29lkrsADZsRSzdTPl/yjWrftDzc4W9rfuc84W9z+xYv621uH+RID7g1FyzQlDQfUvmw4LNhcN+pnEfJ3UFYmy6VNWvWkg+n/FtWLF8sa9Zvkob1z5B+/X8vS5avDr1k3jNMg2ivybQSdnunRzT2bKKY25PI9PNXqVZDalYWOZCXK5WqhHxk2OdZ5eTI4UPSqHYVGTx4qGRVOsFoJjyg+G4AkwcffNAU8aG1kMCHIxmthb9T6UueDX6b3/72t0bEfnq4/bbmyJPBC9YvgwZ6+Nkwh0eOvNG8UJo0ayHhNdwXNmgcevmUN5otL57rrrvOtz65MsHGbky7CH5ZgHScxwk16soNF3WSP735hlTqc5tUCJUh5IcK7HIWPic1qmySCpWryZlnn20eBMCEgVbz+OOPm+xR1H8KMC2J9gMPPGASwcgevuOOO0xSpuZKRd45aI1W4/LDHkM7pXMp9VJ0Lo3ElEkqA+byW2+9JfDi0KPKj7QwCjZ+2FmhOdz/h9/Jzt1/kHGfPiw/VW0kVfN+lF9e1ERaNB0hjz/xhOGOHTZsmNl8RJcwjYgwwWNj38iADtooWg0Hn6HRPWns1kzwye36bhqWOMsPJpTtxU7CJoyLsbAu8KIBcKiJe/bZZw0vjt+aFJQJNggdpFfNxpvnonzFyvJEyAy6e8ViEzVqHGoo17JdF3Pxlq3bmE2EWk3NDHky5EBBRm+dsfZh4Scgg6lFEzKKCxVooq8h+9wPmk1JDul4GBfgwAnvUUUJA2a2X0aZYGMdfWxqHd5JoFWHbsIRPugd3b59e3nxxRflySeflAsvvNDwybIRbdarNZNs9IcaKUDLrxXA3kk0tiuxz3F+p1KzgdoV8jPq2jCFEsmHgsuInCzMa/YLvDikRfhhRNRsCLH6NYXbD8Lzcg4tWrSQP//5zybiRJiWUgWcgYAQIe5wDZSHhnyRPXv2SOfOnb2cZiCvheMdcE5lac6XIcZFeIjQTuCQTmYulhcHE9ry4viBSCtiNAr1mw1rqQsCuZPSaNI4enlT4auh7ICsYTQccizIDeGBYRD+pGwBpyIE6DoiSwCgRnapcqrOnz/f1LNBeIaZ7ASlC1oR/h7OywuKSFWqSc8jgg3oiv2Piqk0E/55ZIkyUfsEoxvhbUuaRCkCUZV9+/aZhD/ML6+7HvpHSrHPhAgf2mEy2kTsVzv+k3PmzDFggKkD0DiZJ8Mzi/ZreXFg/oMXx8lrxHPfUTUbFoJDwSYesbr/WRzAZA9jJmGbP/LIIyapi7YdFG/ie6BXkI7oEuBlyuE12MyYMcNkfVPHRg2bGwm07AO0XzQdrmVzcVLBvhkRbJggi4DfJhFnVfRl1k8kKwEcwJhKJH9Blk1yH9qon5O7kr1np79PKgH73Ms9TokBlB9wBaNtuJ0HRcEuGg5+oeeffz4luTgRwYb4PouA+hVv8yynN4Ser2wJ4Gv4zW9+Y3hPAB18bPGETDNdtqQH4LfxinaDntvQfeK0HTJkiGcRsPBcHNujiq4OXo2oYMPG9QMRtFcCCfJ1MJvIq0BdJhKxfv1647fx8o0dRPnZ/U1ZgNuDzG7a4dr0BbevV/L8RLsgWiM0DuB4mYsT1Yyy5fdeC0Wvl5gEbP8hNjRvTwr5eHu2bds2sRNmwLegl8An6SYok2yJMx+H8IABA+TSSy9NmWQhYrO5OLZHFYmibo+o0SicZkQ3dARHAtj/vDlJDiPSgY2Ozc7flI/45+vI/mafu+U0JUIIayIhbjpfsA6pHrhFCDAQ0cTXh+PY8iC5Nbeo0Sj8NhAy6QieBOAgvvvuuw0lKJoOZtXgwYO1eV2JpWR/k5PkRsQVnydlI2QHo2H6if8Zs5EiTwo4SRTFN4vG5VYWdVSKCdqRUpMTzhIXvMcuc2dMghiteIlY4S945plnTJEmzkm3IyBBkDqRKLcyrUkZQXOgaSBFtG5rDonIG4ClUJeAAi8lm4vjRn5WVPJSwAZhYdemqg9yIkLU7xwvAbKOqQImEgL3CS8QQKhBgwYZLSpMKADH6b1NRjKlArTTIYeGqn2/DjQZ9gKAw94AcEidcNqsjAo2EPagCsIE5vSC+FX46Tov1GY2UatWrUwnhqeeespEq3r27Jmutxz1vtBqcN46mdoBeOEHoX6NrOCzzjor6jz88AH8epYXh0RRIlWReHTinXNUsGERSG+GK4WwmY7gS4B8CyIShMjxJ+DLwXEZC29K8O/++DsgARKnOcWLTgwcrZBe2S6VXkR5nJi3PQfAiIaDVkZonOpzp3JxooING5CF2LJli5P3pOdKsQTQWG+99VaZO3euyWTduHGjcR5nWpU41fG8UJ3IsYHzGaDhnJBXBVWW4bk4Y8eONRoO9LLJjqhgQ70G7T9g7FcncbLi9tf3sdX79u0r0FfgPH755Zeld6h/NHkgXtcJpUIylvEQUEjWWY6bAWJyCM2otob6I8gDzdcScWFSYQ5263Y8z1K89xcVbDghF160aJEJgTulUsU7Uf28exKwhEsUBnLwYhk6dGja01MADHTBTJY+E78PyXE4mwkl+4kdL5ldg/YL4JCNTsYxJiIvo0RHTGCDZkPZAv1rFGwSFbW/v0cIlBwLQuRkusJje8EFF5gjVZQEbksMcweNJlLjvmhzALAAGh7EUaNGmUTKdBr4b9DUyMWBG8fm4iRyjzGBDQhXp04d88ajHF5H+koA25xEwA8++MDkXdhEwGTf/n6UGPdGEWuikSicy5ie5NMANJij6ThwoNtcHKrVARzyhuLNxYkJbLgY2ajkZpA/4ASTWDouSrrcE/kV0B4QIseX8/TTTxutBxXarexSr2WH+bR582bjxI33oWGuaPloNGj8pP2nIxiHrwkaIAEENB3SJmwuTjy+vZjAhouy8Ui5RvXkdx3pLwFoKvHnsLmo7eFlAxFTopqAnyRGdBXTJ5EoCyCFRoN5iY+GHuuZMmAWsLw49KiKJxcnZrCBchK7HlNKwSZTtpYY85lcC3w5ZB+PGTPGAE737t0DLQSAk7cyYBrPYP/DGQQ/N6kDmZjoSi4O2i+OY0Lj7I9Y/F4xgw2bDufwmjVrDPVksqHCeBZYP5t6CZBub0PkZMeuXbvWEGg7kZ/i9d1BlMX8ibLGM3/2Pnk05J6h0TiVCOj1/TtxPcpfLC+OzcWJpoTEDDbY6lyAtxuOMY1KObFkwToHHRrZYLNmzTLOY5sIyL4I0sCE2r59u6HkjHWsXLnSAA3BEoCGl2+mD6wdy4uDSUUpDGyRZY2YwYYTkFlIlALBK9hk5lYjyZNwOCFeQuQkfFFBbtvJBEEqFEcS9Ig1H2bZsmWm1glTgTCwV/ShQZAl4MsLCJMKMMZxXFYuTlxgg2MQNGOx2HBqSgVhO7gzR0yqO++802g4s2fPNiFy+FqIWvp5UFRM6xvAMhbthGRWHiTulxIE5Xb++eqG8+JEysWJC2y4TMeOHc0bjagUNq+OzJWA5cqxIXISAfHnUT3sRlsSJyRtTSjKNKKNhQsXmna4RKyIujhNuRDt+kH6d5uLQ9U4uThE+shCD08riBtsqPnAlILjRsEmSNvBvbmyJzCrp02bZnx6aDnwo/gxJIymAmhE42S27XDxR5HQRvRJR2QJYOmg3Ybn4tDB1YJ03GCD6snmWrp0qeFSVbTXLYgEUKUp1kPLgboCrhyiVX4ijaKLAn210M4jRaGohCeviMJD7kl5m+Pb47hY0HCgGiXxEa0Q3IgbbLgsrUJJ8MNRDDeKDpWAlQD5N5Yrh4ZoNhEwFv+I21IEaMgcZv+WNT799FPDVse+5q2cSHax2/cRhPNT1hTOi4O/KyGwwbmGZx7AUbAJwtJ7O0cCCeFcOZs2bTJmVaSwqNszhI0PE4okvrJqmD755BOTLY02Ritjv/qd3JaVU+fHBKVqnCgVRFwJgQ1oz9uBhSHXIl0L0JwSeqaeB4JvolMEFOB66dWrl6mxcrM/U1myJvOXvYrTsrQoKn5IDsK2FBlqpNWZXRuei5MQ2DAN6A5J7vr8888VbJxZl7Q8i22aZ7lybNO8WHNcnBAKWs1nn31m/AalEUDx0qShH1E0ig3TpdjUCdk5cQ5KOtBwEgYbHGzY5/PmzTM5N5lYI+LEQmTCOShYhP3PtpMhvZ2CPvaNF87XIwf3yXfr18m5vfscF9CgYhutizwhkhJxaOtwRwLgRcJgw5RwAtHljzAhNrkOlUAkCRCpuuuuu4y5gn/EcuU4nUKxNydfpi3aLT/lFoQoIEQa16koN99y63G1TNRHEXHiZYlpRy8tHe5KICmwwRGIWrpgwQJj6zrZ9sHd29azp0oCVFrjE7GJgLZpHkl2TvlJVmw9KH99Z4Pc0Kdh6Daz5JPle6V7q3ryp0G1zW3TDhfmOV6SVLCjYelwXwJJgQ3Twwn4zTffGJtYtRv3FyxdrgBpFZEhcnIg6LJN85wwx/MLi+TslrXk/w09Wjqx80CeDPm/ZfLbgY2louSHulROMHvWb+1w02Vty7qPpMGGSmDC37wlCBmmA7FSui+6X+4Pqga4UCgHwEkLVw5+k2SpZytVKCff782T6Yt3h8yoIpm9cr8MOquBlC86LK+OnyAUVhLajqfq2y8yC/I8kgabcO2GzEtUZB0qgXgkALgQIkXDoRYJLSeZpnkVy2fJ7uwj8uGyvVI+lEK/bNNP0qdjFdm3P1u+37LJAA1heB3eSsARsEH1hb0L7YaNEwtrl7e3qVfzuwToOQ6XL5EhKsltImAijd4O5RVJ1+Y15cmRIRa+EBm5VGop5/9tkXRv1ljuvOMXctLJmUPj6ad1dwRsuKF+/frJkiVLTMUn2aM6VALxSoCMXULi4YmABB6IFMVDrM11sw8XyN4DBRIKOsn+I4ckJxSZqlujcgho6sc7Lf28QxJwDGxImGKjkLfw7bffmmI3HSqBRCRguXIIj5M4SvYvzlzydGIZDetVl32HiuSe8VulOPTfgZxcubF3Q+nZqlYsX9fPuCQBx8CG+fXs2VPgAEG7wekHQboOlUAiEoArB78NdXj4csK5ciI1zYMp7siuzfL66KaSc/iIqW9q2rCeVKtcPpFp6HcclICjYAO4kCkKHym2N0RKOlQCyUgAOhN6MsGVM336dOM8Rsspyy9ICsbMmTOkbu2apuqYgEW1ymo6JbMGTn3XUbBhUlR6nn322aZnNL+rs9ippcrc8wAa0D1YrhxC5AMHDjzKdVuUL3NnzTB9tuEGXvDVV6FGinlSUJQll1x6uTRq0jxzBeezO3ccbLg/0r9pewEvCOzrTmWG+kx2Oh2PJQDTgOXKIRlw+eKF8u7MhTJ7aw0prnqiVNy9TDpW3yZXXX116Lgm7p5QHt9Oxl3OFbDhDcObZ/z48ab2hCxjHSoBJyRAryY6HMybM0tGjP6VbGk2Wk48p49kFRdKQeFQWTjzUbmtZjUFGieE7fA5XAEb5khFOEx+2NpEF9K9F7LD66KniyABKCBOPrGW7K99utRtc65k5e03n65YobKUaz9EPv58roy6XUXoNwm4BjZsCGql6IsMFyltP7ygE/CbgHU+7khg3/79kp9VWSplcf5QaXeo4FKKCiUrBDgHDua5c1E9a1IScA1smBW1L0QDiE5BKwAxkQ6VgBMS6NL1DGlR+SnZsG+H1Kx7UqiU+4gUV6ohhd/9R/oO93fvKifuP4jncBVsEAgRKRjQIJLGuVcaU1oQBadzTq0EftixR7qekiVrPvub7Gx9pZQ7oY4Ubf1Grmi1T+6564HUTk6vXqoEXAcbrkruzdatWw2HCO06NRyuuzEZCVC1TfCheu0T5eFfdJGCrMOyY/c26XnV6XJVKEReLmRK6fCfBDwBGzI+rw6FI+kl9Oabbxq2tnhrXfwnOp1RKiRAhwRavJJXQw3VXffeq73LUrEQCVzTE7BhXvXq1TMNv+CfZbPQuErzbxJYsQz+CoyQlC5QkgDTQHi3xQwWS2Bu3TOwQSLUS+EkBmygEKX2RYdKIBYJ0MUDgq28vDyjFbN3tIVQLJLzz2c8BRtumxRzVGAqeolWmZRzHSqBCBKYM2eOKe6lJQvaMOwClMToCJYEPAcbxAP144EDB+Sdd94xxXKp7JQYrOXKvNkSxeTFRGNEtBpI2rTAN5j7ICVgw9vpqquuMrY3DmOS/dq1axdMCeqsXZMAjeMAG8wmenTT2E6byLkmbtdPnBKw4a4AGJzEL774oukFfMsttxifjg6VABKgzAW6COrseCmRo4VDWDmSgrs/UgY2iAwT6qabbjKAM27cOBk1apQJZ+rIbAlQ0Y1DmE4d+0NlCfATjxgxQkPcAd8WKQUbZAedKFW8zz//vLz88svmd9jZdGSeBHAAQysL2yMAQyCBIAIaMO1bdQRbAikHG8QHbQAk6YANWs7IkSOlbdu2wZaszj4uCdAOl4ABzePIodm7d68xoegrxQtJR/Al4AuwQYxsMFp5vPLKK+ZAbe7UqVPwJax3EFUCth0uZQiYTnv27DFlLewBbekcVXyB+YBvwMZqOAAODmMOmolpPkVg9lJCEz0S6utEzRwdOTCZAJrGjRsb0wmfno70kYCvwAaxsuGITBES58BBePHFF6ePxPVOjkkgNzdXJk2aJCtWrJDq1UPtV0I+GsLb1NFVq1ZNJZVmEvAd2CBf3mg4irHh33//fZMACKN+pBYeabYuaX87hw4dkokTJ8rq1atNOJs8GnrGk0dDAp+O9JOAb1eVDcgbDk0H4i0chiQCqg0f/E2Yk5NjtFZI8QEWolAXXnihXHDBBcG/Ob2DMiXgW7BhxlCL0nqVaARaztNPP20ASJP/grujs7Oz5Y033jAaDQ3kMJfoxgFntY70loCvwcaKHicx0QnUbvJxrrjiCu3YEMB9iTn82muvyapVq4xGQ1YwZhM/daS/BAIBNiwDdAKQbkGezrFp0yZDM6A5GMHYpDh/X3rpJQM0aDO8QHhpaMQpGOvnxCwDAzbcLFmklDfMnDlTPvroI9myZYt5M3bs2NEJWeg5XJLArl27DGkarX0Ia2M29e3b15jJOjJHAoECG5aFinGcidRQ4cfhbXneeeeZvynVqP82LtzT+Nq+++47o80MHz5czSb/LZMnMwoc2FipUD9FLypIlSBXwuFIF07VcjzZNzFd5OuvvzYlKPhqiCRi9mrvsJhEl5YfCizYsBrY/vSlggsHSgLqqnh7Qq4E57GO1EiALGDyo0hZwLF///33a+lJapbCV1cNNNhYSbZv316aN28us2bNktmzZxstB9PqnHPO0Teph9vt8OHD8uWXXxrCq7Vr10qrVq2MU79hw4YezkIv5VcJpAXYIFz8NZhRNMWD4e29994zFcQkinXp0kWdkS7uQJLylixZYsiuNmzYYMiuaEYIPxEV/TpUAkggbcDGLicazm233WY2P9y1kHJRb4Omo9Sjzm96Cijnzp1rIoN2wClNIaWmJTgv7yCfMe3AJnzDw4nz1VdfmTcuyYCATZ8+fZQrx4EdSxgbkMFcatq0qYkOrl+/3gD79ddfb7hodKgEwiWQtmDDTVapUsXkc5xxxhlCg7N58+YZ0AGEevToIfh6tOgv9gciPz/f5MpA2blu3TpjIsELjBk1ffp04zeD7IoKbh0qgZISSGuwsTfL5u/fv7+pvwF0OCDoatKkifTs2dP4dDRHp+yHg8LJpUuXGucveTNEmMj+7dWrlymmpASBDG+ARqkhFGTKkkBGgI29eTKQcRgTpVq+fLl5eOjOSUZy586dDehonc5RaRUXF5uSEEAGWVFugOZy3XXXmTA2WiOgPWHCBKMhwqpXtWpVfdJUAmVKIKPAxkoBLYZ8HDQdwuQknwE8+HZ4Q59++ummsjwT6SzIkUFbWbx4sQEbOITwdZHPhPlp+7NjklKjBkjjo9FkPUWZaBLISLCxQoHigFA5B/U7cOASxULbAZBwepKR3LJly7SOrMAVhA+GyNLGjRsFBr1GjRqZGia0mJLha3KZ6IKAL+zaa69VUrNoT5n+u5FARoNN+B6AaJse0jiUeaPz4KH18BPgwbxC2wGAaDMSZMcyjt5t27YZYCGatHnzZgMwyID2tgAs9wsYlxwzZswQ+jrhYKcEIchyUAzwVgIKNiXkzcODJsNBb2keRCIwFBKSgs/gocR/wXHKKaeYN7+fzQjuA83txx9/NEDKsXv3bpPoCHACMJhKAEykjpPUoRF16t27tzGrrEnl7ZbVqwVVAgo2EVYOAEGb4aDdyM6dO02GLCYHfg0cpDxwUJcCOtAnEKmhLou/pSLCBZcvJPH4Xnbs2GGiR2gxFEMSomZemEhEktDSmG9pGkxJsUydOtVkZhPVgw9ah0ogXgko2MQoMTQeAIXj3HPPNQTdaAc8zGg/aA0ktWGO8PASbufBxsnMTw6iYURsiOQAZBxoEnwe0CpNUwAgOGjiRtsTtBQO6pA4oNkEXDiIGPGTUDWf5zpcF6c3WgtgiBYWDwgSlXr33XdNvRMFrlRu61AJJCIBBZtEpBb6DvkkHDzEaAloPjz4OFvRgNAm0C4AITQhQIjPMAAXwAug4eB3Dvt3fgIWfN7+5HfAhsP+3TjdQt8DVJgLYIazG9MIjYVyAf6WqF8FkJs8ebKJ0uEsHjBgQILS0q+pBNRB7Nge4IFGi+HA32MHwIAGgrYBGPGT/6eVCdoRP3HY8jmrwfD/aDloPgAPByFoAAWtxB5oT4AJf0djShRUShMCIEfzuC+++MJoM5CT6VAJJCMB1WySkV4M3wUA4NnlCArVAmAHuTz5R/hn+vXrF8Od6kdUApEloGCjO+Q4CeAPIiuYfCPaH+Of0qEScEICCjZOSDFNzoFf6fXXXzehfgosyaXRoRJwSgIKNk5JMuDnwX9EQSXObOqfaIWrQyXgpAQUbJyUZkDPheMaoCGjGNIrWPZ0qAScloCCjdMSDdj5yMuBzZAQ/ciRI01hpQ6VgBsSULBxQ6oBOSdJgLRaIS+I5n9QRehQCbglAQUbtyTr8/OS/QzQoNnccssths5Th0rATQko2LgpXZ+ee/v27YapEKcwHRCokdKhEnBbAgo2bkvYZ+fHN0PLYjKWR48ercyEPlufdJ7O/wftkZTjAE1FaQAAAABJRU5ErkJggg==)
Here, ΔPOA and ΔPOB are congruent right angled triangle.
⇒ ∠BOP = 1/2 ∠AOB
= 1/2 × 80°
= 40°
In right angled ΔOBP,
We know that sum of angles in a triangle is 180°.
⇒ ∠BOP + ∠B + ∠OPB = 180°
⇒ 40 + 90 + ∠OPB = 180°
⇒ 130 + ∠OPB = 180°
∴ ∠OPB = 50°
Question 12.A tangent from P, a point in the exterior of a circle, touches the circle at Q. If OP = 13, PQ = 5, then the diameter of the circle is
A. 576
B. 15
C. 8
D. 24
Answer:Given OP = 13 and PQ = 5
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)
In right angled ΔOQP,
By Pythagoras Theorem,
⇒ OP2 = OQ2 + PQ2
⇒ 132 = r2 + 52
⇒ r2 = 169 – 25 = 144
∴ r = 12
Diameter of circle = 2r
= 2 (12)
= 24
∴ Diameter = 24
Question 13.In ΔABC, AB = 3, BC = 4, AC = 5, then the radius of the circle touching all the three sides is ………..
A. 2
B. 1
C. 4
D. 3
Answer:Given in ΔABC, AB = 3, BC = 4, AC = 5.
![](data:image/png;base64,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)
By Pythagoras Theorem,
⇒ AC2 = AB2+ BC2
⇒ 52 = 32 + 42
⇒ 52 = 9 + 16
⇒ 52 = 52
∴ ΔABC is a right angled triangle and ∠B is a right angle.
We know that the radius of the circle touching all the sides is
.
⇒ The required radius of circle = ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAADZQTFRFAAAAAAAAAAA6OgA6Oma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/25A627Zm29u2/7Zm/9uQ//+2Ct+ztwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAQ0lEQVQYV2NgQAd8HIyMLAwMguxcDLxM3GBZAWYIzcMKpniB0iAehOIHUvycDIJsjEDAiWEcLgGQaiAgWj1cIXXdBwD9yAH2spUFWQAAAABJRU5ErkJggg==)
= 1
Question 14.
and
touch the circle with centre 0 at A and B respectively. If m∠OPB = 30 and OP = 10, then radius of the circle =
A. 5
B. 20
C. 60
D. 10
Answer:Given ∠OPB = 30° and OP = 10
![](data:image/png;base64,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)
In right angled ΔOBP,
Consider sin30° = ![](data:image/png;base64,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)
⇒ 1/2 = ![](data:image/png;base64,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)
⇒ OB = 5
∴ Radius of circle = OB = 5
Question 15.The points of contact of the tangents from an exterior point P to the circle with centre 0 are A and B. If m∠OPB = 30, then m∠ AOB = ……
A. 30
B. 60
C. 90
D. 120
Answer:In right angled ΔOBP,
Given ∠OBP = 30°
![](data:image/png;base64,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)
⇒ ∠BOP + ∠OPB + ∠B = 180°
⇒ ∠BOP + 30° + 90° = 180°
⇒ ∠BOP = 180° – 120° = 60°
∴ ∠BOP = 60°
Now, ∠AOB = 2 ∠BOP
= 2 (60°)
= 120°
∴ ∠AOB = 120°
Question 16.A chord of ⨀(O, 5) touches ⨀(O, 3). Therefore the length of the chord =
A. 8
B. 10
C. 7
D. 6
Answer:Given chord of circle (O, 5) touches circle (O, 3).
Radius of smaller circle OM = 3 and radius of bigger circle OB = 5
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)
In right angled ΔOMB,
⇒ OB2 = OM2 + MB2
⇒ 52 = 32 + MB2
⇒ MB2 = 25 – 9 = 16
⇒ MB = 4
Length of chord AB = 2 (MB) = 2 (4) = 8
A circle touches the sides ,
,
of ΔABC at points D, E, F respectively. BD = x, CE = y, AF = z. Prove that the area of ΔABC =
.
Answer:
Given that a circle touches the sides BC, CA and AB of ΔABC at points D, E and F respectively.
Let BD = x, CE = y and AF = z.
We have to prove that area of ΔABC =
Proof:
BD and BF are tangents drawn from B. And D and F are points of contact.
∴ BD = BF = x
Similarly for tangents CE and CD drawn by C and tangents AE and AF drawn from A,
CE = CD = y and AF = AE = z
Sides of ΔABC,
⇒ AB = c = AF + BF = z + x … (1)
⇒ BC = a = BD + DC = x + y … (2)
⇒ CA = b = CE + AE = y + z … (3)
In ΔABC, 2s = AB + BC + AC
= (z + x) + (x + y) + (y + z)
= 2 (x + y + z)
∴ s = x + y + z … (4)
We know that area of ΔABC =
Area =
=
=
Hence proved.
Question 2.
ΔABC is an isosceles triangle in which. A circle touching all the three sides of ΔABC touches
at D. Prove that D is the mid – point of
.
Answer:
Given that ΔABC is an isosceles triangle in which AB ≅ AC and a circle touching all the three sides of ΔABC touches BC at D.
We have to prove that D is the midpoint of BC.
Proof:
We know that if a circle touches all sides of a triangle, the lengths of tangents drawn from each vertex are equal.
∴ AE = AF, BD = BF and CD = CE … (1)
Consider AB = AC,
Subtracting AF from both sides,
⇒ AB – AF = AC – AF
From (1),
⇒ AB – AF = AC – AE
Since A – F – B and A – E – C,
⇒ BF = CE
From (1),
⇒ BD = CD
Since B – D – C and BD = CD,
D is the midpoint of BC.
Question 3.
∠B is a right angle in ΔABC. If AB = 24, BC = 7, then find the radius of the circle which touches all the three sides of ΔABC.
Answer:
Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.
In the figure,
ID = IE = IF = r
Since ΔABC is a right angled triangle, ∠B = 90°.
Also ID ⊥ BC and AB ⊥ BC.
∴ ID || AB and ID || FB
Similarly, IF || BD
∴ IFBD is a parallelogram.
∴ FB = ID = r and BD = IF = r … (1)
∴ Parallelogram IFBD is a rhombus.
Since ∠B = 90°, parallelogram IFBD is a square.
By Pythagoras Theorem,
⇒ AC2 = AB2 + BC2
= 242 + 72
= 576 + 49
= 625
∴ AC = 25
⇒ AB + BC + AC = 24 + 7 + 25
⇒ AF + FB + BD + DC + AC = 56
⇒ AE + r + r + CE + AC = 56
⇒ 2r + (AE + CE) + AC = 56
⇒ 2r + 2AC = 56
⇒ 2r + 2(25) = 56
⇒ r + 25 = 28
⇒ r = 3
∴ The radius of circle is 3.
Question 4.
A circle touches all the three sides of a right angled ΔABC in which LB is right angle. Prove that the radius of the circle is
Answer:
Given that a circle touches all the three sides of a right angled ΔABC.
We have to prove that radius of a circle =
Proof:
Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.
In the figure,
ID = IE = IF = r
Since ΔABC is a right angled triangle, ∠B = 90°.
Also ID ⊥ BC and AB ⊥ BC.
∴ ID || AB and ID || FB
Similarly, IF || BD
∴ IFBD is a parallelogram.
∴ FB = ID = r and BD = IF = r … (1)
∴ Parallelogram IFBD is a rhombus.
Since ∠B = 90°, parallelogram IFBD is a square.
Now AE = AF
⇒ AE = AB – FB
= AB – r [From (1)] … (2)
And CE = CD
⇒ CE = BC – BD
= BC – r [From (1)] … (3)
Now, AC = AE + CE,
⇒ AC = AB – r + BC – r
⇒ AC = AB + BC – 2r
⇒ 2r = AB + BC – AC
⇒ r =
∴ The radius of a circle is .
Question 5.
In â ABCD, m∠D = 90. A circle with centre 0 and radius r touches its sides and
in P, Q, R and S respectively. If BC = 40, CD = 30 and BP = 25, then find the radius of the circle.
Answer:
Given that in â ABCD, ∠D = 90°. BC = 40, CD = 30 and BP = 25
We know that tangents drawn to a circle are perpendicular to the radius of the circle.
⇒ ∠ORD = ∠OSD = 90°
Given ∠D = 90° and OR = OS = radius.
∴ ORDS is a square.
We know that the tangents drawn to a circle from a point in the exterior of the circle are congruent.
∴ BP = BQ, CQ = CR and DR = DS.
Consider BP = BQ,
⇒ BQ = 25 [BP = 25]
⇒ BC – CQ = 40
⇒ CQ = 40 – 25 = 15 [BC = 40]
Consider CQ = CR,
⇒ CR = 15
⇒ CD – DR = 15
⇒ DR = 30 – 15 = 15 [CD = 30]
But ORDS is a square.
∴ OR = DR = 15
∴ Radius of circle OR is 15.
Question 6.
Two concentric circles are given. Prove that all chords of the circle with larger radius which touch the circle with smaller radius are congruent.
Answer:
Given are two concentric circles.
Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius.
We have to prove that PQ ≅ RS.
Proof:
Let two chords PQ and RS of the circle with larger radius touch the circle with smaller radius at points M and N respectively.
PQ and RS are tangents to the circle with smaller radius,
∴ OM = ON = radius of smaller circle.
Thus, chords AB and CD are equidistant from the centre of the circle with larger radius.
∴ PQ = RS
∴ PQ ≅ RS
∴All chords of the circle with larger radius which touch the circle with smaller radius are congruent.
Hence proved.
Question 7.
A circle touches all the sides of â ABCD. If AB = 5, BC = 8, CD = 6. Find AD.
Answer:
We know that if a circle touches all the sides of a quadrilateral, then AB + CD = BC + DA
Given AB = 5, BC = 8, CD = 6
⇒ 5 + 6 = 8 + DA
⇒ 11 = 8 + DA
⇒ DA = 3
∴ AD = 3
Question 8.
A circle touches all the sides of â ABCD. If is the largest side then prove that
is the smallest side.
Answer:
Given that a circle touches all the sides of ABCD and AB is the largest side.
We have to prove that CD is the smallest side.
Proof:
The circle touches all sides of ABCD.
∴ AB + CD = BC + DA … (1)
Given AB is the largest side.
⇒ AB > BC
∴ AB = BC + m
From (1),
⇒ BC + m + CD = BC + DA
⇒ CD + m = DA
∴ CD < DA
Hence CD is smaller than DA. … (2)
But AB is the largest side.
⇒ AB > DA
∴ AB = DA + n
From (1),
⇒ DA + n + CD = BC + DA
⇒ CD + n = BC
∴ CD < BC
Hence CD is smaller than BC. … (3)
AB is largest side, so CD is smaller than AB. … (4)
From (2), (3) and (4), CD is the smallest side of ABCD.
Question 9.
P is a point in the exterior of a circle having centre 0 and radius 24. OP = 25. A tangent from P touches the circle at Q. Find PQ.
Answer:
Given P lies in the exterior of a circle having centre O and PQ is a tangent.
∴ OQ ⊥ PQ
Also given OP = 25 and OQ = 24.
Consider ΔOQP,
∠OQP = 90°
By Pythagoras Theorem,
⇒ OP2 = OQ2 + PQ2
⇒ 252 = 242 + PQ2
⇒ PQ2 = 625 – 576
= 49
∴ PQ = 7
Question 10.
P is in exterior of ⨀(O, 15). A tangent from P touches the circle at T. If PT = 8, then OP = ………..
A. 17
B. 13
C. 23
D. 7
Answer:
Given P is in exterior of circle (O, 15) and a tangent from P touches the circle at T.
Thus, ∠OTP = 90°
By Pythagoras Theorem,
⇒ OP2 = OT2 + TP2
⇒ OP2 = 152 + 82
= 225 + 64
= 289
∴ OP = 17
Question 11.
,
touch ⨀(O, 15) at A and B. If m∠AOB = 80, then m∠OPB =
A. 80
B. 50
C. 10
D. 100
Answer:
Given PA and PB touch circle (O, 15) at A and B and m ∠AOB = 80°.
Here, ΔPOA and ΔPOB are congruent right angled triangle.
⇒ ∠BOP = 1/2 ∠AOB
= 1/2 × 80°
= 40°
In right angled ΔOBP,
We know that sum of angles in a triangle is 180°.
⇒ ∠BOP + ∠B + ∠OPB = 180°
⇒ 40 + 90 + ∠OPB = 180°
⇒ 130 + ∠OPB = 180°
∴ ∠OPB = 50°
Question 12.
A tangent from P, a point in the exterior of a circle, touches the circle at Q. If OP = 13, PQ = 5, then the diameter of the circle is
A. 576
B. 15
C. 8
D. 24
Answer:
Given OP = 13 and PQ = 5
In right angled ΔOQP,
By Pythagoras Theorem,
⇒ OP2 = OQ2 + PQ2
⇒ 132 = r2 + 52
⇒ r2 = 169 – 25 = 144
∴ r = 12
Diameter of circle = 2r
= 2 (12)
= 24
∴ Diameter = 24
Question 13.
In ΔABC, AB = 3, BC = 4, AC = 5, then the radius of the circle touching all the three sides is ………..
A. 2
B. 1
C. 4
D. 3
Answer:
Given in ΔABC, AB = 3, BC = 4, AC = 5.
By Pythagoras Theorem,
⇒ AC2 = AB2+ BC2
⇒ 52 = 32 + 42
⇒ 52 = 9 + 16
⇒ 52 = 52
∴ ΔABC is a right angled triangle and ∠B is a right angle.
We know that the radius of the circle touching all the sides is .
⇒ The required radius of circle =
=
= 1
Question 14.
and
touch the circle with centre 0 at A and B respectively. If m∠OPB = 30 and OP = 10, then radius of the circle =
A. 5
B. 20
C. 60
D. 10
Answer:
Given ∠OPB = 30° and OP = 10
In right angled ΔOBP,
Consider sin30° =
⇒ 1/2 =
⇒ OB = 5
∴ Radius of circle = OB = 5
Question 15.
The points of contact of the tangents from an exterior point P to the circle with centre 0 are A and B. If m∠OPB = 30, then m∠ AOB = ……
A. 30
B. 60
C. 90
D. 120
Answer:
In right angled ΔOBP,
Given ∠OBP = 30°
⇒ ∠BOP + ∠OPB + ∠B = 180°
⇒ ∠BOP + 30° + 90° = 180°
⇒ ∠BOP = 180° – 120° = 60°
∴ ∠BOP = 60°
Now, ∠AOB = 2 ∠BOP
= 2 (60°)
= 120°
∴ ∠AOB = 120°
Question 16.
A chord of ⨀(O, 5) touches ⨀(O, 3). Therefore the length of the chord =
A. 8
B. 10
C. 7
D. 6
Answer:
Given chord of circle (O, 5) touches circle (O, 3).
Radius of smaller circle OM = 3 and radius of bigger circle OB = 5
In right angled ΔOMB,
⇒ OB2 = OM2 + MB2
⇒ 52 = 32 + MB2
⇒ MB2 = 25 – 9 = 16
⇒ MB = 4
Length of chord AB = 2 (MB) = 2 (4) = 8