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,

OQ^{2} = OR^{2} + RQ^{2}

∴ RQ^{2} = OQ^{2 }– OR^{2}

^{ }= (13 cm)^{2} – (8 cm)^{2 }

= 169 cm^{2} – 64 cm^{2}

RQ^{2} = 105 cm^{2}

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,

PR^{2} = PS^{2} + SR^{2}

Thus, **the length of PR = 19 cm.**

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