in two concentric circles with centre O,PQ is the diameter of the outer circle and QS i sthe tangent line to the cinner circle touching it in R and the outer circle in S find the lengt of pr if radii of two circles are 13cm and 8cm
Two concentric circles with centre O and radius 13 cm and 8 cm.
PQ is the diameter of the outer circle, QS is the tangent to the inner circle touching it in R.
Construction: Join PS.
∠ORQ = (Radius is perpendicular to the tangent at point of contact)
∴ RQ = SR (Perpendicular from the centre to the chord, bisects the chord)
In right ΔORQ,
OQ2 = OR2 + RQ2
∴ RQ2 = OQ2 – OR2
= (13 cm)2 – (8 cm)2
= 169 cm2 – 64 cm2
RQ2 = 105 cm2
∠PSQ = [Angle in a semi-circle is ]
In ΔOQR and ΔPQS,
∠OQR = ∠PQS [Common]
∠ORQ = ∠PSQ 
∴ ΔOQR ∼ ΔPQS [By AA similarity]
In right ΔPSR,
PR2 = PS2 + SR2
Thus, the length of PR = 19 cm.