Q. 315.0( 3 Votes )

In Fig. 10.93, there are two concentric circles with centre O. PR and PQS are tangents to the inner circle from point plying on the outer circle. If PR = 7.5 cm, then PS is equal to


A. 10 cm

B. 12 cm

C. 15 cm

D. 18 cm

Answer :

Given:


PR = 7.5 cm



Property 1: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.


Property 2: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal.


By property 1, ∆OSQ is right-angled at OQS (i.e., OQS = 90°) and OPQ is right-angled at OQP (i.e., OQP = 90°).


OQ PS


PO = OS [radius of circle]


POS is an isosceles triangle


Now,


POS is an isosceles triangle and OQ is perpendicular to its base


OQ bisects PS


i.e., PQ = QS


By property 2,


PR = PQ = 7.5 cm (tangent from P)


Now,


PS = PQ + QS


PS = PQ + PQ [ PQ = QS]


PS = 7.5 cm + 7.5 cm


PS = 15 cm


Hence, PS = 15 cm

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