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

Given: 

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

 

 

Now, 

 

∠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.

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