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Applications, , A circle touches the side BC of a AABC, at P and touches AB and AC when, produced at Q and R respectively then, , AQ =} (Perimeter of AABC)., , , , [Tangents drawn from an external, , point to a circle are equal], , Now, perimeter of AABC, , = AB+BC+CA, , = AB+BP+PC+CA, , = (AB +BQ)+(CR+CA) [From (ii) and (iii)], = AQ + AR = AQ+ AQ [From(i)], , AQ =} (Perimeter of AABC)., , Special Note :, , 1. If quadrilateral ABCD is circumscribing a, circle, then AB + CD = AD + BC., , 2. If all the sides of a parallelogram touches, , acircle, then the parallelogram is a rhombus., , , , , , , , , , Special case of secant, with only one point of contact. }, , Two tangents TP and TQ are drawn, soacircle with centre O from an, , , , int T. Prove that, P, , , , Q, ce ZTOP =, gor, Z0Pa= 40, , in APTQ, TP =TQ, hen, , since ZOPT=, 1 (480° - 9), Z0pa= 90-5", , = ZPTQ= 2Z0PO. atte, , Zopt = 112 (180-®), pT — ZTPQ, , A & Vv B, , A tangent to a circle is perpendicular to, the radius through the point of contact., i.e. OPL AB, , , , Length of two tangent drawn, from external point are equal, In AOQP & AORP., , ZOQP = ZORP (Each 90°), OP=OP, (Common, OQ=OR (radius), , -. AOQP = AORP = PQ=PR, , Results :, , (i) If two tangents are drawn toa, circle from an external point,, then they subtend equal angles, at the centre. ZPOQ = ZPOR, , (ii) If two tangents are drawn toa, , circle from an external point,, they are equally inclined to the, segment, joining the centre to, that point ZOPQ = ZOPR, , NCERT / X / Circles