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- 3. Find the 10h term.PQ is a chord of length 8 cm of a circle of radius5 cm. The tangents at P and Q intersect at apoint T (see figure). Find the length of thetangent TP17.ë26BSE-100% SUCCESS IN MATHEMATICS-10 |
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Answer» Given radius, OP = OQ = 5 cmLength of chord, PQ = 4 cmOT ⊥ PQ,∴ PM = MQ =4 cm [Perpendicular draw from the centre of the circle to a chord bisect the chord]In right ΔOPM,OP2= PM2+ OM2⇒ 52= 42+ OM2⇒ OM2= 25 – 16 = 9Hence OM = 3cmIn right ΔPTM,PT2= TM2+ PM2→(1)∠OPT = 90º [Radius is perpendicular to tangent at point of contact]In right ΔOPT,OT2= PT2+ OP2→(2)From equations (1) and (2), we getOT2= (TM2+ PM2) + OP2⇒ (TM + OM)2= (TM2+ PM2) + OP2⇒ TM2+ OM2+ 2 × TM × OM = TM2+ PM2+ OP2⇒ OM2+ 2 × TM × OM = PM2+ OP2⇒ 32+ 2 × TM × 3 = 42+ 52⇒ 9 + 6TM = 16 + 25⇒ 6TM = 32⇒ TM =32/6 = 16/3Equation (1) becomes,PT2= TM2+ PM2 = (16/3)2+ 42 =(256/9) + 16 = (256 + 144)/9 = (400/9) = (20/3)Hence PT = 20/3Thus, the length of tangent PT is(20/3) cm. thanks |
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