# 1 ext of in ade q a p 2 ext of 1 2 a s q a r ps q r p

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(1)(ext. of In ADEqa p........(2)(ext. of (1) (2),as(q+ a)rsqrppsqr13.CIn ABCand CBDBACBCD(given)CBADBC(common angle)BCABACCBA180(sum of BCA180BACCBABDCBCDDBC180(sum of BDC180BCDDBCBCABDCABC~ CBD(equiangular)CDACBDBCCBAB(corr. sides, ~ s)II and III are correct. 14.BP.34
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In ABCand EDC,BACDEC(alt. s, AB // DE)ABCEDC(alt. s, AB // DE)BCADCE(vert. opp. s)ABC~ EDC(equiangular)8436xxDCBCEDAB(corr. sides, ~ s)15.AThe angle bisectors of the interior angles of any triangle intersect at a point. This intersecting point is called the incentre of the triangle. 16.BThe three perpendicular bisectors of the sides of any triangle intersect at a point. This intersecting point is called the circumcentre of the triangle.17.CThe three medians of any triangle intersect at a point. This intersecting point is called the centroid of the triangle.18.DThe three altitudes at any triangle intersect at a point. This intersecting point is called the orthocentre of the triangle. 19.CP.35
Ois the circumcentre of ABCAPBP10 cmRCAR5 cmBQQC9 cmPerimeter [2 (10 + 5 + 9)] cm48 cm20.BOis the centroid of ABCPCBP3 cmAC2QC8 cmPerimeter 3 cm + 8 cm + 6 cm17 cm21.AOis the circumcentre of ABCBPAP4 cmAQQC5 cmBRRC6 cmPerimeter[2 (4 + 5 + 6)] cm30 cm22.BOis the circumcentre of ABCBEEC4 cmArea(214 3) cm6 cm2P.36
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23.DOis the incentre of ABCPAOOAR 50PBOOBQ15In ABOAOB+ BAO+ ABO180(sum of AOB+ 50+ 15 180AOB11524.BOis the incentre ofABC.FBCABF50DCABCD10EACBAEaIn ABCABC+ BAC+ BCA180(sum of 50+ 50+ a+ a+ 10+ 10 1802a60a3025.BThe incentre of any triangle lies inside the triangle.Iis incorrect.The orthocentre of an obtuse-angled triangle lies outside the triangle.II is correct.The circumcentre of a right-angled triangle lies on the vertex of the right angle of the triangle.III is incorrect.26.DP.37
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ACOPTUBQRSIn the figure, APS, BPS, ARU, CRU, BTQ and CTQare right angles.Also, AB2+ BC262+ 82100AC2ABCis a right angle. (converse of Pyth. theorem)ARU, QTCand ABCare right-angled triangles.27.AOis the orthocentre of ABC.BDCBFC= 90°Area of ABC3228)62(2BFACArea of ABCCDCDCDAB42)62(24CD32CD= 8P.38
28.COis the incentre of ABCOBDOBE= 22.5°OCEOCF= 22.5In ABCABC+ ACB+ BAC180(sum of 2 22.5+ 2 22.5+ BAC180BAC90Area of ABC22cm32cm)288(29.DO is the orthocentre of ABCAQCCRA= 90°In ACQQAC+ ACQ+ AQC180(sum of QAC+ 60+ 90 180QAC30In ACRACR+ CAR+ CRA180(sum of ACR+ 80+ 90 180ACR10In AOCAOC+ OAC+ ACO180(sum of AOC+ 30+ 10 180AOC14030.CP.39
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O is the circumcentre of ABCBPZBRY= 90°In ABCABC+ BAC+ BCA180(sum of ABC+ 70+ 60 180ABC50In the quadrilateral BPORPBR+ BPO+ POR+ BRO(4 2) 180(sum of polygon)50+ 90+ POR+ 90 360POR130P.40
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