1. Prove that the parallelogram circumscribing a circle is rhombus.
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1 UNIT-9 1. Prve that the parallelgram circumscribing a circle is rhmbus. Ans Given : ABCD is a parallelgram circumscribing a circle. T prve : - ABCD is a rhmbus r ABBCCDDA Prf: Since the length f tangents frm external are equal in length AS AR..(1) BQ BR..() QC PC..(3) SD DP..(4) Adding (1), (), (3) & (4). AS + SD + BQ + QC AR + BR + PC + DP AD + BC AB + DC AD + AD AB + AB Since BC AD & DC AB (ppsite sides f a parallelgram are equal) AD AB AD AB..(5) BC AD (ppsite sides f a parallelgram) DC AB..(6) Frm (5) and (6) AB BC CD DA Hence prved. A circle tuches the side BC f a triangle ABC at P and tuches AB and AC when prduced at Q and R respectively as shwn in figure. Shw that AQ (perimeter f triangle ABC) A B P C Q R 70
2 Ans: Since the length f tangents frm external pint t a circle are equal. AQ AR BQ BP PC CR Since AQ AR AB + BQ AC + CR AB + BP AC + PC (Since BQ BP & PC CR) Perimeter f ABC AB + AC + BC AB + BP + PC + AC AQ + PC + AC (Since AB + BP AQ) AQ + AB + BP (Since PC + AC AB + BP) AQ + AQ (Since AB + BP AQ) Perimeter f ABC AQ AQ 1 (perimeter f triangle ABC) 3. In figure, XP and XQ are tangents frm X t the circle with centre O. R is a pint n the circle. Prve that XA+ARXB+BR Ans: Since the length f tangents frm external pint t a circle are equal XP XQ PA RA BQ BR 71
3 XP XQ XA + PA XB + BQ XA + AR XB + BR Hence prved PA AR & BQ BR 7
4 4. In figure, the incircle f triangle ABC tuches the sides BC, CA, and AB at D, E, and F respectively. Shw that AF+BD+CEAE+BF+CD triangle ABC), (perimeter f Ans: Since the length f tangents frm an external pint t are equal AF AE FB BD EC CD Perimeter f ABC AB + BC+ AC AF + FB + BD + DC + AE + EC AF + BD + BD + CE + AF + CE ( AFAE, FBBD, ECCD) AF + AF + BD + BD + CE + CE Perimeter f ABC (AF + BD+ CE) AF + BD + CE 1 (perimeter f ABC)..(1) Perimeter f ABC AB + BC + AC AF + FB + BD + DC + AE + EC AE + BF + BF + CD + AE + CD ( AF AE, FB BD, EC CD) AE + AE + BF + BF + CD + CD Perimeter f ABC (AE + BF + CD) AE + BF + CD 1 (perimeter f ABC)..() Frm (1) and () AF + BD + CE AE + BF + CD 1 (perimeter f ABC) 73
5 5. A circle tuches the sides f a quadrilateral ABCD at P, Q, R and S respectively. Shw that the angles subtended at the centre by a pair f ppsite sides are supplementary. Ans: T prve :- AOB + DOC 180 BOC + AOD 180 Prf : - Since the tw tangents drawn frm an external pint t a circle subtend equal angles at centre. 1, 3 4, 5 6, 7 8 but ( ) AOB + DOC 180 Similarly ( ) BOC + AOD 180 Hence prved 6. In figure, O is the centre f the Circle.AP and AQ tw tangents drawn t the circle. B is a pint n the tangent QA and PAB 15, Find POQ. (Ans: 15 ) O P 15 A B Ans: Given PAB 15 T find : - POQ? Cnstructin : - Jin PQ Prf : - PAB + PAQ 180 (Linear pair) PAQ PAQ PAQ 55 Since the length f tangent frm an external pint t a circle are equal. PA QA Frm PAQ APQ AQP Q 74
6 In APQ APQ + AQP + PAQ 180 (angle sum prperty) APQ + AQP APQ ( APQ AQP) APQ APQ AQP OQ and OP are radii QA and PA are tangents OQA 90 & OPA 90 OPQ + QPA OPA 90 (Linear Pair) OPQ OPQ OPQ Similarly OQP + PQA OQA OQP OQP OQP In POQ OQP + OPQ + POQ 180 (angle sum prperty) POQ POQ POQ POQ 50 POQ
7 POQ 15 POQ Tw tangents PA and PB are drawn t the circle with center O, such that APB10. Prve that OPAP. Ans: Given : - APB 10 Cnstructin : -Jin OP T prve : -OP AP Prf :- APB 10 APO OPB 60 Cs 60 AP OP 1 AP OP OP AP Hence prved 8. Frm a pint P, tw tangents PA are drawn t a circle with center O. If OPdiameter f the circle shw that triangle APB is equilateral. Ans: PAPB (length f tangents frm an external pint Frm OAP, OA 1 sin APO OP Since OP OA (Since OPDiameter) APO 30 since APO BPO APO BPO 30 APB 60 APB is equilateral 9. In the given fig OPQR is a rhmbus, three f its vertices lie n a circle with centre O If the area f the rhmbus is 3 3 cm. Find the radius f the circle. Ans: QP OR OP OQ OPQ is a equilateral. area f rhmbus (ar f OPQ) 3 3 3r 4 Q P O 3r 3 3 R 76
8 r 3 x 64 r 8cm Radius 8cm 10. If PA and PB are tangents t a circle frm an utside pint P, such that PA10cm and APB60. Find the length f chrd AB. Self Practice 11. The radius f the in circle f a triangle is 4cm and the segments int which ne side is divided by the pint f cntact are 6cm and 8cm. Determine the ther tw sides f the triangle. O Ans: a BC x + 8 b AC cm c AB x + 6 Semi perimeter BC x x 8 x + 14 AC 14 8 AB x 6 a b c (Ans: 15, 13) Area f ABC s( s a)( s b)( s c) n substituting we get ( x 14 )(6)( x)(8) ( x 14 )(48 x)..(1) Area f ABC area AOB + area BOC + area AOC 77
9 area AOC 1 b. h 1 x 4 x 14 8 On substituting we get area ABC area AOC + area BOC + area AOB 4x + 56 () Frm (1) and () 4x + 56 ( x 14 )(48 x) Simplify we get x 7 AB x cm BC x cm 1. A circle is inscribed in a triangle ABC having sides 8cm, 10cm and 1cm as shwn in the figure. Find AD, BE and CF. (Ans :7cm,5cm,3cm) Self Practice 13. Prve that the intercept f a tangent between tw parallel tangents t a circle subtends a right angle at the centre. Since ADF DFC ADF CDF ADC CDF Similarly we can prve CEB CEF Since l m ADC + CEB 180 CDF + CEF 180 CDF + CEF 90 In DFE DFE 90 78
10 14. Prve that ppsite sides f a quadrilateral circumscribing a circle subtend supplementary angles at the centre f the circle. Ans: Same as questin N QR is the tangent t the circle whse centre is P. If QA RP and AB is the diameter, prve that RB is a tangent t the circle. A Q R Self Practice B 79
= ( +1) BP AC = AP + (1+ )BP Proved UNIT-9 CIRCLES 1. Prove that the parallelogram circumscribing a circle is rhombus. Ans Given : ABCD is a parallelogram circumscribing a circle. To prove : - ABCD is
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